This document is a thesis presented by Larry Huang to the University of Waterloo in fulfillment of the requirements for a Master's degree in Pure Mathematics. The thesis studies ways to calculate the dimensions of symmetry classes of finite dimensional complex tensor product spaces. It presents general results for calculating these dimensions, as well as several specific methods including using Freese's theorem, disjoint cycle decompositions of the symmetric group Sm, and examining subspaces of the orbital subspaces.
This document provides an analytic-combinatoric proof of Pólya's recurrence theorem for simple random walks on lattices Zd. It introduces generating functions to represent combinatorial classes of random walks. The generating function for simple random walks yields a formula for the probability of being at a given position at a given time. Laplace's method is then used to estimate these probabilities asymptotically, showing the random walk is recurrent if d = 1, 2 and transient if d ≥ 3, proving Pólya's theorem.
Understanding lattice Boltzmann boundary conditions through momentsTim Reis
This document provides an overview of the key points that will be covered in a lecture about understanding lattice Boltzmann boundary conditions through moments. The lecture will focus on connecting LB boundary conditions to LB distribution function moments. It will discuss interpreting other boundary condition methods in terms of moments and analyzing boundary condition accuracy, including for slip and no-slip flow conditions. The goal is to provide a clear perspective on LB boundary conditions and how to determine the appropriate conditions for different applications.
This document introduces differential forms as an alternative approach to vector calculus. It provides a brief overview of 1-forms and 2-forms, including how to calculate line integrals and surface integrals of differential forms. The author explains that differential forms are similar to vector fields but written in a "funny notation" that is ultimately quite powerful. The document is intended as a supplement for teaching multivariable calculus using differential forms at the undergraduate level in a informal way without advanced linear algebra or manifolds.
The document discusses the eigenvalues of finite CMV matrices, which are matrices that represent multiplication by z in an orthonormal basis of polynomials on the unit circle called the CMV basis. It explores how the eigenvalues of truncated CMV matrices behave asymptotically for different classes of measures, such as Lebesgue measure or measures with inserted mass points. In particular, it shows that for measures with a single inserted mass point, the eigenvalues exhibit "clock" behavior, clustering uniformly around the unit circle in the limit as the matrix size increases.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
The document summarizes the lattice Boltzmann equation (LBE) method for computational fluid dynamics. It begins by discussing how the LBE is derived from kinetic theory and discrete kinetic theory, rather than directly from fluid equations. Taking moments of the discrete Boltzmann equation recovers the compressible Navier-Stokes equations through Chapman-Enskog expansion. The document then derives an analytic solution for the LBE governing Poiseuille flow between stationary walls under the influence of gravity.
The document discusses algorithms for solving single-source shortest path problems on directed graphs. It begins by defining the problem and describing Bellman-Ford's algorithm, which can handle graphs with negative edge weights but runs in O(VE) time. It then describes how Dijkstra's algorithm improves this to O(ElogV) time using a priority queue, but requires non-negative edge weights. It also discusses how shortest paths can be found in O(V+E) time on directed acyclic graphs by relaxing edges topologically. Finally, it discusses how difference constraints can be modeled as shortest path problems on a corresponding constraint graph.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
This document provides an analytic-combinatoric proof of Pólya's recurrence theorem for simple random walks on lattices Zd. It introduces generating functions to represent combinatorial classes of random walks. The generating function for simple random walks yields a formula for the probability of being at a given position at a given time. Laplace's method is then used to estimate these probabilities asymptotically, showing the random walk is recurrent if d = 1, 2 and transient if d ≥ 3, proving Pólya's theorem.
Understanding lattice Boltzmann boundary conditions through momentsTim Reis
This document provides an overview of the key points that will be covered in a lecture about understanding lattice Boltzmann boundary conditions through moments. The lecture will focus on connecting LB boundary conditions to LB distribution function moments. It will discuss interpreting other boundary condition methods in terms of moments and analyzing boundary condition accuracy, including for slip and no-slip flow conditions. The goal is to provide a clear perspective on LB boundary conditions and how to determine the appropriate conditions for different applications.
This document introduces differential forms as an alternative approach to vector calculus. It provides a brief overview of 1-forms and 2-forms, including how to calculate line integrals and surface integrals of differential forms. The author explains that differential forms are similar to vector fields but written in a "funny notation" that is ultimately quite powerful. The document is intended as a supplement for teaching multivariable calculus using differential forms at the undergraduate level in a informal way without advanced linear algebra or manifolds.
The document discusses the eigenvalues of finite CMV matrices, which are matrices that represent multiplication by z in an orthonormal basis of polynomials on the unit circle called the CMV basis. It explores how the eigenvalues of truncated CMV matrices behave asymptotically for different classes of measures, such as Lebesgue measure or measures with inserted mass points. In particular, it shows that for measures with a single inserted mass point, the eigenvalues exhibit "clock" behavior, clustering uniformly around the unit circle in the limit as the matrix size increases.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
The document summarizes the lattice Boltzmann equation (LBE) method for computational fluid dynamics. It begins by discussing how the LBE is derived from kinetic theory and discrete kinetic theory, rather than directly from fluid equations. Taking moments of the discrete Boltzmann equation recovers the compressible Navier-Stokes equations through Chapman-Enskog expansion. The document then derives an analytic solution for the LBE governing Poiseuille flow between stationary walls under the influence of gravity.
The document discusses algorithms for solving single-source shortest path problems on directed graphs. It begins by defining the problem and describing Bellman-Ford's algorithm, which can handle graphs with negative edge weights but runs in O(VE) time. It then describes how Dijkstra's algorithm improves this to O(ElogV) time using a priority queue, but requires non-negative edge weights. It also discusses how shortest paths can be found in O(V+E) time on directed acyclic graphs by relaxing edges topologically. Finally, it discusses how difference constraints can be modeled as shortest path problems on a corresponding constraint graph.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
This document summarizes research on simplifying calculations of scattering amplitudes, especially for tree-level amplitudes. It introduces the spinor-helicity formalism for writing compact expressions for amplitudes. It then discusses color decomposition in SU(N) gauge theory and the Yang-Mills Lagrangian. Specific techniques explored include BCFW recursion relations, an inductive proof of the Parke-Taylor formula, the 4-graviton amplitude and KLT relations, multi-leg shifts, and the MHV vertex expansion. The goal is to develop recursion techniques that vastly simplify calculations compared to traditional Feynman diagrams.
The document discusses issues with pinning and facetting in lattice Boltzmann simulations of multiphase flows. It presents a lattice Boltzmann model for propagating sharp interfaces using a phase field approach. Sharpening the phase field interface causes it to become pinned to the lattice or develop facets. Introducing randomness via a random projection method or random threshold prevents pinning and delays facetting, allowing the interface to propagate at the correct speed even for very sharp boundaries.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
This document discusses harmonic and subharmonic functions. It begins by defining harmonic functions and providing examples to show properties like the maximum/minimum principle and mean-value property. It then introduces the Dirichlet problem for the disk and proves there is always a solution. Harnack's inequality relating values of positive harmonic functions is presented. Finally, it defines subharmonic functions and provides examples to show properties like the submean-value property.
The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Bregman divergences from comparative convexityFrank Nielsen
This document discusses generalized divergences and comparative convexity. It introduces Jensen divergences, Bregman divergences, and their generalizations to quasi-arithmetic and weighted means. Quasi-arithmetic Bregman divergences are defined for strictly (ρ,τ)-convex functions using two strictly monotone functions ρ and τ. Power mean Bregman divergences are obtained as a subfamily when ρ(x)=xδ1 and τ(x)=xδ2. A criterion is given to check (ρ,τ)-convexity by testing the ordinary convexity of the transformed function G=Fρ,τ.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
(a) E-ZPass cannot prove that the driver was speeding. E-ZPass records entry and exit times and locations, but does not continuously track speed. It cannot determine the driver's exact speed at any point during the trip, so it cannot prove a specific speeding violation occurred. The best it could show is an average speed that may or may not indicate speeding depending on the specific speed limit(s) along the route.
Bowser e Eggman discutem um plano para obter todas as Esmeraldas do Caos. Bowser sugere terminar o robô de Eggman, mas ele ainda precisa de 4 meses para completá-lo. Bowser então sugere pedir ajuda a Sharlasur, mas Eggman informa que ele está morto. Eggman diz ter um novo plano sombrio.
This document discusses multilinear twisted paraproducts, which are generalizations of classical paraproduct operators to higher dimensions. It begins by reviewing classical paraproducts on the real line and their generalization to higher dimensions using dyadic squares. It then discusses complications that arise, such as twisted paraproducts. The document presents a unified framework for studying such operators using bipartite graphs and selections of vertices. It proves a main boundedness result and discusses special cases like classical dyadic paraproducts and dyadic twisted paraproducts. It introduces tools like Bellman functions and calculus of finite differences to analyze estimates for paraproduct-like operators on finite trees of dyadic squares.
The lattice Boltzmann equation: background, boundary conditions, and Burnett-...Tim Reis
An overview of the lattice Boltzmann equation and a discussion of moment-based boundary conditions. Includes applications to the slip flow regime and the Burnett stress. Some analysis sheds insight into the physical and numerical behaviour of the algorithm.
This document summarizes research on simplifying calculations of scattering amplitudes, especially for tree-level amplitudes. It introduces the spinor-helicity formalism for writing compact expressions for amplitudes. It then discusses color decomposition in SU(N) gauge theory and the Yang-Mills Lagrangian. Specific techniques explored include BCFW recursion relations, an inductive proof of the Parke-Taylor formula, the 4-graviton amplitude and KLT relations, multi-leg shifts, and the MHV vertex expansion. The goal is to develop recursion techniques that vastly simplify calculations compared to traditional Feynman diagrams.
The document discusses issues with pinning and facetting in lattice Boltzmann simulations of multiphase flows. It presents a lattice Boltzmann model for propagating sharp interfaces using a phase field approach. Sharpening the phase field interface causes it to become pinned to the lattice or develop facets. Introducing randomness via a random projection method or random threshold prevents pinning and delays facetting, allowing the interface to propagate at the correct speed even for very sharp boundaries.
Prof. Vishnu Jejjala (Witwatersrand) TITLE: "The Geometry of Generations"Rene Kotze
This document summarizes Vishnu Jejjala's talk "The Geometry of Generations" given at the University of the witwatersrand on 23 September 2014. It discusses using the geometry of vacuum moduli spaces of supersymmetric gauge theories to understand phenomenological aspects of the Standard Model, such as the number of generations. It provides examples of calculating the vacuum moduli space for simple supersymmetric gauge theories and outlines a process for obtaining the moduli space from the F-term and D-term equations.
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
Stochastic Gravity in Conformally-flat SpacetimesRene Kotze
The National Institute for Theoretical Physics, and the Mandelstam Institute for Theoretical Physics, School of Physics, would like to invite to its coming talk in the theoretical physics seminar series, entitled:
"Stochastic Gravity in Conformally-flat Spacetimes"
to be presented by Prof. Hing-Tong Cho (Tamkang University, Taiwan)
Abstract: The theory of stochastic gravity takes into account the effects of quantum field fluctuations onto the classical spacetime. The essential physics can be understood from the analogous Brownian motion model. We shall next concentrate on the case with conformally-flat spacetimes. Our main concern is to derive the so-called noise kernels. We shall also describe our on-going program to investigate the Einstein-Langevin equation in these spacetimes.
Dates: Tuesday, 17th February 2015
Venue: The Frank Nabarro lecture theatre, P216
Time: 13.20 - 14.10 - TODAY
This document discusses harmonic and subharmonic functions. It begins by defining harmonic functions and providing examples to show properties like the maximum/minimum principle and mean-value property. It then introduces the Dirichlet problem for the disk and proves there is always a solution. Harnack's inequality relating values of positive harmonic functions is presented. Finally, it defines subharmonic functions and provides examples to show properties like the submean-value property.
The document discusses Chebyshev polynomials. It defines Chebyshev polynomials as orthogonal polynomials related to de Moivre's formula that can be defined recursively. It provides key properties of Chebyshev polynomials including that they are the extremal polynomials for many properties and important in approximation theory. The document also provides formulas for generating Chebyshev polynomials, their orthogonality properties, and their use in representing functions through orthogonal series expansions.
NITheP UKZN Seminar: Prof. Alexander Gorokhov (Samara State University, Russi...Rene Kotze
This document discusses the application of dynamical groups and coherent states in quantum optics and molecular spectroscopy. It provides an introduction to using Lie groups and algebras to describe quantum systems and defines coherent states. Specific applications discussed include using dynamical symmetries to calculate energy levels of systems like the harmonic oscillator and hydrogen atom. Coherent states are used to derive classical equations of motion and represent open quantum systems. Examples of coherent state dynamics are shown for two-level and three-level atoms interacting with laser fields.
