Or: Cell theory for algebras
In this series of talks we explain how representation theory of monoids, algebras and monoidal categories can all be based on the theory of cells.
This is part 2 of 3.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Slides (if I need to update the slides): tubbenhauer.com/talks.html
UGC NET Paper 1 Mathematical Reasoning & Aptitude.pdf
Representation theory of algebras
1. Representation theory of algebras
Or: Cell theory for algebras
Daniel Tubbenhauer
Part 1: Reps of monoids; Part 3: Reps of monoidal cats
August 2022
Cell theory for algebras Representation theory of algebras August 2022 1 / 7
2. Where do we want to go?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
Group reps Monoid reps
Fusion reps Fiat reps
generalize
“categorify”
rep theory
I Green, Clifford, Munn, Ponizovskiı̆ ∼1940+
+ + many others
Representation theory of (finite) monoids
I Goal Find some categorical analog
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
3. Where do we want to go?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
Group reps Monoid reps
Fusion reps Fiat reps
generalize
“categorify”
rep theory
I Green, Clifford, Munn, Ponizovskiı̆ ∼1940+
+ + many others
Representation theory of (finite) monoids
I Goal Find some categorical analog
“Categorify”
!
is motivated by
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
4. Where do we want to go?
I Today: off track Interlude on H-reduction for algebras
I We will discover the H-reduction for algebras in real time!
I Examples we discuss The good, the ugly and the bad
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
5. The good
Connect 4 points at the bottom with 4 points at the top, potentially turning back:
{1, −4}, {2, 4}, {3, −2}, {−1, −3}
!
or
{1, 2}, {3, 4}, {−1, −3}, {−2, −4}
!
We just invented the Brauer monoid Br4 on {1, ..., 4} ∪ {−1, ..., −4}
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
6. The good
× =
= δ ·
Fix some field K and δ ∈ K, evaluate circles to δ ⇒ Brauer algebra Br4(δ)
The Brauer monoid is the non-linear version of Br4(1)
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
7. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
8. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Crucial notion in the linear world:
Pseudo idempotent ee = δe for δ 6= 0
◦ = δ·
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
9. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Crucial notion in the linear world:
Pseudo idempotent ee = δe for δ 6= 0
◦ = δ·
Pseudo idempotent can be normalized to idempotents
ee = δe ⇒ e
δ
e
δ
= e
δ
Green, Clifford, Munn, Ponizovskiı̆ ⇒ pseudo idempotents are good!
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
10. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Example (cells of Br3(δ) for δ 6= 0, pseudo idempotent cells colored)
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
11. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Example (cells of Br3(δ) for δ 6= 0, pseudo idempotent cells colored)
Example (cells of Br3(δ) for δ = 0, pseudo idempotent cells colored)
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
12. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Theorem (Brown ∼1953, Fishel–Grojnowski ∼1995)
H-reduction works for the Brauer algebra
(a) J-cells ! through strands, all J-cells are pseudo idempotent for δ 6= 0
and H(e) are symmetric groups in the through strands
(b) For δ = 0 similar but with J0 having no pseudo idempotent
The top cell:
etc. are nilpotent for δ = 0
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
13. The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Theorem (Brown ∼1953, Fishel–Grojnowski ∼1995 and folklore)
H-reduction works for other diagram algebra and quantum versions
More details on the exercise sheets
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
14. The ugly
S4 :
I Symmetric groups Sn of order n! are the symmetry groups of (n−1) simplices
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
15. The ugly
S4 : , S5 :
I Kazhdan–Lusztig (KL) ∼1979, Lusztig ∼1984, folklore Use a J-reduction approach
I Lusztig’s approach gives an H-reduction way of classifying simple Sn reps
I Aside The character table of Sn is integral ⇒ categorification!
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
16. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
17. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Green cells in linear
B = {x, y, z, ...} fix basis of an algebra A
⊂
+ = has a nonzero coefficient when expressed via B
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
18. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Example (groups and monoids)
For B=monoid basis: linear cells = Green cells
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
19. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Example (groups and monoids)
For B=monoid basis: linear cells = Green cells
Example (group-like)
All invertible basis elements form the smallest cell (well, kind of...)
