Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

1. David M. Bressoud
Macalester College, St. Paul, MN
Talk given at University of Florida
October 29, 2004

2. 1. The Vandermonde determinant
2. Weyl’s character formulae
3. Alternating sign matrices
4. The six-vertex model of statistical mechanics
5. Okada’s work connecting ASM’s and character
formulae

3. x1
n−1
x2
n−1
L xn
n−1
M M O M
x1 x2 L xn
1 1 L 1
= −1( )I σ( )
xi
n−σ i( )
i=1
n
∏σ ∈Sn
∑
Cauchy 1815
“Memoir on functions whose
values are equal but of opposite
sign when two of their variables
are interchanged”
(alternating functions)
Augustin-Louis Cauchy
(1789–1857)

4. Cauchy 1815
“Memoir on functions whose
values are equal but of opposite
sign when two of their variables
are interchanged”
(alternating functions)
This function is 0 when so it is divisible byxi = xj
xi − xj( )i< j
∏
x1
n−1
x2
n−1
L xn
n−1
M M O M
x1 x2 L xn
1 1 L 1
= −1( )I σ( )
xi
n−σ i( )
i=1
n
∏σ ∈Sn
∑

5. Cauchy 1815
“Memoir on functions whose
values are equal but of opposite
sign when two of their variables
are interchanged”
(alternating functions)
This function is 0 when so it is divisible byxi = xj
xi − xj( )i< j
∏
But both polynomials have same degree, so ratio is constant, = 1.
= xi − xj( )i< j
∏
x1
n−1
x2
n−1
L xn
n−1
M M O M
x1 x2 L xn
1 1 L 1
= −1( )I σ( )
xi
n−σ i( )
i=1
n
∏σ ∈Sn
∑

6. Cauchy 1815
Any alternating function in divided by the
Vandermonde determinant yields a symmetric function:
x1,x2,K ,xn
x1
λ1 +n−1
x2
λ1 +n−1
L xn
λ1 +n−1
M M O M
x1
λn−1 +1
x2
λn−1 +1
L xn
λn−1 +1
x1
λn
x2
λn
L xn
λn
x1
n−1
x2
n−1
L xn
n−1
M M O M
x1
1
x2
1
L xn
1
x1
0
x2
0
L xn
0
= sλ x1,x2,K ,xn( )

7. Cauchy 1815
Any alternating function in divided by the
Vandermonde determinant yields a symmetric function:
x1,x2,K ,xn
Called the Schur function.
I.J. Schur (1917) recognized
it as the character of the
irreducible representation of
GLn indexed by λ.
x1
λ1 +n−1
x2
λ1 +n−1
L xn
λ1 +n−1
M M O M
x1
λn−1 +1
x2
λn−1 +1
L xn
λn−1 +1
x1
λn
x2
λn
L xn
λn
x1
n−1
x2
n−1
L xn
n−1
M M O M
x1
1
x2
1
L xn
1
x1
0
x2
0
L xn
0
= sλ x1,x2,K ,xn( )
Issai
Schur
(1875–
1941)

8. sλ 1,1,K ,1( ) is the dimension of the representation
sλ 1,1,K ,1( )=
ρ + λ( )⋅r
ρ ⋅rr∈An−1
+
∏
where ρ =
n −1
2
,
n − 3
2
,K ,
1− n
2



 =
1
2
r
r∈An−1
+
∑ ,
λ = λ1,λ2,K ,λn( ),
An−1
+
= ei − ej 1 ≤ i < j ≤ n{ },
ei is the unit vector with 1 in the ith coordinate
Note that the symmetric group on n letters is the group of
transformations of An−1 = ei − ej 1 ≤ i ≠ j ≤ n{ }

9. Weyl 1939 The Classical Groups: their invariants and
representations
Sp2n λ;
r
x( )=
x1
λ1 +n
− x1
−λ1 −n
L xn
λ1 +n
− xn
−λ1 −n
M O M
x1
λn +1
− x1
−λn −1
L xn
λn +1
− xn
−λn −1
x1
n
− x1
−n
L xn
n
− xn
−n
M O M
x1
1
− x1
−1
L xn
1
− xn
−1
Sp2n λ;
r
x( ) is the character of the irreducible representation,
indexed by the partition λ, of the symplectic group
(the subgoup of GL2n of isometries).
Hermann Weyl (1885–1955)

10. Sp2n λ;1
r
( )=
ρ + λ( )⋅r
ρ ⋅rr∈Cn
+
∏
where ρ = n,n −1,K ,1( )=
1
2
r
r∈Cn
+
∑ ,
λ = λ1,λ2,K ,λn( ),
Cn
+
= ei ± ej 1 ≤ i < j ≤ n{ }U 2ei 1 ≤ i ≤ n{ },
ei is the unit vector with 1 in the ith coordinate
The dimension of the representation is