Wits Node Seminar: Dr Sunandan Gangopadhyay (NITheP Stellenbosch)
TITLE: Path integral action of a particle in the noncommutative plane and the Aharonov-Bohm effect
The document discusses Euler's generalization of Fermat's Little Theorem to composite moduli called the Theorem of Euler-Fermat. It explains that for any integer a coprime to a composite number m, a raised to the totient function of m (φ(m)) is congruent to 1 modulo m. It also provides formulas for calculating the totient function for prime powers and products of coprime integers. The Chinese Remainder Theorem, which states that a system of congruences with coprime moduli always has a solution, is introduced as well.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
Bregman divergences from comparative convexityFrank Nielsen
This document discusses generalized divergences and comparative convexity. It introduces Jensen divergences, Bregman divergences, and their generalizations to quasi-arithmetic and weighted means. Quasi-arithmetic Bregman divergences are defined for strictly (ρ,τ)-convex functions using two strictly monotone functions ρ and τ. Power mean Bregman divergences are obtained as a subfamily when ρ(x)=xδ1 and τ(x)=xδ2. A criterion is given to check (ρ,τ)-convexity by testing the ordinary convexity of the transformed function G=Fρ,τ.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
(a) E-ZPass cannot prove that the driver was speeding. E-ZPass records entry and exit times and locations, but does not continuously track speed. It cannot determine the driver's exact speed at any point during the trip, so it cannot prove a specific speeding violation occurred. The best it could show is an average speed that may or may not indicate speeding depending on the specific speed limit(s) along the route.
Bowser e Eggman discutem um plano para obter todas as Esmeraldas do Caos. Bowser sugere terminar o robô de Eggman, mas ele ainda precisa de 4 meses para completá-lo. Bowser então sugere pedir ajuda a Sharlasur, mas Eggman informa que ele está morto. Eggman diz ter um novo plano sombrio.
El documento describe los actos del juez en un proceso legal. Explica que los actos del juez son decisiones procesales por escrito que resuelven cuestiones incidentales o de fondo de la relación procesal. Estos actos deben cumplir con ciertas formalidades legales como ser emitidos por la autoridad competente, firmados por los funcionarios responsables, y por lo general deben ser escritos.
El documento describe cómo se celebra el Día de los Difuntos en diferentes culturas alrededor del mundo, incluyendo México, Francia, Nepal, China, Egipto y las Islas Canarias. En México se celebra el Día de Muertos para honrar a los antepasados, mientras que en países como Francia, Nepal y China también incluyen rituales para recordar a los fallecidos recientemente. Las culturas antiguas como Egipto y China creían que los espíritus de los muertos necesitaban ser recordados y alimentados a través de of
This document contains activities for students to learn English vocabulary related to weather. It includes exercises where students check the weather, discuss weather conditions and plans in their town through email exchanges, and review weather vocabulary in a game. The goal is for students to practice describing weather and making plans based on conditions.
Este documento presenta una lista de temas para una formación en Tecnologías de la Información y Comunicación (TIC). Los temas incluyen sistemas operativos, software libre, navegadores web, herramientas de ofimática como LibreOffice, programas de imagen como Gimp, habilidades para formadores, redes sociales como Twitter, plataformas web 2.0 y recursos en línea como Google Drive y Dropbox. También cubre comandos básicos de sistemas como mkdir, ls y apt-get para gestionar software en sistemas Ubuntu.
Gabriel concluiu o pré-escolar e quer se tornar médico para cuidar de sua avó e mecânico de carros. Ele agradece seus pais e educadores pelo apoio em sua educação. Seus pais ficaram orgulhosos por terem educado Gabriel com dedicação e paciência.
Certificates are important documents that recognize an individual's achievements or qualifications. They provide proof that a person has completed a course or training program. Certificates help individuals advance their careers or education by demonstrating they have acquired new skills to potential employers or schools.
Kosta Djordjevic is a Serbian film director and producer born in 1981 in Belgrade. He graduated from the Faculty of Dramatic Arts in Belgrade in 2007 with a master's degree in film directing. Some of his films include the short film "Street Walker" which screened at the Locarno Film Festival, the documentary series "Together" about children with disabilities, and his feature film debut "Trolling" from 2013. In addition to directing films, Djordjevic has extensive experience in television, commercials, music videos and other projects. He is currently working on developing new screenplays and directing episodes of a television series in Belgrade.
This thesis examines methods for resynchronizing a belief-based software supervisor after failure detection. The supervisor monitors a private branch exchange (PBX) control system using its SDL specifications. Five resynchronization methods are implemented and evaluated experimentally using trace files with seeded failures: the All-States method, Last-Signal method, Next-Signal method, Known-State Resynchronization (KSR), and Generalized KSR. The KSR methods generate candidate beliefs based on the supervisor's state before failure. The evaluation compares the effectiveness and memory costs of the methods.
1) This document investigates 3D printing techniques in liquid-like solids (LLS), which are granular microgels that form a solid when dispersed in water but transition to a fluid state under shear stresses.
2) Preliminary experiments demonstrated stable printing of soft materials at high tip velocities up to 1.05 m/s.
3) The results provide initial evidence that the maximum stable printing speed in an LLS occurs at a Reynolds number of around 5, as predicted by the hypothesis. However, more testing is needed to define the precise Reynolds number ranges where printing becomes unstable.
Senota Singh is a passionate and customer-focused retail professional seeking to grow her career with a reputable company. She has a strong track record of success in store operations and uses data-driven analysis to improve business performance. Her key strengths include delivering quality customer service and putting customers first.
Este documento presenta los resultados de un experimento para observar y medir las propiedades físicas de varios minerales. Los estudiantes midieron la forma cristalina, color, dureza, densidad y magnetismo de minerales como talco, esfalerita, calcita y hematita utilizando un microscopio y otras herramientas. En general, encontraron que las propiedades de los minerales variaban y no siempre coincidían con sus hipótesis iniciales.
The document describes two projects conducted by the UEL London Scholars Project - Youth Employment Radio (YER) and Primary School Entrepreneurship Training. For YER, the scholars attended training at a local radio station, prepared and recorded a radio program to promote entrepreneurship. For the Primary School project, they taught 30 primary students about entrepreneurship through workshops and a school fair, gaining experience in facilitation and presentation skills. Both projects aimed to empower young people and develop the scholars' communication and career skills.
There are infinitely many maximal primitive positive clones in a diagonalizable algebra M* that serves as an algebraic model for provability logic GL. The paper constructs M* as the subalgebra of infinite binary sequences generated by the zero element (0,0,...). It defines primitive positive clones as sets of operations closed under existential definitions, and shows there are infinitely many maximal clones K1, K2, etc. that preserve the relations x = ¬Δi, x = ¬Δ2, etc. in M*. This provides a simple example of infinitely many maximal primitive positive clones.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
This document discusses the convexity of the set of k-admissible functions on a compact Kähler manifold. It begins by introducing k-admissible functions and some necessary convex analysis concepts. It then proves four main results: 1) the log of elementary symmetric functions of eigenvalues is a convex function, 2) the set of matrices with eigenvalues in a convex set is convex, 3) certain functions of eigenvalues are convex, and 4) the set of k-admissible functions is convex. It uses results on conjugation of spectral functions from convex analysis to prove these results. The proofs rely on properties of convex, lower semicontinuous functions and indicator functions.
The document discusses weighted secure domination in graphs. It begins by defining domination number and weighted domination number. It proposes a greedy algorithm that provides a 1 + log(n) approximation for weighted domination number. The algorithm works by iteratively selecting the unselected vertex with the minimum ratio of weight to number of uncovered neighbors. This achieves an approximation ratio of H(n), which is at most 1 + log(n). The algorithm runs in polynomial time.
This document provides an introduction to tensor notation and algebra. It defines scalars, vectors, and tensors, and how they transform under changes of reference frame. Vectors have direction and magnitude, and tensors generalize this to have multiple directions/indices. Tensors of different orders are discussed, along with common examples like the velocity gradient tensor. Frame rotations are described using orthogonal transformation matrices, and how vectors and tensors transform under these changes of basis.
This document discusses equations of motion for higher-order Legendre transforms in quantum field theory. It begins by introducing Legendre transforms and their use in expressing potentials in terms of Green's functions. It then:
1) Derives Schwinger's equation and constraint equations for the generating functional of Green's functions. These determine the Green's functions uniquely.
2) Shows that iterating the equations expresses the generating functional as the sum of all vacuum loops, demonstrating the connected Green's functions are generated.
3) Notes that generalizing this proof to higher-order Legendre transforms would demonstrate their m-irreducibility properties, in line with previous work.
ENUMERATION OF 1-REGULAR SEMIGRAPHS OF ORDER p WITH THREE MUTUALLY ADJACENT m...IAEME Publication
For a vertex 𝑣v in a semigraph 𝐺 = (𝑉, 𝑋), 𝑑𝑒𝑔 𝑣 G=(V,X), deg v is the number of edges having v𝑣 as an end vertex. A semigraph G𝐺 is said to be k-regular if all its vertices have degree k. In this paper, we classify the class of all 1-regular semigraphs with exactly three 𝑚m-vertices into four categories and we enumerate the nonisomorphic semigraphs in one of these categories.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
As many things in the history of analysis of algorithms the all-pairs shortest path has a long history (From the point of view of Computer Science). We can see the initial results from the book “Studies in the Economics of Transportation” by Beckmann, McGuire, and Winsten (1956) where the notation that we use for
the matrix multiplication alike was first used.
In this slides, we go trough several all pairs Shortest path problem solutions from a slow version to the Johnson algorithm.
Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed.
MSC classes: 16T05, 16T25, 17A42, 20N15, 20F29, 20G05, 20G42, 57T05
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1) An infinite series of positive terms converges if the terms decrease to 0. A necessary condition is that the limit of the terms as n approaches infinity is 0.
2) Two positive term series can be compared - if the terms of one series are always less than or equal to the terms of another series, and one converges, then the other must also converge. If one diverges, the other must also diverge.
3) If the limit of the ratio of corresponding terms of two positive term series is a finite, non-zero number as n approaches infinity, then the two series converge or diverge together.
This document provides an introduction to group theory with applications to quantum mechanics and solid state physics. It begins with definitions of groups and examples of groups that are important in physics. It then discusses several applications of group theory in classical mechanics, quantum mechanics, and solid state physics. Specifically, it explains how group theory can be used to evaluate matrix elements, understand degeneracies of energy eigenvalues, classify electronic states in periodic potentials, and construct models that respect crystal symmetries. It also briefly discusses the use of group theory in nuclear and particle physics.
This document is a master's thesis that studies independence complexes constructed from independent sets of sequences of graphs. It examines generating functions, closed formulas, and homology groups of independence complexes for different graph sequences. The thesis provides formulas for the f-polynomials, Euler characteristics, and dimensions of homology groups for each independence complex studied. It also compares the results to known number sequences and establishes bijections to other problems when possible.
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
Average Polynomial Time Complexity Of Some NP-Complete ProblemsAudrey Britton
This document discusses the average polynomial time complexity of some NP-complete problems. Specifically:
- It proves that when N(n) = [n^e] for 0 < e < 2, the CLIQUEN(n) problem (finding a clique of size k in a graph with n vertices and at most N(n) edges) is NP-complete, yet can be solved in average and almost everywhere polynomial time.
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Average Polynomial Time Complexity Of Some NP-Complete Problems
MMath
1. Dimensions of Symmetry Classes of Tensors
by
Larry Huang
A thesis
presented to the University of Waterloo
in fulfilment of the
thesis requirement for the degree of
Master of Mathematics
in
Pure Mathematics
Waterloo, Ontario, Canada, 1993
c Larry Huang 1993
2. I hereby declare that I am the sole author of this thesis.
I authorize the University of Waterloo to lend this thesis to other institutions
or individuals for the purpose of scholarly research.
I further authorize the University of Waterloo to reproduce this thesis by pho-
tocopying or by other means, in total or in part, at the request of other institutions
or individuals for the purpose of scholarly research.
ii
3. The University of Waterloo requires the signatures of all persons using or pho-
tocopying this thesis. Please sign below, and give address and date.
iii
4. Abstract
This paper will study ways to calculate dimensions of symmetry classes of finite
dimensional complex tensor products spaces. General results as well as several
methods for explicitly calculating the dimensions of the symmetry classes are pre-
sented.
iv
5. Acknowledgements
I would like to thank my supervisor Dr. Larry J. Cummings who gave me
excellent guidance and provided me with most of the research material. Also of
invaluable help during preparation of this thesis is fellow graduate student Jun
Wu, who gave me useful pointers.
I would like to thank the Department of Pure Mathematics, University of Wa-
terloo, for their financial support which made my graduate study possible.
v
8. Chapter 1
Introduction
1.1 Preliminary Definitions and Notations
Let V be a finite dimensional inner product space of dimension n over a field F (for
the purpose of this paper F is the field of complex numbers). The usual Cartesian
Product space of m copies of V is denoted by V m
, and the space of mth
tensor
power over V is denoted by ⊗m
V . The inner product on ⊗m
V is the one induced
from V .
Let Γm,n be the set of all the functions from {1, 2, . . . , m} to {1, 2, . . . , n}. For
α ∈ Γm,n, α is considered as a sequence of positive integers of length m chosen from
{1, 2, . . . , n}.
If V has an ordered basis E = {e1, e2, . . . , en}, then the set of decomposable ten-
sors {eα(1) ⊗ . . . ⊗ eα(m) | α ∈ Γm,n} form a basis of ⊗m
V . If E is orthonormal then
so is {eα(1) ⊗ . . . ⊗ eα(m) | α ∈ Γm,n}. (A common short hand for the decomposable
tensor eα(1) ⊗ . . . ⊗ eα(m) is e⊗
α .)
1
9. CHAPTER 1. INTRODUCTION 2
Let Sm be the symmetry group on {1, 2, . . . , m}. That is, Sm is the set of all
bijections from {1, 2, . . . , m} to {1, 2, . . . , m}.