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
20. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Example (groups and monoids)
For B=monoid basis: linear cells = Green cells
Example (group-like)
All invertible basis elements form the smallest cell (well, kind of...)
Example (diagram algebras)
For B=diagram basis: linear cells = Green cells
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
21. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
22. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (no specific algebra)
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
23. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (cells for S2 = h1i and S3 = h1, 2i over C, pseudo idempotent cells colored)
For B=KL basis bw , ∼
=s = up to scaling (more on the exercise sheets)
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
24. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (cells for S2 = h1i and S3 = h1, 2i over Z/2Z, pseudo idempotent cells colored)
For B=KL basis bw , ∼
=s = up to scaling (more on the exercise sheets)
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
25. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
The notion of an apex makes sense for the pair (Sn, BKL) J-reduction
“Apex = fish” means that the KL basis elements in the red bubble do not annihilate your rep and the others do
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
26. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
27. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
For the record
(b) works over any field but with p-restricted partitions
There is also a quantum version
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
28. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
Example (symmetric group S3)
The simple reps over C corresponding to the three J (e) cells via
χ2 ! J∅, χ3 ! Jm, χ1 ! Jw0 ,
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
29. The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
Example (symmetric group S3)
The simple reps over K with char(K) = 2 corresponding to the two J (e) cells via
χ1 = χ2 ! J∅, χ3 ! Jm,
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
30. The bad
D4
,
D4
I Dihedral groups Dn of order 2n are the symmetry groups of n gons
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
31. The bad
D4 : , D5 :
I Kazhdan–Lusztig (KL) ∼1979, Lusztig ∼1984, folklore Use a J-reduction approach
I Lusztig’s approach almost gives an H-reduction way of classifying simple Dn reps
I Aside The character table of Dn is not integral ⇒ looks bad for categorification
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
32. The bad
The (linear) cell orders and equivalences for a fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I H-cells = intersections of left and right cells
I Slogan Cells measure information loss
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
33. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
34. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (cells for D4, D5 = h1, 2i over C, pseudo idempotent cells colored)
D4:
D5:
For B=KL basis bw , ∼
=s = up to scaling (more on the exercise sheets)
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
35. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (Lusztig ∼80ish, folklore)
For Dn, n odd there exists a basis B such that:
(a) All simple Dn modules have a unique apex
(b) There are three J-cells, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to C or C[Z/(n−1
2
)Z]
There is also a slightly more involved version for general K
(c) H-reduction works for Dn
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
36. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (dihedral group D5)
The simple reps over C corresponding to the three J (e) cells via
χ2 ! J∅, χ3, χ4 ! Jm, χ1 ! Jw0 ,
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
37. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Example (dihedral group D5)
The simple reps over C corresponding to the three J (e) cells via
χ2 ! J∅, χ3, χ4 ! Jm, χ1 ! Jw0 ,
Example (dihedral group D5)
The simple reps over K with char(K) = 2 corresponding to the two J (e) cells via
χ2 ! J∅, χ3, χ4 ! Jm, χ1 ! Jw0 ,
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
38. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (Lusztig ∼80ish, folklore)
The J-reduction (notion of an apex) makes sense for all Coxeter groups
But H-reduction only works in type A and odd dihedral type
H-reduction Powerful, but doesn’t work in general
J-reduction General, but not so powerful
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
39. The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
J-reduction often doesn’t get you far
A middle cell in type E8 where dim SJ = 202671840 and zillions of associated simples:
Ak,l = H-cells of size A arranged in a (k × l)-matrix
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
40. The good, the ugly and the bad – comparison
I For monoids all H-cells within one J-cell are of the same size
I For the good, the ugly and the odd bad the same is true
I For the even bad this is false
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