11. Weyl 1939 The Classical Groups: their invariants and
representations
x1
n
− x1
−n
L xn
n
− xn
−n
M O M
x1
1
− x1
−1
L xn
1
− xn
−1
x1x2 L xn( )n
xi − xj( )i< j
∏
is a symmetric polynomial. As a polynomial in x1 it
has degree n + 1 and roots at ±1, xj
−1
for 2 ≤ j ≤ n

12. Weyl 1939 The Classical Groups: their invariants and
representations
x1
n
− x1
−n
L xn
n
− xn
−n
M O M
x1
1
− x1
−1
L xn
1
− xn
−1
x1x2 L xn( )n
xi − xj( )i< j
∏
= xi
2
−1( )
i
∏ xi xj −1( )i< j
∏
is a symmetric polynomial. As a polynomial in x1 it
has degree n + 1 and roots at ±1, xj
−1
for 2 ≤ j ≤ n

13. Weyl 1939 The Classical Groups: their invariants and
representations: The Denominator Formulas
x1
n− 1
2
− x1
−n+ 1
2
L xn
n− 1
2
− xn
−n+ 1
2
M O M
x1
1
2
− x1
− 1
2
L xn
1
2
− xn
− 1
2
x1x2 L xn( )n− 1
2
xi − xj( )i< j
∏
= xi −1( )
i
∏ xi xj −1( )i< j
∏
x1
n−1
+ x1
−n+1
L xn
n−1
+ xn
−n+1
M O M
x1
0
+ x1
−0
L xn
0
+ xn
−0
x1x2 L xn( )n−1
xi − xj( )i< j
∏
= 2 xi xj −1( )i< j
∏

15. Desnanot-Jacobi adjoint matrix thereom (Desnanot for
n 6 in 1819, Jacobi for general case in 1833≤
M j
i
is matrix M with row i and column j removed.
det M =
det M1
1
⋅det Mn
n
− det Mn
1
⋅det M1
n
det M1,n
1,n
Given that the determinant of the
empty matrix is 1 and the
determinant of a 1×1 is the entry
in that matrix, this uniquely
defines the determinant for all
square matrices.
Carl Jacobi (1804–1851)

16. det M =
det M1
1
⋅det Mn
n
− det Mn
1
⋅det M1
n
det M1,n
1,n
detλ M =
det M1
1
⋅det Mn
n
+ λ det Mn
1
⋅det M1
n
det M1,n
1,n
det−1 M = det M( )
detλ aj
i−1
( )i, j=1
n
= ai + λ aj( )1≤i< j≤n
∏
David Robbins (1942–2003)

17. det M =
det M1
1
⋅det Mn
n
− det Mn
1
⋅det M1
n
det M1,n
1,n
detλ M =
det M1
1
⋅det Mn
n
+ λ det Mn
1
⋅det M1
n
det M1,n
1,n
detλ
a b
c d



 = ad + λbc
detλ
a b c
d e f
g h j








= aej + λ bdj + afh( )+ λ2
bfg + cdh( )+ λ3
ceg
+ λ 1+ λ( )bde−1
fh

18. detλ
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4












= a1,1a2,2a3,3a4,4
+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )
+L sums over other permutations × λinversion number
+ λ3
1+ λ−1
( )a1,2a2,1a2,2
−1
a2,3a3,4a4,2 +L
+ λ3
1+ λ−1
( )
2
a1,2a2,1a2,2
−1
a2,3a3,2a3,3
−1
a3,4a4,3 +L

19. detλ
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4












= a1,1a2,2a3,3a4,4
+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )
+L sums over other permutations × λinversion number
+ λ3
1+ λ−1
( )a1,2a2,1a2,2
−1
a2,3a3,4a4,2 +L
+ λ3
1+ λ−1
( )
2
a1,2a2,1a2,2
−1
a2,3a3,2a3,3
−1
a3,4a4,3 +L
0 1 0 0
1 −1 1 0
0 1 −1 1
0 0 1 0












0 1 0 0
1 −1 1 0
0 0 0 1
0 1 0 0













20. detλ
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4












= a1,1a2,2a3,3a4,4
+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )
+L sums over other permutations × λinversion number
+ λ3
1+ λ−1
( )a1,2a2,1a2,2
−1
a2,3a3,4a4,2 +L
+ λ3
1+ λ−1
( )
2
a1,2a2,1a2,2
−1
a2,3a3,2a3,3
−1
a3,4a4,3 +L
0 1 0 0
1 −1 1 0
0 1 −1 1
0 0 1 0