For any fixed σ ∈ Sm, define φ : V m
→ ⊗m
V by φ(v1, . . . , vm) = v⊗
σ−1 . By
straight forward calculations it can be easily shown that φ is multilinear. By the
Universal Factorization Property of ⊗m
V , there is a unique linear operator P(σ) on
⊗m
V such that for all v1, . . . , vm ∈ V , φ(v1, . . . , vm) = P(σ)(v1 ⊗ . . . ⊗ vm). That
is, the following diagram is commutative:
V m -⊗
⊗m
V
⊗m
V
?
P(σ) (unique)
HHH
HHHHj
φ
Obviously for all v1, . . . , vm ∈ V , P(σ)(v1 ⊗ . . . ⊗ vm) = vσ−1(1) ⊗ . . . ⊗ vσ−1(m).
For any α ∈ Γm,n, let µi = vα(i) for i = 1, . . . , m. we have:
P(σ)(v⊗
α ) = P(σ)(vα(1) ⊗ . . . ⊗ vα(m))
= P(σ)(µ1 ⊗ . . . ⊗ µm)
= µσ−1(1) ⊗ . . . ⊗ µσ−1(m). (1.1)
Since µi = vα(i) for i = 1, . . . , m, we have µσ−1(i) = vα(σ−1(i)) = vασ−1(i), for i =
1, . . . , m. Thus (1.1) above becomes
P(σ)(v⊗
α ) = vασ−1(1) ⊗ . . . ⊗ vασ−1(m)
= v⊗
ασ−1 .
Note that for all σ, τ ∈ Sm, P(σ) ◦ P(τ) = P(στ). Further, with respect to the
induced inner product, the adjoint of P(σ) is P(σ)∗
= P(σ−1
) = (P(σ))−1
(where
P(σ)∗
denotes the adjoint of P(σ)).
10. CHAPTER 1. INTRODUCTION 3
To show this, let u1 ⊗ . . . ⊗ um and v1 ⊗ . . . ⊗ vm be two decomposable tensors
in ⊗m
V . Observe that
(P(σ)(v1 ⊗ . . . ⊗ vm), u1 ⊗ . . . ⊗ um) = (v1 ⊗ . . . ⊗ vm, P(σ−1
)(u1 ⊗ . . . ⊗ um))
since by expanding the left side, we have
(vσ−1(1) ⊗ . . . ⊗ vσ−1(m), u1 ⊗ . . . ⊗ um) =
m
i=1
(vσ−1(i), ui),
and by expanding the right side we have
(v1 ⊗ . . . ⊗ vm, uσ(1) ⊗ . . . ⊗ uσ(m)) =
m
i=1
(vi, uσ(i)),
and we see that the multipliers and inner products match. Thus P(σ)∗
= P(σ−1
) on
the set of decomposables. But the decomposables span ⊗m
V , so P(σ)∗
= P(σ−1
).
Definition 1.1.1 Let G be a subgroup of Sm, and λ be an irreducible character of
G. Define the linear operator T(G, λ) on ⊗m
V by
T(G, λ) =
λ(e)
|G| σ∈G
λ(σ)P(σ)
(where e is the identity element in G). The image of ⊗m
V under T(G, λ) is denoted
by Vλ(G).
In the trivial case where G = {e} (and so the only irreducible character of G is
λ = 1), we have:
T(G, λ) =
λ(e)
|G| σ∈G
λ(σ)P(σ) = I⊗mV .
So in this case Vλ(G) is just ⊗m
V .
By straight-forward calculations, it can be easily shown that for any σ ∈ G,
P(σ) ◦ T(G, λ) = T(G, λ) ◦ P(σ).
We have the following property for T(G, λ).
11. CHAPTER 1. INTRODUCTION 4
Proposition 1.1.2 With respect to the induced inner product, T(G, λ) is an or-
thogonal projection.
Proof: We use the following result from group representation theory: Let λ and
χ be irreducible characters of the finite group G afforded by representations A and
B of G respectively. Then for any π ∈ G,
σ∈G
λ(σ−1
)χ(σπ) =
|G|λ(π)/λ(e) if A ≡ B,
0 otherwise
(1.2)
(see [37], page 11).
We have:
T(G, λ)T(G, λ) =
λ(e)2
|G|2
σ∈G
λ(σ)P(σ)
τ∈G
λ(τ)P(τ)
=
λ(e)2
|G|2
σ,τ∈G
λ(σ)λ(τ)P(στ)
=
λ(e)2
|G|2
π∈G σ∈G
λ(σ)λ(σ−1
π) P(π) by writing π = στ
=
λ(e)
|G| π∈G
λ(π)P(π) by (1.2)
= T(G, λ).
Thus T(G, λ) is a projection.
By straight-forward calculations, it is easily shown that
λ(e)
|G| σ∈G
λ(σ)P(σ)∗
(where λ(σ) denotes the complex conjugate of λ(σ)) has all the properties of
T(G, λ)∗
. By the uniqueness of adjoints and the fact that P(σ)∗
= P(σ−1
), we
have
T(G, λ)∗
=
λ(e)
|G| σ∈G
λ(σ)P(σ)∗
=
λ(e)
|G| σ∈G
λ(σ−1
)P(σ−1
) = T(G, λ). 2
12. CHAPTER 1. INTRODUCTION 5
Theorem 1.1.3 Let I(G) denote the set of all irreducible characters of G, where
G is a subgroup of Sm. If λ, χ ∈ I(G) and λ = χ, then T(G, λ) ◦ T(G, χ) = 0.
Moreover,
λ∈I(G)
T(G, λ) is the identity map on ⊗m
V .
Proof: Using (1.2), we have:
T(G, λ)T(G, χ) =
λ(e)χ(e)
|G|2
π∈G σ∈G
λ(σ)χ(σ−1
π) P(π)
= 0.
To show that
λ∈I(G)
T(G, λ) = I⊗mV , we use the following result from group
representation theory:
Let σ ∈ G, where G is any finite group. Then
χ∈I(G)
χ(e)χ(σ) =
|G| if σ = e,
0 otherwise
(1.3)
(see [37], page 17).
Using (1.3), we get:
λ∈I(G)
T(G, λ) =
λ∈I(G)
(
λ(e)
|G| σ∈G
λ(σ)P(σ))
=
1
|G| λ∈I(G)
λ(e)
σ∈G
λ(σ)P(σ)
=
1
|G| σ∈G
(
λ∈I(G)
λ(e)λ(σ))P(σ)
= P(e)
= I⊗mV . 2
Definition 1.1.4 Vλ(G) is a subspace of ⊗m
V (since T(G, λ) is linear), and is
called the symmetry class of tensors associated with G and λ.
13. CHAPTER 1. INTRODUCTION 6
Here is an easy consequence of the two preceding results:
Corollary 1.1.5 For any subgroup G of Sm, ⊗m
V can be written as
λ∈I(G)
Vλ(G),
an orthogonal direct sum. 2
Thus the study of ⊗m
V can be reduced to the study of the subspaces Vλ(G).
1.2 Symmetry Classes of Tensors
The following result will be used when we attempt to find a basis of Vλ(G) in the
next chapter.
Proposition 1.2.1 If λ(e) = 1, that is, λ is one-dimensional, then x ∈ Vλ(G) if
and only if P(σ−1
)(x) = λ(σ)x for all σ ∈ G.
Proof: Let E = {e1, . . . , en} be a basis of V . Then {e⊗
α | α ∈ Γm,n} is a basis of
⊗m
V .
Suppose x ∈ Vλ(G). Write x in general form:
x = T(G, λ)(
α∈Γm,n
cαe⊗
α )
=
λ(e)
|G| τ∈G
λ(τ)P(τ)
α∈Γm,n
cαe⊗
α
=
λ(e)
|G| τ∈G
λ(τ)
α∈Γm,n
cαe⊗
ατ−1 .
Using straight-forward calculations, we have:
P(σ−1
)(x) = P(σ−1
)(
λ(e)
|G| τ∈G
λ(τ)
α∈Γm,n
cαe⊗
ατ−1 )
14. CHAPTER 1. INTRODUCTION 7
=
λ(e)
|G| τ∈G
λ(τ)
α∈Γm,n
cαe⊗
ατ−1σ
=
λ(e)
|G| τ∈G
λ(στ)
α∈Γm,n
cαe⊗
ατ−1
=
λ(e)
|G| τ∈G
λ(σ)λ(τ)
α∈Γm,n
cαe⊗
ατ−1 since λ is one-dimensional
= λ(σ)
λ(e)
|G| τ∈G
λ(τ)
α∈Γm,n
cαe⊗
ατ−1
= λ(σ)x.
The converse is easily shown by observing that x = T(G, λ)(x) ∈ Vλ(G):
T(G, λ)(x) =
λ(e)
|G| σ∈G
λ(σ)P(σ)(x)
=
1
|G| σ∈G
λ(σ)λ(σ−1
)(x)
=
1
|G| σ∈G
λ(e)x
=
1
|G| σ∈G
x
= x. 2
Definition 1.2.2 Denote T(G, λ)(v1 ⊗ . . . ⊗ vm) by v1 ∗ . . . ∗ vm. It is called a
decomposable symmetrized tensor. (For α ∈ Γm,n, a common short hand for
vα(1) ∗ . . . ∗ vα(m) is v∗
α.)
Note that in the above definition, the ∗ in v1 ∗ . . . ∗ vm depends on G and λ.
We can consider v1 ∗ . . . ∗ vm as the part of v1 ⊗ . . . ⊗ vm that belongs to Vλ(G).
Also note that Vλ(G) is spanned by, but not necessarily equal to, {v1 ∗ . . . ∗ vm |
v1, . . . , vm ∈ V }.
15. CHAPTER 1. INTRODUCTION 8
By straight-forward calculations, it is easily shown that P(σ−1
)(v1 ∗ . . . ∗ vm) =
v∗
σ, for all σ ∈ G.
Definition 1.2.3 Let G be a subgroup of Sm, λ an irreducible character of G, W
a vector space over F. A multilinear map φ : V m
→ W is said to be symmetric
with respect to G and λ if for all v1, . . . , vm ∈ V ,
φ(v1, . . . , vm) =
λ(e)
|G| σ∈G
λ(σ−1
)φ(vσ(1), . . . , vσ(m)).
Proposition 1.2.4 Let G be a subgroup of Sm, λ be a one-dimensional character
of G, and φ : V m
→ Vλ(G) by φ(v1, . . . , vm) = v1 ∗ . . . ∗ vm. Then φ is multilinear
and symmetric with respect to G and λ (This is where the term “symmetry class”
comes from).
Proof: That φ is multilinear comes from straight-forward calculations.
Let v1, . . . , vm ∈ V , we have:
λ(e)
|G| σ∈G
λ(σ−1
)φ(vσ(1), . . . , vσ(m))
=
λ(e)
|G| σ∈G
λ(σ−1
)(vσ(1) ∗ . . . ∗ vσ(m))
=
λ(e)
|G| σ∈G
λ(σ−1
)(
λ(e)
|G| τ∈G
λ(τ)P(τ)(vσ(1) ⊗ . . . ⊗ vσ(m)))
=
λ(e)
|G|2
σ∈G τ∈G
λ(σ−1
)λ(τ)(vστ−1(1) ⊗ . . . ⊗ vστ−1(m))
=
λ(e)
|G|2
σ∈G τ∈G
λ(τσ−1
)P(τσ−1
)(v1 ⊗ . . . ⊗ vm) since λ is one-dimensional
=
λ(e)
|G|2
σ∈G τ∈G
λ(τ)P(τ)(v1 ⊗ . . . ⊗ vm)
=
λ(e)
|G| τ∈G
λ(τ)P(τ)(v1 ⊗ . . . ⊗ vm)
16. CHAPTER 1. INTRODUCTION 9
= T(G, λ)(v1 ⊗ . . . ⊗ vm)
= v1 ∗ . . . ∗ vm
= φ(v1, . . . , vm) 2
The following result is known as the Universal Factorization Property for Sym-
metric Multilinear Functions:
Theorem 1.2.5 Let G be a subgroup of Sm, λ be an irreducible character of G, W
a vector space over F, and φ : V m
→ W be multilinear and symmetric with respect
to G and λ. Then there is a unique linear map hλ : Vλ(G) → W such that for all
v1, . . . , vm ∈ V, hλ(v1 ∗ . . . ∗ vm) = φ(v1, . . . , vm). That is, the following diagram is
commutative:
V m -∗ Vλ(G)
W
?
hλ
HHH
HHHHj
φ
Proof: By the universal factorization property of ⊗m
V , there is a unique linear
map h : ⊗m
V → W such that h(v1 ⊗ . . . ⊗ vm) = φ(v1, . . . , vm), for all v1, . . . , vm ∈
V . In particular, for all v1, . . . , vm ∈ V and for all σ ∈ G, we have:
h(vσ(1) ⊗ . . . ⊗ vσ(m)) = φ(vσ(1), . . . , vσ(m)).
Multiply both sides by
λ(e)
|G|
λ(σ−1
), and sum on σ ∈ G, we get:
σ∈G
λ(e)
|G|
λ(σ−1
)h(vσ(1) ⊗ . . . ⊗ vσ(m)) =
σ∈G
λ(e)
|G|
λ(σ−1
)φ(vσ(1), . . . , vσ(m)).