41. The good, the ugly and the bad – comparison
I For monoids all H-cells within one J-cell are of the same size
I For the good, the ugly and the odd bad the same is true
I For the even bad this is false
Strategy (Brown ∼1953, König–Xi ∼1999, folklore)
Find axioms ensuring that
H-cells within one J-cell are of the same size
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
42. The good, the ugly and the bad – comparison
Sandwich datum:
Sλ = sandwiched algebras
A sandwich cellular algebra A: sandwich datum + some axioms such as
xcλ
D,b,U ≡
X
S∈Mλ,a∈B
r(S, D) · cλ
S,a,U + higher order friends
Cellular = all Sλ
∼
= K = all H-cells are of size one
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
43. The good, the ugly and the bad – comparison
Sandwich datum:
Sλ = sandwiched algebras
A sandwich cellular algebra A: sandwich datum + some axioms such as
xcλ
D,b,U ≡
X
S∈Mλ,a∈B
r(S, D) · cλ
S,a,U + higher order friends
Cellular = all Sλ
∼
= K = all H-cells are of size one
Approximate picture to keep in mind
D0
U0
b0
D
U
b
≡ r(U0
, D)·
D0
U
b0
b
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
44. The good, the ugly and the bad – comparison
Sandwich datum:
Sλ = sandwiched algebras
A sandwich cellular algebra A: sandwich datum + some axioms such as
xcλ
D,b,U ≡
X
S∈Mλ,a∈B
r(S, D) · cλ
S,a,U + higher order friends
Cellular = all Sλ
∼
= K = all H-cells are of size one
Approximate picture to keep in mind
D0
U0
b0
D
U
b
≡ r(U0
, D)·
D0
U
b0
b
H-cells are of equal size, by definition! As free K vector spaces:
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
45. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
46. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
For sandwich cellular algebras
The J-reduction (notion of an apex) makes sense
The H-reduction works
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
47. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example
All algebras are sandwich cellular with Λ = {•} and S• = A
We get the fantastic H-reduction tautology:
(
simples of
A
)
=
(
simples with
apex •
)
one-to-one
←
−
−
−
−
→
(
simples of
S•
)
=
(
simples of
A
)
The point is to find a good sandwich datum!
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
48. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example (cf. talk 1)
Many monoid algebras with the monoid basis
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
49. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example (cf. talk 1)
Many monoid algebras with the monoid basis
Example
Diagram algebras with the diagram basis
The good
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
50. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example (cf. talk 1)
Many monoid algebras with the monoid basis
Example
Diagram algebras with the diagram basis
The good
Example
Sn with the KL basis
The ugly
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
51. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example (cf. talk 1)
Many monoid algebras with the monoid basis
Example
Diagram algebras with the diagram basis
The good
Example
Sn with the KL basis
The ugly
Example
Dn, n odd with the KL basis
The odd bad
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
52. The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Example (cf. talk 1)
Many monoid algebras with the monoid basis
Example
Diagram algebras with the diagram basis
The good
Example
Sn with the KL basis
The ugly
Example
Dn, n odd with the KL basis
The odd bad
Nonexample
Dn, n even with the KL basis
The even bad
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
53. Where do we want to go?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
Group reps Monoid reps
Fusion reps Fiat reps
generalize
“categorify”
rep theory
I Green, Clifford, Munn, Ponizovskiı̆ ∼1940+
+ + many others
Representation theory of (finite) monoids
I Goal Find some categorical analog
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
Where do we want to go?
I Today: off track Interlude on H-reduction for algebras
I We will discover the H-reduction for algebras in real time!
I Examples we discuss The good, the ugly and the bad
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Example (cells of Br3(δ) for δ 6= 0, pseudo idempotent cells colored)
Example (cells of Br3(δ) for δ = 0, pseudo idempotent cells colored)
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
The ugly
S4 :
I Symmetric groups Sn of order n! are the symmetry groups of (n−1) simplices
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
Example (symmetric group S3)
The simple reps over C corresponding to the three J (e) cells via
χ2 ! J∅, χ3 ! Jm, χ1 ! Jw0 ,
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
The bad
D4
,
D4
I Dihedral groups Dn of order 2n are the symmetry groups of n gons
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (Lusztig ∼80ish, folklore)
The J-reduction (notion of an apex) makes sense for all Coxeter groups
But H-reduction only works in type A and odd dihedral type
H-reduction Powerful, but doesn’t work in general
J-reduction General, but not so powerful
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
There is still much to do...