0 1 0 0
1 −1 1 0
0 0 0 1
0 1 0 0













21. detλ
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4












= a1,1a2,2a3,3a4,4
+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )
+L sums over other permutations × λinversion number
+ λ3
1+ λ−1
( )a1,2a2,1a2,2
−1
a2,3a3,4a4,2 +L
+ λ3
1+ λ−1
( )
2
a1,2a2,1a2,2
−1
a2,3a3,2a3,3
−1
a3,4a4,3 +L
detλ xi, j( )= λInv A( )
A= ai, j( )
∑ 1+ λ−1
( )
N A( )
xi, j
ai, j
i, j
∏
Sum is over all alternating sign matrices, N(A) = # of –1’s

22. detλ
a1,1 a1,2 a1,3 a1,4
a2,1 a2,2 a2,3 a2,4
a3,1 a3,2 a3,3 a3,4
a4,1 a4,2 a4,3 a4,4












= a1,1a2,2a3,3a4,4
+ λ a1,2a2,1a3,3a4,4 + a1,1a2,3a3,2a4,4 + a1,1a2,2a3,4a4,3( )
+L sums over other permutations × λinversion number
+ λ3
1+ λ−1
( )a1,2a2,1a2,2
−1
a2,3a3,4a4,2 +L
+ λ3
1+ λ−1
( )
2
a1,2a2,1a2,2
−1
a2,3a3,2a3,3
−1
a3,4a4,3 +L
detλ xi, j( )= λInv A( )
A= ai, j( )
∑ 1+ λ−1
( )
N A( )
xi, j
ai, j
i, j
∏
xi + λxj( )= λInv A( )
1+ λ−1
( )
N A( )
xj
n−i( )ai, j
i, j
∏
A∈An
∑
1≤i< j≤n
∏

23. n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
= 2 × 3 × 7
= 3 × 11 × 13
= 22
× 11 × 132
= 22
× 132
× 17 × 19
= 23
× 13 × 172
× 192
= 22
× 5 × 172
× 193
× 23
How many n × n alternating sign
matrices?

24. n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
= 2 × 3 × 7
= 3 × 11 × 13
= 22
× 11 × 132
= 22
× 132
× 17 × 19
= 23
× 13 × 172
× 192
= 22
× 5 × 172
× 193
× 23

25. n
1
2
3
4
5
6
7
8
9
An
1
2
7
42
429
7436
218348
10850216
911835460
There is exactly one 1
in the first row

26. n
1
2
3
4
5
6
7
8
9
An
1
1+1
2+3+2
7+14+14+7
42+105+…
There is exactly one 1
in the first row

27. 1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429

28. 1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
+ + +

29. 1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
+ + +
1 0 0 0 0
0 ? ? ? ?
0 ? ? ? ?
0 ? ? ? ?
0 ? ? ? ?

















30. 1
1 2/2 1
2 2/3 3 3/2 2
7 2/4 14 14 4/2 7
42 2/5 105 135 105 5/2 42
429 2/6 1287 2002 2002 1287 6/2 429

31. 1
1 2/2 1
2 2/3 3 3/2 2
7 2/4 14 5/5 14 4/2 7
42 2/5 105 7/9 135 9/7 105 5/2 42
429 2/6 1287 9/14 2002 16/16 2002 14/9 1287 6/2 429

32. 2/2
2/3 3/2
2/4 5/5 4/2
2/5 7/9 9/7 5/2
2/6 9/14 16/16 14/9 6/2

33. 2
2 3
2 5 4
2 7 9 5
2 9 16 14 6

34. 1+1
1+1 1+2
1+1 2+3 1+3
1+1 3+4 3+6 1+4
1+1 4+5 6+10 4+10 1+5
Numerators:

35. 1+1
1+1 1+2
1+1 2+3 1+3
1+1 3+4 3+6 1+4
1+1 4+5 6+10 4+10 1+5
Conjecture 1:
Numerators:
An,k
An,k+1
=
n − 2
k −1



 +
n −1
k −1




n − 2
n − k −1



 +
n −1
n − k −1





36. Conjecture 1:
Conjecture 2 (corollary of Conjecture 1):
An,k
An,k+1
=
n − 2
k −1



 +
n −1
k −1




n − 2
n − k −1



 +
n −1
n − k −1




An =
3j +1( )!
n + j( )!j=0
n−1
∏ =
1!⋅4!⋅7!L 3n − 2( )!
n!⋅ n +1( )!L 2n −1( )!