Noting that h is linear and φ is symmetric with respect to G and λ, we re-arrange
the above and get:
h(T(G, λ)(v1 ⊗ . . . ⊗ vm)) = φ(v1, . . . , vm),
17. CHAPTER 1. INTRODUCTION 10
or:
h(v1 ∗ . . . ∗ vm) = φ(v1, . . . , vm).
Define hλ to be the restriction of h to Vλ(G). 2
18. Chapter 2
Dimensions of Symmetry Classes
In last chapter we showed that ⊗m
V is an orthogonal direct sum of the symmetry
classes Vλ(G) over all irreducible characters λ of G. Indeed, ⊗m
V itself is Vλ(G) in
the case where G = {e} (and hence λ = 1). Thus to study ⊗m
V we need to study
Vλ(G).
One of the most basic concerns about Vλ(G) is how to find its dimension. In
the rest of this thesis we will discuss results on this topic.
2.1 General Results
We will find a spanning set of Vλ(G) in this section. In addition, a basis of Vλ(G)
is presented in the simpliest case, where the irreducible character λ of G is one-
dimensional.
Let α ∈ Γm,n, and σ ∈ Sm. Note that since α : {1, . . . , m} → {1, . . . , n} and
σ : {1, . . . , m} → {1, . . . , m}, α ◦ σ : {1, . . . , m} → {1, . . . , n}, and so α ◦ σ ∈ Γm,n.
11
19. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 12
Definition 2.1.1 Given a subgroup G of Sm, define an equivalence relation ≡ on
Γm,n as follows: for α, β ∈ Γm,n, α ≡ β (mod G) if there is σ ∈ G such that
β = α ◦ σ.
It is easy to show that ≡ is indeed an equivalence relation on Γm,n.
Consider each α ∈ Γm,n as a sequence of length m of positive integers chosen
from the set {1, . . . , n}, i.e., α = (α(1), α(2), . . . , α(m)). We can order elements of
Γm,n lexicographically, i.e., α precedes β if the first non-zero difference β(i) − α(i)
is positive.
Let be the set of distinct representatives of the equivalence classes (mod G)
of Γm,n so chosen that α ∈ if and only if α is the first in lexicographical order in
its equivalence class. (Note that depends on m, n, and G.)
From chapter 1 we know that if {e1, . . . , en} is a basis of V , then Vλ(G) has
a spanning set {T(G, λ)(e⊗
α ) | α ∈ Γm,n}. Some elements of this spanning set,
however, are 0, and the rest are not all linearly independent. Thus to find a basis
of Vλ(G) we need to eliminate these elements from our spanning set. We start with
a definition and a very important lemma.
Definition 2.1.2 Let G be a subgroup of Sm, and α ∈ Γm,n. The stabilizer sub-
group of α is Gα = {σ ∈ G | ασ = α}.
It is easy to show that Gα is indeed a subgroup of G.
Lemma 2.1.3 Let G be a subgroup of Sm, λ be an irreducible character of G,
α, β ∈ Γm,n, and {ei | i = 1, . . . , n} be an orthonormal basis of V . Then
(e∗
α, e∗
β) =
λ(e)
|G| σ∈Gα
λ(τ−1
σ) if β = ατ for some τ ∈ G
0 otherwise.
20. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 13
Proof:
(e∗
α, e∗
β) = (T(G, λ)(e⊗
α ), T(G, λ)(e⊗
β ))
= (T(G, λ)∗
◦ T(G, λ)(e⊗
α ), e⊗
β ).
By Proposition (1.1.2), T(G, λ) is idempotent and Hermitian, so by straight-
forward calculations the above reduces to:
(e∗
α, e∗
β) =
λ(e)
|G| σ∈G
λ(σ)(e⊗
ασ−1 , e⊗
β )
=
λ(e)
|G| σ∈G
λ(σ)
m
i=1
(eασ−1(i), eβ(i)) (2.1)
since the inner product on ⊗m
V is the one induced from V . If α ≡ β (mod G),
then (2.1) above is 0 because the ei’s are orthonormal. If β = ατ for some τ ∈ G,
then
m
i=1
(eασ−1(i), eβ(i)) = 1 if and only if ασ−1
= β, which means ασ−1
= ατ, which
is equivalent to τσ ∈ Gα. Therefore (2.1) above becomes
λ(e)
|G| τσ∈Gα
λ(σ),
and can be written as
λ(e)
|G| σ∈Gα
λ(τ−1
σ). 2
By Lemma 2.1.3, e∗
α
2
=
λ(e)
|G| σ∈Gα
λ(σ), and since e∗
α = 0 if and only if
e∗
α = 0, it follows that e∗
α = 0 if and only if
σ∈Gα
λ(σ) = 0.
Define Ω = {α ∈ Γm,n |
σ∈Gα
λ(σ) = 0}. It follows that {e∗
α | α ∈ Ω} spans
Vλ(G), since {e∗
α | α ∈ Γm,n} does.
Having eliminated the zero elements from our spanning set of Vλ(G), we now
want to find a subset of this new spanning set that is a basis of Vλ(G).
21. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 14
Definition 2.1.4 Let = ∩ Ω = {α ∈ |
σ∈Gα
λ(σ) = 0}.
Note that in the case of λ being identically l (i.e., λ being the principal character
of G), we have
σ∈Gα
λ(σ) =
σ∈Gα
1 = |Gα| = 0
for all α ∈ , which gives = .
The following is an easier and more useful way to look at Ω:
Proposition 2.1.5 Ω = α∈ {ασ | σ ∈ G} = {ασ | α ∈ , σ ∈ G}.
Proof: That α∈ {ασ | σ ∈ G} = {ασ | α ∈ , σ ∈ G} is obvious. We will
show that Ω = {ασ | α ∈ , σ ∈ G}.
Let β ∈ Ω. If β ∈ , then β ∈ Ω ∪ = , and certainly
β = βe ∈ {ασ | α ∈ , σ ∈ G}. (2.2)
If β ∈ , then there is β ∈ and σ ∈ G such that β = β σ. We have:
β ∈ Ω ⇐⇒
τ∈{π∈G|βπ=β}
λ(τ) = 0
⇐⇒
τ∈{π∈G|β σπ=β σ}
λ(τ) = 0
⇐⇒
τ∈{π∈G|β (σπσ−1)=β }
λ(τ) = 0
⇐⇒
τ∈G
β
λ(τ) = 0
⇐⇒ β ∈ Ω.
But β ∈ , so β ∈ Ω ∩ = . Thus
β = β σ ∈ {ασ | α ∈ , σ ∈ G}. (2.3)
22. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 15
Results (2.2) and (2.3) give Ω ⊆ {ασ | α ∈ , σ ∈ G}.
Now let β ∈ {ασ | α ∈ , σ ∈ G}. Then β = ατ, for some α ∈ and some
τ ∈ G. We have:
σ∈Gβ
λ(σ) =
βσ=β
λ(σ)
=
ατσ=ατ
λ(σ)
=
α(τστ−1)=α
λ(σ)
=
α(τστ−1)=α
λ(τστ−1
)
=
ασ=α
λ(σ)
=
σ∈Gα
λ(σ)
= 0 since α ∈ = ∩ Ω ⊆ Ω.
So β ∈ Ω, and hence {ασ | α ∈ , σ ∈ G} ⊆ Ω. 2
The following theorem gives the relationship between Vλ(G) and the induced
basis of ⊗m
V .
Theorem 2.1.6 Let G be a subgroup of Sm, λ be an irreducible character of G,
and E = {e1, ..., en} be a basis of V . Then Vλ(G) is the direct sum of the subspaces
span{e∗
ασ | σ ∈ G}, for α ∈ . That is, Vλ(G) =
α∈
span{e∗
ασ | σ ∈ G}, the sum
being direct.
Proof: From the discussion after lemma 2.1.3, {e∗
α | α ∈ Ω} spans Vλ(G). By
Proposition 2.1.5, Ω = {ασ | α ∈ , σ ∈ G}, so it follows that Vλ(G) is the sum of
the subspaces span{e∗
ασ | σ ∈ G}, for α ∈ .
23. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 16
To see that the sum is direct, take the inner product on V that makes E or-
thonormal. By lemma 2.1.3, the subspaces span{e∗
ασ | σ ∈ G}, α ∈ Γm,n become
orthogonal, and the sum is therefore direct. 2
In next section we will show a formula by R. Freese that calculates the dimension
of span{e∗
ασ | σ ∈ G}, for α ∈ Γm,n.
We can now prove the major result of this section:
Theorem 2.1.7 Let E = {e1, . . . , en} be a basis of V . Then {e∗
α | α ∈ } is a
linearly independent set in Vλ(G). If E is orthogonal then so is {e∗
α | α ∈ }. If
λ(e) = 1 then {e∗
α | α ∈ } is a basis of Vλ(G).
Proof: From theorem 2.1.6 and the discussion after lemma 2.1.3, the e∗
α’s are
non-zero vectors from distinct subspaces that form a direct sum of Vλ(G). So the
e∗
α’s must be linearly independent.
If E is orthogonal, then by lemma 2.1.3, {e∗
α | α ∈ } is an orthogonal set
(since for α, β ∈ , α ≡ β (mod G) if and only if α = β).
If λ(e) = 1, by proposition 1.2.1 e∗
ασ = λ(σ)e∗
α for all σ ∈ G and α ∈ Γm,n. Thus
span{e∗
ασ | σ ∈ G} = span{λ(σ)e∗
α | σ ∈ G} = span{e∗
α},
which means the subspaces span{e∗
ασ | σ ∈ G} making up the direct sum of Vλ(G)
are all one-dimensional. This shows {e∗
α | α ∈ } spans Vλ(G), and so must be a
basis of Vλ(G). 2
Theorem 2.1.6 tell us that Vλ(G) is the direct sum of span{e∗
ασ | σ ∈ G}, for
α ∈ . It is therefore important to find the dimension of span{e∗
ασ | σ ∈ G}.
In the remaining sections of this chapter we will discuss methods for calculating
the dimension of span{e∗
ασ | σ ∈ G}.
24. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 17
2.2 Freese’s Theorem
Definition 2.2.1 Let {e1, . . . , en} be a basis of V , and let α ∈ Γm,n. The subspace
spanned by {e∗
ασ | σ ∈ G} is called an orbital subspace of Vλ(G) corresponding to
α. The dimension of span{e∗
ασ | σ ∈ G} is denoted sα.
Proposition 2.2.2 sα defined above has the following properties:
1. sα = 0 if and only if α ∈ Ω
2. if λ(e) = 1 then sα = 1 for all α ∈ Ω
3. dim Vλ(G) =
α∈
sα
Proof:
1. sα = 0 if and only if {e∗
ασ | σ ∈ G} = {0}, which holds if and only if e∗
ασ = 0
for some σ ∈ G. By the discussion after lemma 2.1.3, e∗
ασ = 0 if and only if
τ∈Gασ
λ(τ) = 0. But we have:
τ∈Gασ
λ(τ) =
αστ=ασ
λ(τ)
=
αστσ−1=α
λ(τ)
=
α(στσ−1)=α
λ(στσ−1
)
=
ατ=α
λ(τ)
=
τ∈Gα
λ(τ),
so
τ∈Gα
λ(τ) = 0, which gives α ∈ Ω.
25. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 18
Conversely, α ∈ Ω means
τ∈Gα
λ(τ) = 0 which, by the discussion after lemma
2.1.3, is equivalent to e∗
α = 0, i.e., e∗
αe = 0. Thus {e∗
ασ | σ ∈ G} = {0}, giving
sα = 0.
2. This is the last part in the proof of theorem 2.1.7.
3. By theorem 2.1.6, Vλ(G) =
α∈
span{e∗
ασ | σ ∈ G}, the sum being direct.
This immediately gives us the result. 2
In [9], R. Freese gave the following general formula for sα:
Theorem 2.2.3 (Freese) For α ∈ Γm,n, sα = λ(e)(λ, 1)Gα , where
(λ, 1)Gα =
1
|Gα| σ∈Gα
λ(σ).
To prove Freese’s theorem, we first prove the following well known result in
linear algebra:
Lemma 2.2.4 The rank of an idempotent linear operator on a vector space is its
trace.
Proof: Let L : V → V be an idempotent linear operator on the n-dimensional
space V . Let R and N denote the image and kernel of L respectively.
It is easy to see that v ∈ R if and only if v = L(v), for if v = L(u) for some
u ∈ V , then since L is idempotent, L(v) = L2
(u) = L(u) = v; and if v = L(v), then
(obviously) v ∈ R.
Thus we have R ∩ N = {0}. Further, since every vector v ∈ V can be written
as v = L(v) + (v − L(v)), and v − L(v) ∈ N, we conclude that V is a direct sum of
R and N.
26. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 19
Let {e1, . . . , et} be a basis of R and {et+1, . . . , en} be a basis of N. Then E =
{e1, . . . , en} is a basis of V and the matrix representing L with respect to E is:
It 0
0 0
,
where It is the t × t identity matrix. This shows rank(L) = trace(L) = t. 2
Now we can prove Freese’s theorem:
Proof: Since Gα is a subgroup of G, we can decompose G by picking τi ∈ G, i =
1, . . . , r such that G = Gατ1 ∪ . . . ∪ Gατr, where the Gατi’s are all disjoint.