Cell theory for algebras Representation theory of algebras August 2022 7 / 7
54. Where do we want to go?
Groups Monoids
Fusion mon-
oidal cats
Fiat mon-
oidal cats
Group reps Monoid reps
Fusion reps Fiat reps
generalize
“categorify”
rep theory
I Green, Clifford, Munn, Ponizovskiı̆ ∼1940+
+ + many others
Representation theory of (finite) monoids
I Goal Find some categorical analog
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
Where do we want to go?
I Today: off track Interlude on H-reduction for algebras
I We will discover the H-reduction for algebras in real time!
I Examples we discuss The good, the ugly and the bad
Cell theory for algebras Representation theory of algebras August 2022 2 / 7
The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
The good
Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence
simples with
apex J (e)
one-to-one
←
−
−
−
−
→
simples of (any)
H(e) ⊂ J (e)
Reps of monoids are controlled by H(e)-cells
I Brn(δ) is not a monoid ⇒ H-reduction does not apply
I My favorite strategy Ignore problems and just go for it
Example (cells of Br3(δ) for δ 6= 0, pseudo idempotent cells colored)
Example (cells of Br3(δ) for δ = 0, pseudo idempotent cells colored)
Cell theory for algebras Representation theory of algebras August 2022 3 / 7
The ugly
S4 :
I Symmetric groups Sn of order n! are the symmetry groups of (n−1) simplices
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
The ugly
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (KL ∼1979, Lusztig ∼1984, folklore)
For Sn there exists a basis B such that:
(a) All simple Sn modules have a unique apex
(b) For char(K) = 0, J-cells ! partitions of n, all J-cells are pseudo idempotent
and all corresponding SH are isomorphic to K
(c) H-reduction works for Sn
Example (symmetric group S3)
The simple reps over C corresponding to the three J (e) cells via
χ2 ! J∅, χ3 ! Jm, χ1 ! Jw0 ,
Cell theory for algebras Representation theory of algebras August 2022 4 / 7
The bad
D4
,
D4
I Dihedral groups Dn of order 2n are the symmetry groups of n gons
I Frobenius ∼1895+
+ Their (complex) rep theory is well understood
I They are groups ⇒ Green’s cell theory is boring
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
The bad
The (linear) cell orders and equivalences for fixed basis B:
Left, right and two-sided cells (a.k.a. L, R and J-cells): equivalence classes
I Get algebras SJ , SH by killing higher order terms
I Pseudo idempotents make SJ , SH unital
Theorem (Lusztig ∼80ish, folklore)
The J-reduction (notion of an apex) makes sense for all Coxeter groups
But H-reduction only works in type A and odd dihedral type
H-reduction Powerful, but doesn’t work in general
J-reduction General, but not so powerful
Cell theory for algebras Representation theory of algebras August 2022 5 / 7
The good, the ugly and the bad – comparison
In spirit of Clifford, Munn, Ponizovskiı̆ ∼1940+
+ H-reduction
There is a one-to-one correspondence (under some conditions on K and Sλ)
simples with
apex λ
one-to-one
←
−
−
−
−
→
simples of
Sλ
Reps are controlled by the sandwiched algebras
I Each simple has a unique maximal λ ∈ Λ where having a pseudo idempotent is
replaced by a paring condition Apex
I In other words (smod means the category of simples):
S-smodλ ' Sλ-smod
Cell theory for algebras Representation theory of algebras August 2022 6 / 7
Thanks for your attention!
Cell theory for algebras Representation theory of algebras August 2022 7 / 7