Observe that for all π ∈ G,
e∗
απ = T(G, λ)(e⊗
απ)
=
λ(e)
|G| σ∈G
λ(σ)P(σ)(e⊗
απ)
=
λ(e)
|G| σ∈G
λ(σ)e⊗
απσ−1
=
λ(e)
|G| σ∈G
λ(σ−1
π)e⊗
ασ
=
λ(e)
|G|
r
i=1 τ∈Gα
λ((ττi)−1
π)e⊗
αττi
=
λ(e)
|G|
r
i=1
(
τ∈Gα
λ(τ−1
i τ−1
π))e⊗
ατi
since ατ = α.
In particular,
e∗
ατj
=
λ(e)
|G|
r
i=1
(
τ∈Gα
λ(τ−1
i ττj))e⊗
ατi
. (2.4)
For each σ ∈ G, σ ∈ Gατi for some i ∈ {1, . . . , r}. Thus σ = σ τi for some
σ ∈ Gα. So for σ ∈ G,
e∗
ασ = T(G, λ)(e⊗
ασ)
27. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 20
= T(G, λ)(e⊗
ασ τi
)
= T(G, λ)(e⊗
ατi
)
= e∗
ατi
.
Therefore we have
span{e∗
ασ | σ ∈ G} ⊆ span{e∗
ατi
| i = 1, . . . , r},
hence
span{e∗
ασ | σ ∈ G} = span{e∗
ατi
| i = 1, . . . , r},
and by (2.4),
span{e∗
ασ | σ ∈ G} ⊆ span{e⊗
ατi
| i = 1, . . . , r}.
Since {e⊗
β | β ∈ Γm,n} is a basis of ⊗m
V , {e⊗
ατi
| i = 1, . . . , r} is a linearly indepen-
dent set if the ατi’s are distinct. The ατi’s are indeed distinct because:
If ατi = ατj, then α = ατjτ−1
i , so τjτ−1
i ∈ Gα. This gives τj = (τjτ−1
i )τi ∈ Gατi.
But τj = eτj ∈ Gατj, so by the choice of the τi’s in the decomposition of G we
must have Gατi = Gατj, which means τi = τj. This shows that the ατi’s are indeed
distinct, and hence {e⊗
ατi
| i = 1, . . . , r} is a linearly independent set.
Now let C be the r×r matrix with
λ(e)
|G| σ∈Gα
λ(τ−1
i στj) at the (i, j)th
position (ith
row and jth
column). Consider C as a linear operator on span{e⊗
ατi
| i = 1, . . . , r}.
Clearly for k ∈ {1, . . . , r},
C(e⊗
ατk
) =
r
i=1
(
λ(e)
|G| σ∈Gα
λ(τ−1
i στk))e⊗
ατi
= e∗
ατk
by (2.4).
Thus the image of span{e⊗
ατi
| i = 1, . . . , r} under C is span{e∗
ατi
| i = 1, . . . , r}, so
the dimension of span{e∗
ατi
| i = 1, . . . , r} is just rank(C).
28. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 21
C is idempotent because the (i, j)th
element of C2
is:
(C2
)i,j =
r
k=1
(
λ(e)
|G| σ∈Gα
λ(τ−1
i στk))(
λ(e)
|G| σ∈Gα
λ(τ−1
k στj))
= (
λ(e)
|G|
)2
r
k=1
(
σ,τ∈Gα
λ(τ−1
i στk)λ(τ−1
k ττj))
= (
λ(e)
|G|
)2
r
k=1
(
σ,π∈Gα
λ(τ−1
i στk)λ(τ−1
k σ−1
πτj))
= (
λ(e)
|G|
)2
r
k=1
(
σ,π∈Gα
λ(τ−1
i (στk))λ((στk)−1
πτj))
= (
λ(e)
|G|
)2
π∈Gα σ∈G
λ(τ−1
i σ)λ(σ−1
πτj)
= (
λ(e)
|G|
)2
π∈Gα σ∈G
λ(σ)λ(σ−1
τ−1
i πτj)
= (
λ(e)
|G|
)2
π∈Gα
(
|G|λ(τ−1
i πτj)
λ(e)
) by (1.2)
=
λ(e)
|G| π∈Gα
λ(τ−1
i πτj)
= (C)i,j.
By lemma 2.2.4,
rank(C) = trace(C)
=
r
i=1
(C)i,i
=
r
i=1
λ(e)
|G| σ∈Gα
λ(τ−1
i στi)
=
λ(e)
|G|
r
i=1 σ∈Gα
λ(σ)
=
λ(e)
|G|
r
σ∈Gα
λ(σ)
=
λ(e)
|Gα| σ∈Gα
λ(σ). 2
29. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 22
2.3 Using Disjoint Cycle Decomposition of Sm
In [26], M. Marcus gave a formula for the rank of the linear operator T(G, λ), that
is, the dimension of Vλ(G). The method uses the disjoint cycle decomposition of
elements of Sm. To prove that result we first prove the following lemma.
Lemma 2.3.1 Let σ ∈ Sm and let c(σ) denote the number of cycles in the dis-
joint cycle decomposition of σ, including cycles of length one. Then for the linear
operation P(σ) on ⊗m
V , trace(P(σ)) = nc(σ)
.
Proof: Let {e1, . . . , en} be a basis of V , {f1, . . . , fn} be the basis of V ∗
dual to
{e1, . . . , en}, i.e.,
fi(ej) =
1 if i = j
0 otherwise
extended linearly to all of V ∗
.
Let E = {e⊗
α | α ∈ Γm,n} be the corresponding (ordered) basis of ⊗m
V . For
each β ∈ Γm,n, define a linear functional hβ : ⊗m
V → F by extending
hβ(eα(1) ⊗ . . . ⊗ eα(m)) = fβ(1)(eα(1)) . . . fβ(m)(eα(m)) =
m
i=1
fβ(i)(eα(i))
To all of ⊗m
V . We have
hβ(e⊗
α ) =
1 if α = β
0 otherwise.
This means that {hβ | β ∈ Γm,n} is the basis of (⊗m
V )∗
dual to E.
The elements on the main diagonal of the matrix representing P(σ) with respect
to E are the numbers
hα(P(σ)(e⊗
α )) = hα(e⊗
ασ−1 )
30. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 23
=
m
i=1
fα(i)(eασ−1(i))
=
1 if α = ασ−1
0 otherwise
= δα,ασ−1 .
Thus the trace of P(σ) is
α∈Γm,n
δα,ασ−1 , which is the number of α’s in Γm,n
satisfying α = ασ−1
, or equivalently, α = ασ.
Let (t σ(t) σ2
(t) . . . σl−1
(t)) be a cycle of length l in σ. Then any α ∈ Γm,n
satisfying α = ασ must satisfy
α(t) = ασ(t),
ασ(t) = ασ2
(t),
ασ2
(t) = ασ3
(t),
...
ασl−1
(t) = ασl
(t) = α(t).
This shows α(t) = ασ(t) = ασ2
(t) = . . . = ασl−1
(t) for all α with α = ασ.
Consider each α ∈ Γm,n as a sequence (α(1) . . . α(m)). Any α satisfying
α = ασ must have the same integer k at positions t, σ(t), . . . , σl−1
(t) in the above
sequence. There are n choices for the integer k for each disjoint cycle in σ, and
since there are c(σ) disjoint cycles in σ, there are nc(σ)
choices for α. Hence
trace(P(σ)) =
α∈Γm,n
δα,ασ−1 = nc(σ)
. 2
We can now prove the main result mentioned at the beginning of this section:
Theorem 2.3.2 The dimension of Vλ(G) is
λ(e)
|G| τ∈G
λ(τ)nc(τ)
.
31. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 24
Proof: By proposition 1.1.2 T(G, λ) is idempotent. We have:
dim Vλ(G) = rank(T(G, λ))
= trace(T(G, λ)) by lemma 2.2.4
= trace(
λ(e)
|G| τ∈G
λ(τ)P(τ))
=
λ(e)
|G| τ∈G
λ(τ)trace(P(τ))
=
λ(e)
|G| τ∈G
nc(τ)
by lemma 2.3.1. 2
2.4 Subspaces of the Orbital Subspaces
In [46] B. Y. Wang and M. P. Gong described methods to construct subspaces of
the orbital subspaces span{e∗
ασ | σ ∈ G}, α ∈ , and calculate their dimensions.
Some of their results are summarized in this section.
As before, let G be a subgroup of Sm with identity element e, and let {e1, . . . , en}
be a fixed orthonormal basis of V , so that {e⊗
α | α ∈ Γm,n} is an orthonormal basis
of ⊗m
V . Let A be an irreducible representation of G which affords the irreducible
character λ. For each σ ∈ G and i, j ∈ {1, . . . , λ(e)}, let aij(σ) denote the (i, j)th
element of the matrix A(σ) with respect to the basis {e1, . . . , en}.
Define the operators Tij on ⊗m
V by
Tij =
λ(e)
|G| σ∈G
aij(σ)P(σ)
for each i, j = 1, . . . , λ(e).
By straight-forward calculations we see that
T(G, λ) =
λ(e)
i=1
Tii.
32. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 25
We further have the following identities:
T2
ii = Tii (2.5)
and
TijTkl =
Til if j = k,
0 otherwise
(2.6)
for all i, j, k, l ∈ {1, . . . , λ(e)} (see [37], page V-8).
If A is a unitary representation (that is, A(σ)A(σ)∗
= A(σ)∗
A(σ) = I, or that
A(σ) is a unitary matrix, for every σ ∈ G), then
T∗
ij = Tji (2.7)
for all i, j ∈ {1, . . . , λ(e)}.
Let V ij
A (G) = Tij(⊗m
V ). We have the following theorem:
Theorem 2.4.1 The symmetry class of tensors, Vλ(G), is equal to
λ(e)
i=1
V ii
A (G), a
direct sum, and dim V ii
A (G) =
dim Vλ(G)
λ(e)
. If A is a unitary representation of G,
then the sum is orthogonal. 2
(see [37], page V-8).
Let α ∈ Γm,n. To divide span{e∗
ασ | σ ∈ G} into subspaces, we first introduce a
lemma:
Lemma 2.4.2 Let V i
α = span{Ti1(e⊗
α ), . . . , Tiλ(e)(e⊗
α )}, for i = 1, . . . , λ(e). Then
V i
α = span{Tij(e⊗
ασ) | σ ∈ G, j = 1, . . . , λ(e)}, for i = 1, . . . , λ(e) (2.8)
and
V i
ατ = V i
α, for all τ ∈ G and i = 1, . . . , λ(e). (2.9)
33. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 26
Proof: Note that we only need to prove (2.8), since (2.9) follows directly from
(2.8).
Let k ∈ {1, . . . , λ(e)}. By (2.6) we have
Tik(e⊗
α ) = Tij(Tjk(e⊗
α )) for any j ∈ {1, . . . , λ(e)}
= Tij
λ(e)
|G| σ∈G
ajk(σ)e⊗
ασ−1
=
λ(e)
|G| σ∈G
ajk(σ)Tij(e⊗
ασ−1 )
=
λ(e)
|G| σ∈G
ajk(σ−1
)Tij(e⊗
ασ)
∈ span{Tij(e⊗
ασ) | σ ∈ G}.
So we have V i
α ⊆ span{Tij(e⊗
ασ) | σ ∈ G, j = 1, . . . , λ(e)}. To show inclusion in the
other direction, first note that for any i, j ∈ {1, . . . , λ(e)} and any π, σ ∈ G,
aij(πσ) =
λ(e)
l=1
ail(π)alj(σ). (2.10)
For i, j ∈ {1, . . . , λ(e)} and σ ∈ G, We have:
Tij(e⊗
ασ) =
λ(e)
|G| τ∈G
aij(τ)P(τ)(e⊗
ασ)
=
λ(e)
|G| τ∈G
aij(τ)P(τ)P(σ−1
)(e⊗
α )
=
λ(e)
|G| τ∈G
aij(τ)P(τσ−1
)(e⊗
α ). (2.11)
By writing π = τσ−1
, (2.11) above becomes:
Tij(e⊗
ασ) =
λ(e)
|G| π∈G
aij(πσ)P(π)(e⊗
α )
=
λ(e)
|G| π∈G
λ(e)
l=1
ail(π)alj(σ)
P(π)(e⊗
α ) by (2.10)
34. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 27
=
λ(e)
l=1
λ(e)
|G| π∈G
ail(π)alj(σ)P(π)(e⊗
α )
=
λ(e)
l=1
alj(σ)
λ(e)
|G| π∈G
ail(π)P(π)(e⊗
α )
=
λ(e)
l=1
alj(σ)Til(e⊗
α )
∈ V i
α. 2
Now we can divide span{e∗
ασ | σ ∈ G} into subspaces:
Theorem 2.4.3 For any fixed α ∈ Γm,n,
span{e∗
ασ | σ ∈ G} =
λ(e)
i=1
V i
α = span{Tij(e⊗
α ) | i, j = 1, . . . , λ(e)}; (2.12)
and
Vλ(G) =
α∈
λ(e)
i=1
V i
α =
α∈
span{Tij(e⊗
α ) | i, j = 1, . . . , λ(e)}. (2.13)
If A is a unitary representation of G, then the above sums are orthogonal.
Proof: From straight-forward calculations we know that span{e⊗
ασ | σ ∈ G} is
invariant under both T(G, λ) and Tij, for all i, j = 1, . . . , λ(e). That is,
T(G, λ)(span{e⊗
ασ | σ ∈ G}) ⊆ span{e⊗
ασ | σ ∈ G}
and
Tij(span{e⊗
ασ | σ ∈ G}) ⊆ span{e⊗
ασ | σ ∈ G}
for all i, j = 1, . . . , λ(e).
We have:
span{e∗
ασ | σ ∈ G} = span{T(G, λ)(e⊗
ασ) | σ ∈ G}
35. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 28
= span{
λ(e)
i=1
Tii(e⊗
ασ) | σ ∈ G}
= {
σ∈G
(fσ
λ(e)
i=1
Tii(e⊗
ασ)) | fσ ∈ F}
= {
λ(e)
i=1 σ∈G
fσTii(e⊗
ασ) | fσ ∈ F}
=
λ(e)
i=1
span{Tii(e⊗
ασ) | σ ∈ G}
=
λ(e)
i=1
span{Tii(e⊗
ασ) | σ ∈ G}
⊆
λ(e)
i=1
V i
α
= span{Tij(e⊗
α ) | i, j = 1, . . . , λ(e)}. (2.14)
Since span{e⊗
ασ | σ ∈ G} is invariant under Tij for all i, j = 1, . . . , λ(e), and
Tij = TiiTij, we have:
Tij(span{e⊗
ασ | σ ∈ G}) = Tii(Tij(span{e⊗
ασ | σ ∈ G}))
⊆ Tii(span{e⊗
ασ | σ ∈ G}).
This mean the inclusion in (2.14) is also valid in the other direction. This proves
(2.12). Theorem 2.1.6 and (2.12) leads directly to (2.13).
If A is unitary, then by (2.6) and (2.7) the direct sums are orthogonal. 2
Lemma 2.4.4 Let λ ∈ Γm,n. If {T1j1 (e⊗
α ), T1j2 (e⊗
α ), . . . , T1jk
(e⊗
α )} is a basis of V 1
α ,
then
{Tij1 (e⊗
α ), Tij2 (e⊗
α ), . . . , Tijk
(e⊗
α ) | i = 1, . . . , λ(e)}
is a basis of span{e⊗
ασ | σ ∈ G}.
36. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 29
Proof: Denote {Tij1 (e⊗
α ), Tij2 (e⊗
α ), . . . , Tijk
(e⊗
α )} by Ei, for all i = 1, . . . , λ(e).
By theorem 2.4.3, it suffices to show that Ei is a basis of V i
α, for all i =
2, . . . , λ(e).
Let i ∈ {2, . . . , λ(e)}. Suppose
k
l=1
ClTijl
(e⊗
α ) = 0, for some Cl ∈ F. Then
k
l=1
ClT1jl
(e⊗
α ) =
k
l=1
ClT1i(Tijl
(e⊗
α ))
= T1i(
k
l=1
ClTijl
(e⊗
α ))
= T1i(0)
= 0.
Since E1 is linearly independent, we must have Cl = 0 for all l = 1, . . . , k. This
shows that Ei is linearly independent for all i = 2, . . . , λ(e).
Now let Tij(e⊗
α ) ∈ V i
α for some j ∈ {1, . . . , λ(e)}. We have:
Tij(e⊗
α ) = Ti1(T1j(e⊗
α )).
But T1j(e⊗
α ) ∈ V 1
α = span E1, so we can write
T1j(e⊗
α ) =
k
l=1
ClT1jl
(e⊗
α )
for some suitable values of Cl ∈ F. We have:
Tij(e⊗
α ) = Ti1(T1j(e⊗
α ))
= Ti1(
k
l=1
ClT1jl
(e⊗
α ))
=
k
l=1
ClTi1(T1jl
(e⊗
α ))
=
k
l=1
ClTijl
(e⊗
α ))
∈ span Ei.
37. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 30
Thus Ei also spans V i
α. Hence Ei is a basis of V i
α. 2
The above lemma shows that if we know how to calculate dim V i
α, we can then
easily find the dimension of span{e∗
ασ | σ ∈ G}. The following theorem does exactly
that:
Theorem 2.4.5 dim V i
α = (λ, 1)Gα , where
(λ, 1)Gα =
1
|Gα| σ∈Gα
λ(σ),
for all i = 1, . . . , λ(e).
Proof: By lemma 2.4.4 all subspaces V i
α have the same dimensions. Using this
result and theorem 2.4.3, we have:
dim V i
α =
dim span{e∗
ασ | σ ∈ G}
λ(e)
for each i = 1, . . . , λ(e). Freese’s theorem gives
dim span{e∗
ασ | σ ∈ G} = λ(e)(λ, 1)Gα
which leads directly to our result. 2
If the irreducible representation A of G is unitary, we can then construct or-
thonormal bases of all the orbital subspaces of Vλ(G). Before we do that, first we
introduce a couple of notation:
If A is a p × q matrix and i1, . . . , ik, j1, . . . , jl are positive integers with i1 <
. . . < ik ≤ p and j1 < . . . < jl ≤ q, let A[i1, . . . , ik | j1, . . . , jl] denote the k × l
matrix whose element at row s and column t is the element at row is and column
jt of the matrix A.
38. CHAPTER 2. DIMENSIONS OF SYMMETRY CLASSES 31
For α ∈ Γm,n, let
dj =
λ(e)
|G| σ∈Gα
ajj(σ)
for j = 1, . . . , λ(e). We have the following:
Theorem 2.4.6 Let A be an irreducible and unitary representation of G. Let α ∈
Γm,n and k = (λ, 1)Gα . If there exist j1, . . . , jk such that
σ∈Gα
A(σ)[j1, . . . , jk | j1, . . . , jk]
is a diagonal matrix, and djl
= 0 for all l = 1, . . . , k, then
E =
1
djl
Tijl
(e⊗
α ) | i = 1, . . . , λ(e), l = 1, . . . , k
is an orthonormal basis of the orbital subspace span{e∗
ασ | σ ∈ G}.
Proof: By the proof of theorem 2.4.5 above, we know that E contains the
right number of elements. So all we need to do is to show that elements in E are
orthonormal.
Using the identities (2.6) and (2.7), we obtain
1
djt
Tijt (e⊗
α ),
1
djs
Trjs (e⊗
α )
=
1
djt djs
(Tijt (e⊗
α ), Trjs (e⊗
α ))
=
1
djt djs
(TjsrTijt (e⊗
α ), e⊗
α )
=
1
djt djs
δir(Tjsjt (e⊗
α ), e⊗
α )
=
δir
djt djs
λ(e)
|G| σ∈G
ajsjt (σ)P(σ)(e⊗
α ), e⊗
α
=
δir
djt djs
λ(e)
|G| σ∈G
ajsjt (σ)(e⊗
ασ−1 ), e⊗
α
40. Chapter 3
Some Special Results
3.1 General Results on Cyclic Groups
In [13] R. Holmes and T. Tam discussed some results regarding cyclic groups. Their
results are presented here.
Let r ∈ Sm be an m-cycle (e.g., the cycle (1 2 . . . m)). Then the subgroup
of Sm generated by r is the cyclic group Cm. The elements of Cm are rk
, for
k = 0, . . . , m − 1.
For convenience in notation, in the following discussions we assume that r is
the m-cycle (1 2 . . . m). The result is the same for other m-cycles.
We the have the following result:
Proposition 3.1.1 Using the above notation, for any rk
∈ Cm, c(rk
) = gcd(m, k).
(where c(g) is the number of cycles in the disjoint cycle decomposition of g, as
defined in lemma 2.3.1)
33
41. CHAPTER 3. SOME SPECIAL RESULTS 34
Proof: Let R(N, D) denote the remainder of N when divided by D, where both
N and D are positive integers. Define M : {1, 2, . . .} → {1, . . . , m} by
M(i) = 1 + R(i − 1, m).
If k = 0, then rk
= r0
= e = (1)(2) . . . (m). So c(rk
) = m = gcd(m, k).
If gcd(m, k) = 1, then the cycle in rk
containing 1 is
(1 M(1 + k) M(1 + 2k) . . . M(1 + (m − 1)k)),
because if M(1 + tk) = 1 for some t ∈ {1, . . . , m − 1}, then R(tk, m) = 0, which
implies m divides tk. But gcd(m, k) = 1, and therefore m divides t, contradicting
t ∈ {1, . . . , m − 1}. Thus (1 M(1 + k) M(1 + 2k) . . . M(1 + (m − 1)k)) must be
the only cycle in rk
. This gives c(rk
) = 1 = gcd(m, k).
If k = 0 and gcd(m, k) = g = 1, write m = gm1 and k = gk1, where
gcd(m1, k1) = 1. Then the distinct cycle decomposition of rk
is
(1 M(1 + k) . . . M(1 + (
m
g
− 1)k))
(2 M(2 + k) . . . M(2 + (
m
g
− 1)k))
...
(g M(g + k) . . . M(g + (
m
g
− 1)k)).
The cycle containing 1 is as shown above because:
Clearly M(1+(
m
g
)k) = 1. Suppose M(1+tk) = 1 for some t ∈ {1, . . . ,
m
g
−1}.
Then R(tk, m) = 0, i.e., m divides tk, which implies gm1 divides tgk1, which implies
m1 divides tk1. But gcd(k1, m1) = 1, and so m1 must divide t. This contradicts
t ∈ {1, . . . ,
m
g
− 1} = {1, . . . , m1 − 1}. So the cycle containing 1 must be the one
shown above.
42. CHAPTER 3. SOME SPECIAL RESULTS 35
By the same reasoning, the cycles containing 2, 3, . . . , g are also as shown above.
Thus c(rk
) = g = gcd(m, k). 2
For π ∈ Sm, define bi(π) to be the number of cycles of length i in the disjoint
cycle decomposition of π. Clearly for any i > m, bi(π) = 0, and the function c(π)
in the proposition above is precisely
m
i=1
bi(π)
and is called the type of π.
Let r ∈ Sm be an m−cycle. Then the subgroup of Sm generated by r is the
cyclic group Cm. The elements of Cm are rk
, for k = 0, . . . , m − 1.
By proposition 3.1.1, c(rk
) = gcd(m, k) for k = 0, . . . , m − 1, so c(rk
) divides
m. On the other hand, for any k that divides m, k = gcd(m, k) = c(rk
). Thus the
types of elements of G constitute exactly the set of divisors of m, i.e.,
{c(σ) | σ ∈ G} = {d | d divides m}.
Since Cm is Abelian, from group representation theory we know that all of its
irreducible representations are of degree one. Thus for any irreducible character λ
of Cm, λ(Cm) is a group of complex roots of unity. Therefore the m irreducible
characters of Cm are:
λl(rk
) = e2πilk/m
, for l = 0, . . . , m − 1 (3.1)
(see [45], page 35). Since each λl is of degree one, by theorem 2.1.7, {e∗
α | α ∈ }
is a basis of Vλl
(Cm), so dimVλl
(Cm) = | |.
Another way to find the dimension of Vλl
(Cm) is to use the following formula
given by R. Holmes and T. Tam in [13]:
43. CHAPTER 3. SOME SPECIAL RESULTS 36
Theorem 3.1.2 dim Vλl
(Cm) =
1
m
m−1
k=1
e2πilk/m
ngcd(m,k)
, for l = 0, . . . , m − 1.
Proof: By theorem 2.3.2,
dim Vλl
(Cm) =
λl(e)
|Cm| σ∈Cm
λl(σ)nc(σ)
=
1
m
m−1
k=0
λl(rk
)nc(rk)
=
1
m
m−1
k=0
e2πilk/m
ngcd(m,k)
by (3.1) and proposition 3.1.1. 2
3.2 Simplified Formulas for Cyclic Groups
Although the results presented in the last chapter and the previous section are of
theoretical importance, they are difficult to use to actually calculate the dimen-
sion of some symmetry class. In [5], L. J. Cummings gave three formulas that
greatly simplify the necessary calculations in some special cases. Those results are
presented in this section.
Definition 3.2.1 For each positive integer p, let φ(p) denote the number of integers
in {1, 2, . . . , p} that are coprime with p. The function φ is called the Euler Totient
Function.
Theorem 3.2.2 Let G = Cm be a subgroup of Sm generated by an m−cycle, and
let λ be the character identically 1. Then
dimVλ(G) =
1
m d|m
φ(m/d)nd
,
44. CHAPTER 3. SOME SPECIAL RESULTS 37
where φ is the Euler totient function, and d|m means the positive integer d divides
m.
Proof: Let G = Cm be generated by the m−cycle r. By theorem 2.3.2,
dim Vλ(G) =
λ(e)
|G| σ∈G
λ(σ)nc(σ)
=
1
m σ∈G
nc(σ)
. (3.2)
By the discussion after proposition 3.1.1, the set of types of elements of G
constitute exactly the set of divisors of m. So the sum in (3.2) can be rewritten to
sum by the divisors of m:
dim Vλ(G) =
1
m d|m
#(d)nd
where #(d) stands for the number of elements in G having type d, that is, having
d disjoint cycles.
By proposition 3.1.1, for any rk
∈ G = Cm (where k ∈ {0, . . . , m − 1}), c(rk
) =
gcd(m, k). So the number of elements in G having d disjoint cycles is the number
of k’s in {1, . . . , m} satisfying gcd(k, m) = d, or equivalently, the number of k/d’s
satisfying gcd(k/d, m/d) = 1. This number is precisely φ(m/d). So #(d) = φ(m/d)
and the result follows. 2
Definition 3.2.3 Let G be a cyclic subgroup of Sm, and let λ be a one-dimensional
character of Sm. λ is said to be primitive if its value at a generator of G is a
primitive mth
root of unity.
Definition 3.2.4 The M¨obius function µ : {1, 2, . . .} → {−1, 0, 1} is defined to be:
µ(n) =
1 if n = 1,
(−1)t
if n is a product of t disctinct primes,
0 if p2
devides n for some prime p.
45. CHAPTER 3. SOME SPECIAL RESULTS 38
Theorem 3.2.5 Let G be a cyclic subgroup of Sm generated by an m−cycle and λ
be a primitive one dimensional character of Sm. Then
dim Vλ(G) =
1
m d|m
µ(m/d)nd
,
where µ is the M¨obius function.
Proof: We will use a result by K. Gauss: If r is a primitive mth
root of unity,
then
gcd(k,m)=1
rk
= µ(m). (3.3)
Let G be generated by the m−cycle π. By theorem 2.3.2 and proposition 3.1.1,
dim Vλ(G) =
λ(e)
|G| σ∈G
λ(σ)nc(σ)
=
1
m
m−1
k=0
λ(πk
)nc(πk)
=
1
m
m−1
k=0
λ(πk
)ngcd(k,m)
. (3.4)
We know from the discussion after proposition 3.1.1 that the divisors of m
constitute exactly the set {c(πk
) | k = 0, . . . , m − 1}. So (3.4) above can be re-
written to sum by the divisors of m, as follows:
dim Vλ(G) =
1
m d|m
(
c(πk)=d
λ(πk
))nd
=
1
m d|m
(
gcd(k,m)=d
λ(πk
))nd
. (3.5)
Writing l = k/d for the bracketed sum in (3.5), we get:
dim Vλ(G) =
1
m d|m
(
gcd(k/d,m/d)=1
λ(πld
))nd
=
1
m d|m
(
gcd(l,m/d)=1
λ(πd
)l
)nd
. (3.6)
46. CHAPTER 3. SOME SPECIAL RESULTS 39
Since λ(π) is a primitive mth
root of unity, λ(πd
) = λ(π)d
is a primitive (m/d)th
root of unity. By (3.3), the bracketed sum in (3.6) is µ(m/d). The desired result
follows immediately. 2
Theorem 3.2.6 Let G be a subgroup of Sm generated by disjoint cycles π1, . . . , πk
of orders m1, . . . , mk, respectively, and let λ be a character of G. If m =
k
i=1
mi,
then
dim Vλ(G) =
k
i=1
dim Vλ( πi ). (3.7)
Proof: Let
F = {f : {1, . . . , k} →
k
i=1
{1, . . . , mi − 1} | f(i) ∈ {1, . . . , mi − 1}, ∀i = 1, . . . , k},
and for each f ∈ F, define rf to be
rf =
k
i=1
c(π
f(i)
i ).
By theorem 2.3.2,
dim Vλ( πi ) =
1
mi
mi−1
l=0
λ(πl
i)nc(πl
i)
for i = 1, . . . , k. So the right side of (3.7) becomes:
k
i=1
dim Vλ( πi ) =
k
i=1
(
1
mi
mi−1
l=0
λ(πl
i)nc(πl
i)
)
= (
k
j=1
1
mj
)
k
i=1
(
mi−1
l=0
λ(πl
i)nc(πl
i)
)
= (
1
m
)
f∈F
(
k
i=1
λ(π
f(i)
i ))nrf
=
1
m f∈F
(nrf
λ(π
f(1)
1 . . . π
f(k)
k )). (3.8)
47. CHAPTER 3. SOME SPECIAL RESULTS 40
Since the πi’s are disjoint and G = π1, . . . , πk , we must have
{π
f(1)
1 . . . π
f(k)
k | f ∈ F} = G,
and
rf =
k
i=1
c(π
f(i)
i ) = c(π
f(1)
1 . . . π
f(k)
k ).
So equation (3.8) above becomes
k
i=1
dim Vλ( πi ) =
1
m g∈G
nc(g)
λ(g)
= dim Vλ(G). 2
3.3 Dihedral Groups
In [13] R. Holmes and T. Tam analyzed the dihedral subgroup Dm of Sm (m ≥ 3).
Consider the elements r, s ∈ Sm with
r = (1 2 . . . m)
and
s =
(2 m)(3 m − 1) . . . (
m
2
m
2
+ 2) if m is even
(2 m)(3 m − 1) . . . (
m + 1
2
m + 3
2
) if m is odd.
The subgroup of Sm generated by r and s is Dm, the dihedral group of degree m.
Clearly, it contains the cyclic subgroup Cm (since Cm = r ). By straight-forward
calculations we see that r, s satisfy
rm
= s2
= e and srs = r−1
. (3.9)
Further,
Dm = {rk
, srk
| k = 0, . . . , m − 1} (3.10)
48. CHAPTER 3. SOME SPECIAL RESULTS 41
(see [14], pages 50-51).
For each integer l with 0 < l < m/2, Dm has an irreducible character χl of
degree 2 which is induced from the irreducible character λl of Cm in (3.1):
χl(rk
) = 2 cos(
2πlk
m
)
χl(srk
) = 0 (3.11)
for k = 0, . . . , m − 1. All other irreducible characters of Dm are of degree 1 (see
[45], page 37).
Also note that for any integer k, by (3.9), we have:
srk
= sr(ss)rk−1
= (srs)srk−1
= r−1
srk−1
...
= r−k
s. (3.12)
Theorem 3.3.1 Let {e1, . . . , en} be an orthonormal basis of V with n ≥ 2, G =
r, s = Dm with m ≥ 3, and χ = χl for some 0 < l < m/2. There is a subset S of
Γm,n for which
{e∗
γ | γ ∈ S}
is an orthogonal basis of Vχ(G), if and only if
m ≡ 0 (mod 4l2)
where l = l2l2 with l2 being the largest power of 2 dividing l, and l2 being odd.
Proof: First suppose Vχ(G) has an orthogonal basis {e∗
γ | γ ∈ S} for some subset
S of Γm,n.
49. CHAPTER 3. SOME SPECIAL RESULTS 42
Let γ = (1, 2, . . . , 2) ∈ Γm,n. Clearly, γs = γ, and so s ∈ Gγ.
For any g ∈ G, by (3.10) either g = rk
or g = srk
for some k ∈ {1, . . . , m − 1, }.
If e = g = rk
∈ Cm, we have
γrk
(1) = 2 = 1 = γ(1),
so g = rk
∈ Gγ, and hence Gγ ∩ Cm = {e}.
If g = srk
∈ Gγ then
rk
= erk
= (ss)rk
= s(srk
) ∈ Gγ,
since both s and srk
are in Gγ. But rk
∈ Cm and we already know that Gγ ∩ Cm =
{e}. So we must have rk
= e, and so srk
= s. This gives Gγ = {e, s}.
For all τ, µ ∈ G, we can use straight-forward calculations with the identities
(3.9) and (3.12) to get:
τ−1
Gγµ =
{srj+k
, rk−j
}, if τ = srj
, µ = rk
{rk−j
, srj+k
}, if τ = rj
, µ = rk
{srj+k
, rk−j
}, if τ = rj
, µ = srk
{rk−j
, srj+k
}, if τ = srj
, µ = srk
.
(3.13)
This clearly shows that for all τ, µ ∈ G, |τ−1
Gγµ ∩ Cm| = |{rk−j
}| = 1.
From lemma 2.1.3, we have:
(e∗
γτ , e∗
γµ) =
χ(e)
|G| σ∈Gγµ
χ((µ−1
τ)−1
σ)
=
χ(e)
|G| σ∈Gγµ
χ(τ−1
µσ). (3.14)
For σ ∈ Gγµ, we have:
σ ∈ Gγµ ⇐⇒ γµσ = γµ
50. CHAPTER 3. SOME SPECIAL RESULTS 43
⇐⇒ γµσµ−1
= γ
⇐⇒ µσµ−1
∈ Gγ
⇐⇒ σ ∈ µ−1
Gγµ.
So (3.14) above becomes
(e∗
γτ , e∗
γµ) =
χ(e)
|G| σ∈µ−1Gγµ
χ(τ−1
µσ)
=
χ(e)
|G| σ∈τ−1Gγµ
χ(σ)
=
χ(e)
|G|
χ(rj
) (3.15)
for some j ∈ {0, . . . , m − 1} by (3.13) and the fact that χ(srk
) = 0 for all k (by
(3.9)).
Since Vχ(G) is a direct sum of the subspaces span{e∗
ασ | σ ∈ G}, for α ∈
(see theorem 2.1.6), and Vχ(G) has an orthogonal basis consisting of decomposable
symmetrized tensors, each subspace span{e∗
ασ | σ ∈ G}, where α ∈ must also
have an orthogonal basis consisting of decomposable symmetrized tensors.
Clearly γ ∈ . So by Freese’s theorem,
dim span{e∗
γσ | σ ∈ G} =
χ(e)
|Gγ| σ∈Gγ
χ(σ)
=
2
2 σ∈{e,s}
χ(σ)
= χ(e) + χ(s)
= 2 + 0
= 2. (3.16)
Thus span{e∗
γσ | σ ∈ G} must have an orthogonal basis consisting of exactly
two decomposable symmetrized tensors, say e∗
γτ and e∗
γµ, for some τ = µ in G.
51. CHAPTER 3. SOME SPECIAL RESULTS 44
If χ = 0 on Cm, then by (3.15) (e∗
γτ , e∗
γµ) = 0 and so e∗
γτ and e∗
γµ can not be
orthogonal to each other. This contradiction shows that χ must vanish for some
element in Cm.
Consequently, we must have
χ(rk
) = 2 cos(
2πlk
m
) = 0
for some k. That is,
2πlk
m
=
(2h + 1)π
2
for some integer h. This means 4lk = (2h + 1)m, or 4l2l2k = (2h + 1)m, where
l2l2 = l and l2 is the largest power of 2 dividing l. Since (2h + 1) is odd, 4l2 must
divide m. Hence m ≡ 0 (mod 4l2).
Conversely, suppose m ≡ 0 (mod 4l2). Let γ ∈ , and let H = Gγ ∩ Cm. We
can easily check that H is a subgroup of Cm.
Recall that χ = χl is the induced character λG
where λ = λl (see (3.11)).
Denote the restrictions to H of χ and λ by χH and λH respectively. Using Mackey’s
subgroup theorem (see [45], page 58) we get
χH = (λG
)H = λH + (λH)s
where (λH)s
is the character of H defined by:
(λH)s
(rk
) = λH(s−1
rk
s)
= λH(s−1
sr−k
) by (3.12)
= λH(r−k
)
= λH(rk
)−1
52. CHAPTER 3. SOME SPECIAL RESULTS 45
for any rk
∈ H. This implies λH is identically 1 if and only if (λH)s
is identically
1. Taking the Hermitian inner product on characters, we get:
(λH + (λH)s
, 1)H = (χH, 1)H
=
1
|H| σ∈H
χ(σ)
=
1
|H| σ∈Gγ∩Cm
χ(σ)
=
1
|H| σ∈Gγ
χ(σ) since χ(x) = 0 ∀x ∈ GCm
= 0
since γ ∈ . It follows that λH = 1.
Let rk
∈ H for some k. Then
λ(rk
) = λH(rk
) = 1.
But
λ(rk
) = λl(rk
) = e(2πilk/m)
,
so 2πlk/m = 2πh for some integer h, that is, lk = hm. Setting
l =
l
gcd(m, l)
and m =
m
gcd(m, l)
,
we easily get l k = hm .
If d is a positive integer that divides both m and l , then m = dm and l = dl
for some integers m and l , which means
m
gcd(m, l)
= dm and
l
gcd(m, l)
= dl .
This gives
m = (d gcd(m, l))m and l = (d gcd(m, l))l ,
53. CHAPTER 3. SOME SPECIAL RESULTS 46
which implies (d gcd(m, l)) is a common factor of m and l. Thus d must be 1, and
so m and l are coprime.
Since l k = hm , we have k = (hm )/l = m (h/l ). If (h/l ) is not an integer,
then there is a prime factor p of l which divides m but does not divide h. This
cannot happen because m and l are coprime. So (h/l ) must be an integer.
Since k = m (h/l ), and (h/l ) is an integer, we have rk
∈ rm
. But rk
is
arbitrary in H, so we conclude that H is a subgroup of rm
.
Suppose Gγ = H. Then Gγ contains some t ∈ GγCm ⊆ GCm, which means
t must be in the form srk
for some integer k. By (3.12) we have t2
= e, and so
T = {t, e} is a subgroup of Gγ.
For any g ∈ Gγ, if g = rj
for some j, then clearly gH = Hg. If g = srj
for some
j, then for any h = rk
∈ H, by (3.9) and (3.12) we have:
gh = (ghg−1
)g = (srj
rk
(srj
)−1
)g = (srj+k
r−j
s−1
)g = (srk
s−1
)g = r−k
g = h−1
g,
so again we have gH = Hg. This indicates that H is a normal subgroup of Gγ,
and hence HT is a subgroup of Gγ. Since T = {t, e}, we have |HT| = 2|H|.
For any g ∈ G, either g = rk
or g = srk
for some integer k. If g = rk
then
g = rk
e ∈ CmGγ. If g = srk
, let t = srj
∈ Gγ for some integer j (t = rj
for any j
because t ∈ Cm). By (3.9) and (3.12) we have:
g = srk
= srk−j
rj
= srk−j
(ss)rj
= (srk−j
s)(srj
) = (rj−k
ss)t = rj−k
t ∈ CmGγ.
So G ⊆ CmGγ, which gives G = CmGγ.
By the Second Group Isomorphism Theorem (see [14], page 44),
Gγ/H = Gγ/(Cm ∩ Gγ) ∼= CmGγ/Cm = G/Cm.
54. CHAPTER 3. SOME SPECIAL RESULTS 47
Since |G/Cm| = 2, |Gγ/H| = 2, hence |Gγ| = 2|H|. But we already showed that
HT is a subgroup of Gγ with |HT| = 2|H|, so we must have Gγ = HT.
Thus we have established that either Gγ = H, or Gγ = HT where T = t with
t2
= e and t ∈ Cm.
By theorem 2.1.6, Vχ(G) is the direct sum of the subspaces span{e∗
γσ | σ ∈ G}
for γ ∈ . Thus to show the existence of a set S ⊆ Γm,n with {e∗
α | α ∈ S} being an
orthogonal basis of Vχ(G), it suffices to show that for each γ ∈ , there is a subset
Kγ ⊆ G such that {e∗
γσ | σ ∈ Kγ} is an orthogonal basis of span{e∗
γσ | σ ∈ G}, and
set S =
γ∈
{γσ | σ ∈ Kγ}.
Let γ ∈ . Recall that on H, χH = λH + (λH)s
= 1 + 1 = 2. So if Gγ = H,
then by Freese’s theorem
dim span{e∗
γσ | σ ∈ G} =
χ(e)
|Gγ| σ∈Gγ
χ(σ)
=
2
|H| σ∈H
χH(σ)
=
2
|H|
· 2|H|
= 4.
On the other hand, if Gγ = HT, then we easily see that Gγ = H Ht, and
H Ht = {}. So again by Freese’s theorem
dim span{e∗
γσ | σ ∈ G} =
χ(e)
|Gγ| σ∈Gγ
χ(σ)
=
2
2|H|
(
σ∈H
χ(σ) +
σ∈Ht
χ(σ))
=
1
|H|
(
σ∈H
χH(σ))
=
1
|H|
· 2|H|
55. CHAPTER 3. SOME SPECIAL RESULTS 48
= 2.
Let
Kγ =
{e, rm /4
, s, srm /4
} if Gγ = H,
{e, rm /4
} if Gγ = HT.
First we have to make sure the above definition is valid by showing m /4 is an
integer. Since m ≡ 0 (mod 4l2), there is a positive integer p with m = 4pl2. We
have
m =
m
gcd(m, l)
=
4pl2
gcd(4pl2, l2l2)
=
4pl2
l2 gcd(4p, l2)
=
4p
gcd(4p, l2)
.
Since l2 is odd, gcd(4p, l2) must also be odd. Therefore 4 is a factor of m , and m /4
is an integer.
To show that {e∗
γσ | σ ∈ Kγ} is an orthogonal basis of span{e∗
γσ | σ ∈ G}, it
suffices to show that e∗
γτ and e∗
γµ are orthogonal if τ, µ ∈ Kγ and τ = µ.
By (3.15), to show that e∗
γτ and e∗
γµ are orthogonal, it suffices to show that
χ(τ−1
Gγµ) = {0}. We will do this for all combinations of τ and µ in Kγ.
Consider the case Gγ = H first. If τ = rk
∈ Cm and µ = srj
∈ Cm, then
τ−1
Gγµ Cm = {}, which means τ−1
Gγµ ⊆ GCm, and so χ(τ−1
Gγµ) = {0}.
If τ = e and µ = rm /4
, then
τ−1
Gγµ = eHrm /4
= Hrm /4
⊆ rm
= {rm (1+4k)/4
| k = . . . , −1, 0, 1, . . .}.
Thus
χ(rm (1+4k)/4
) = 2 cos(
2πlm (1 + 4k)/4
m
)
= 2 cos(
lm (1 + 4k)π
2m
)
= 2 cos(
l gcd(m, l)m (1 + 4k)π
2m gcd(m, l)
)
= 2 cos(
l (1 + 4k)π
2
). (3.17)
56. CHAPTER 3. SOME SPECIAL RESULTS 49
But l is odd since
l =
l
gcd(m, l)
=
l
gcd(4l2k, l2l2)
for some integer k
=
1
l2 gcd(4k, l2)
and l2 is the largest power of 2 that divides l. Hence (3.17) above evaluates to 0.
This again gives χ(τ−1
Gγµ) = {0}.
If τ = s and µ = srm /4
, then
τ−1
Gγµ ⊆ s−1
rm
srm /4
= {s−1
rkm
srm /4
| k = . . . , −1, 0, 1, . . .}
= {r−km +m /4
| k = . . . , −1, 0, 1, . . .}
= {rm (1−4k)/4
| k = . . . , −1, 0, 1, . . .}
and
χ(rm (1−4k)/4
) = 2 cos(
2πlm (1 − 4k)/4
m
)
= 2 cos(
πl gcd(m, l)m (1 − 4k)
2m gcd(m, l)
)
= 2 cos(
l (1 − 4k)π
2
)
= 0,
since we have determined that l is odd. So again χ(τ−1
Gγµ) = {0}.
Finally consider the second case, Gγ = HT = H Ht. We only need to consider
τ = e and µ = rm /4
:
τ−1
Gγµ = τ−1
Hµ τ−1
Htµ = Hrm /4
Htrm /4
.
57. CHAPTER 3. SOME SPECIAL RESULTS 50
By (3.17), χ(Hrm /4
) = {0}. Since Htrm /4
Cm = {}, χ(Htrm /4
) = {0}. This
gives χ(τ−1
Gγµ) = {0} and completes the proof. 2
The tensor space ⊗m
V is said to have an orthogonal basis of decomposable sym-
metrized tensors if for each irreducible character λ of G, there is a subset Sχ of Γm,n
for which {e∗
γ | γ ∈ Sχ} is an orthogonal basis of Vλ(G). We have the following
result from theorem 3.3.1:
Corollary 3.3.2 Let G = Dm with m ≥ 3, and suppose dimV = n ≥ 2. Then
⊗m
V has an orthogonal basis of decomposable symmetrized tensors if and only if
m is a power of 2.
Proof: Let m2 denote the largest power of 2 dividing m, and let χ = χm2 . If m
is not a power of 2, then 0 < m2 < m/2, and m ≡ 0 (mod 4m2). Thus by (3.11)
χ is an irreducible character of G. Since m ≡ 0 (mod 4m2), theorem 3.3.1 implies
that there is no subset Sχ of Γm,n for which {e∗
γ | γ ∈ Sχ} is an orthogonal basis of
Vχ(G). This is a contradiction and so m must be a power of 2.
Conversely, suppose m is a power of 2. Then (3.11) indicates that for all l with
0 < l < m/2, χl is an irreducible character of G of degree 2, and all other irreducible
characters of G = Dm are of degree 1.
If λ is a one dimensional character of G, by theorem 2.1.7, {e∗
α | α ∈ } is an
orthogonal basis of Vλ(G). If λ is two dimensional, then λ = χl for some l with
0 < l < m/2. Setting l2 to be the largest power of 2 dividing l, we get l2 ≤ m/4,
and hence m ≡ 0 (mod 4l2). By theorem 3.3.1 there is a subset Sλ of Γm,n for
which {e∗
γ | γ ∈ Sλ} is an orthogonal basis of Vλ(G) = Vχl
(G). 2
58. CHAPTER 3. SOME SPECIAL RESULTS 51
3.4 Linear Symmetry Classes
In [7] L. J. Cummings and R. W. Robinson used P´olya’s counting theorem to
calculate the dimension of V1(G), the linear symmetry class associated with the
principal character (i.e., identically 1). The results from their work is summarized
in this section.
We first introduce some notations and P´olya’s counting theorem.
Let G be a subgroup of Sm acting on Γm,n by
g · α = α ◦ g−1
for all g ∈ G and α ∈ Γm,n.
For each g ∈ G, denote the number of cycles of length i in the cyclic decompo-
sition of g by bi(g) (as defined after proposition 3.1.1).
The cycle type of g ∈ G is the monomial
Z(g) = Z(g; a1, . . . , am) =
m
k=1
a
bk(g)
k
in the variables a1, . . . , am. The cycle index of G is the polynomial
Z(G) = Z(G; a1, . . . , am) =
1
|G| g∈G
Z(g).
If A is a commutative ring and w : {1, . . . , n} → A, then the elements w(i) are
called weights. The weight of a function α ∈ Γm,n is
W(α) =
m
i=1
w(α(i)).
It is easy to show that if two functions α, β ∈ Γm,n are in the same orbit of Γm,n
under the action of G then W(α) = W(β).
59. CHAPTER 3. SOME SPECIAL RESULTS 52
Note that the orbits of Γm,n under the action of G are precisely the equivalence
classes of Γm,n defined in definition 2.1.1, and is a traversal of these equivalence
classes or orbits.
If W(α) is the weight of a representative of an orbit, then the result of summing
the weights of representatives over all orbits (e.g., summing over ,
α∈
W(α)) is
called the pattern inventory of the equivalence classes. This pattern inventory can
be calculated using P´olya’s Counting Theorem:
Theorem 3.4.1 (P´olya’s Counting Theorem)
α∈
W(α) = Z(G;
n
i=1
w(i), . . . ,
n
i=1
(w(i))m
)
=
1
|G| g∈G
(
m
k=1
(
n
i=1
(w(i))k
)bk(g)
). 2
(For a complete proof of P´olya’s Counting Theorem, see [23], chapter 5.)
Corollary 3.4.2 If G is a subgroup of Sm then the number of G-orbits in Γm,n
(i.e., | |) is Z(G; n, . . . , n).
Proof: Let w : {1, . . . , n} → {1}. Then for all α ∈ Γm,n,
W(α) =
m
i=1
w(α(i)) = 1.
By P´olya’s counting theorem we have
| | =
α∈
1
=
α∈
W(α)
= Z(G;
n
i=1
w(i), . . . ,
n
i=1
(w(i))m
)
= Z(G; n, . . . , n). 2
60. CHAPTER 3. SOME SPECIAL RESULTS 53
Theorem 3.4.3 If λ is the principal character of G (i.e., identically 1), then
dim Vλ(G) =
1
|G| g∈G
Z(g; n, . . . , n).
Proof: Since λ is identically 1, by the remark after definition 2.1.4, we have
= . By theorem 2.1.7 and corollary 3.4.2,
dim Vλ(G) = | | = | | = Z(G; n, . . . , n),
which by definition is
1
|G| g∈G
Z(g) =
1
|G| g∈G
Z(g; n, . . . , n). 2
3.5 Counting Restricted Orbits
In [7] section 3 L. J. Cummings and R. W. Robinson studied the action of a subgroup
of Sm acting on a finite set X equipped with a linear (one-dimensional) character,
λ. They developed a formula for counting special orbits of X under the action of
G, and applied that formula to calculate dim Vλ(G) when X = Γm,n. Their results
are presented in this section.
Let G be a subgroup of Sm acting on a finite set X (later the results obtained
here will be applied to X = Γm,n, with g(α) = g · α = α ◦ g−1
for all g ∈ G and
α ∈ Γm,n). For α ∈ X, let Gα = {g ∈ G | g · α = α} be the stabilizer subgroup
of α. For any g ∈ G, let F(g) denote the number of elements in X left fixed by g.
Burnside’s lemma (see [10], pages 134-135) states that the number of orbits of X
under the action of G is
1
|G| g∈G
F(g).
61. CHAPTER 3. SOME SPECIAL RESULTS 54
For any positive integer k, denote the group of complex kth
roots of unity by
Ck. If λ is a linear character of G, then λ(G) = Cc for some c. For each α ∈ X,
let σ(α) be determined by λ(Gα) = Cσ(α). If α and β are in the same G-orbit (i.e.,
α = g · β for some g ∈ G) then we can easily show that Gα and Gβ are conjugate
(i.e., Gα = gGβg−1
for some g ∈ G), so λ(Gα) = λ(Gβ), and hence σ(α) = σ(β).
For any g ∈ G, denote the order of λ(g) by λ(g)◦
. A new character λk of G can be
obtained by defining
λk(g) = (λ(g))k
for each g ∈ G. We have the following theorem:
Theorem 3.5.1 With the above set-up, suppose λ(G) = Cc with k|c. Then the
number of G-orbits consisting of α ∈ X with σ(α) = k is
ak =
1
|G| g∈G
F(g)(
v|k
µ(λv(g)◦
)
φ(λv(g)◦
)
)µ(k/v),
where µ is the M¨obius function, and φ is the Euler totient function. 2
(See [7], pages 1315-1317 for a complete proof.)
Now let G act on Γm,n by g(α) = g · α = α ◦ g−1
for all g ∈ G and α ∈ Γm,n.
The basic result of L. J. Cummings and R. W. Robinson (see [7]) implies
dim Vλ(G) = a1. (3.18)
By the second half of the proof of lemma 2.3.1, F(g) = nc(g)
, where c(g) is the
number of cycles in the disjoint cycle decomposition of g. Thus using (3.18) and
straight-forward calculations we obtain the following result:
Theorem 3.5.2 If G is a subgroup of Sm and λ is a one-dimensional character of
G, then
dim Vλ(G) =
1
|G| g∈G
µ(λ(g)◦
)
φ(λ(g)◦
)
nc(g)
. 2
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