37. Conjecture 2 (corollary of Conjecture 1):
An =
3j +1( )!
n + j( )!j=0
n−1
∏ =
1!⋅4!⋅7!L 3n − 2( )!
n!⋅ n +1( )!L 2n −1( )!
Exactly the formula found
by George Andrews for
counting descending plane
partitions.
George Andrews
Penn State

38. Conjecture 2 (corollary of Conjecture 1):
An =
3j +1( )!
n + j( )!j=0
n−1
∏ =
1!⋅4!⋅7!L 3n − 2( )!
n!⋅ n +1( )!L 2n −1( )!
Exactly the formula found
by George Andrews for
counting descending plane
partitions. In succeeding
years, the connection would
lead to many important
results on plane partitions.
George Andrews
Penn State

39. A n;x( )= xN A( )
A∈An
∑
A 1;x( )= 1,
A 2;x( )= 2,
A 3;x( )= 6 + x,
A 4;x( )= 24 +16x + 2x2
,
A 5;x( )= 120 + 200x + 94x2
+14x3
+ x4
,
A 6;x( )= 720 + 2400x + 2684x2
+1284x3
+ 310x4
+ 36x5
+ 2x6
A 7;x( )= 5040 + 24900x + 63308x2
+ 66158x3
+ 38390x4
+13037x5
+ 2660x6
+ 328x7
+ 26x8
+ x9

40. A n;x( )= xN A( )
A∈An
∑
A 1;x( )= 1,
A 2;x( )= 2,
A 3;x( )= 6 + x,
A 4;x( )= 24 +16x + 2x2
,
A 5;x( )= 120 + 200x + 94x2
+14x3
+ x4
,
A 6;x( )= 720 + 2400x + 2684x2
+1284x3
+ 310x4
+ 36x5
+ 2x6
A 7;x( )= 5040 + 24900x + 63308x2
+ 66158x3
+ 38390x4
+13037x5
+ 2660x6
+ 328x7
+ 26x8
+ x9
xi + λxj( )= λInv A( )
1+ λ−1
( )
N A( )
xj
n−i( )ai, j
i, j
∏
A∈An
∑
1≤i< j≤n
∏
A n;0( )= n!
A n;1( )= An =
3i +1( )!
n + i( )!i=0
n−1
∏




A n;2( )= 2n(n−1)/2

41. A n;x( )= xN A( )
A∈An
∑
A 1;x( )= 1,
A 2;x( )= 2,
A 3;x( )= 6 + x,
A 4;x( )= 24 +16x + 2x2
,
A 5;x( )= 120 + 200x + 94x2
+14x3
+ x4
,
A 6;x( )= 720 + 2400x + 2684x2
+1284x3
+ 310x4
+ 36x5
+ 2x6
A 7;x( )= 5040 + 24900x + 63308x2
+ 66158x3
+ 38390x4
+13037x5
+ 2660x6
+ 328x7
+ 26x8
+ x9
A n;3( )=
3n n−1( )
2n n−1( )
3 j − i( )+1
3 j − i( )1≤i, j≤n
j−i odd
∏Conjecture:
(MRR, 1983)
A n;0( )= n!
A n;1( )= An =
3i +1( )!
n + i( )!i=0
n−1
∏




A n;2( )= 2n(n−1)/2

42. Mills & Robbins (suggested by Richard Stanley) (1991)
Symmetries of ASM’s
A n( )=
3j +1( )!
n + j( )!j=0
n−1
∏
AV 2n +1( )= −3( )n2 3 j − i( )+1
j − i + 2n +11≤i, j≤2n+1
2 j
∏










A n( )
AHT 2n( )= −3( )n n−1( )/2 3 j − i( )+ 2
j − i + ni, j
∏



 A n( )
AQT 4n( )= AHT 2n( )⋅ A n( )2
Vertically symmetric
ASM’s
Half-turn symmetric
ASM’s
Quarter-turn
symmetric ASM’s

43. December, 1992
Zeilberger announces a
proof that # of ASM’s
equals
3j +1( )!
n + j( )!j=0
n−1
∏
Doron Zeilberger
Rutgers University

44. December, 1992
Zeilberger announces a
proof that # of ASM’s
equals
3j +1( )!
n + j( )!j=0
n−1
∏
1995 all gaps removed, published as “Proof of
the Alternating Sign Matrix Conjecture,” Elect.
J. of Combinatorics, 1996.

45. Zeilberger’s proof is an 84-page
tour de force, but it still left open
the original conjecture:
An,k
An,k+1
=
n − 2
k −1



 +
n −1
k −1




n − 2
n − k −1



 +
n −1
n − k −1





46. 1996 Kuperberg
announces a simple proof
“Another proof of the alternating
sign matrix conjecture,”
International Mathematics
Research Notices
Greg Kuperberg
UC Davis

47. “Another proof of the alternating
sign matrix conjecture,”
International Mathematics
Research Notices
Physicists have been studying ASM’s for
decades, only they call them square ice
(aka the six-vertex model ).
1996 Kuperberg
announces a simple proof

48. H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H
H H H H H
H O H O H O H O H O H

50. Horizontal → 1
Vertical → –1

51. southwest
northeast
northwest
southeast

52. N = # of vertical
I = inversion
number
= N + # of SW
x2, y3

53. Anatoli Izergin
Vladimir Korepin
SUNY Stony Brook
1980’s

54. det
1
xi − yj( )axi − yj( )






xi − yj( )axi − yj( )i, j=1
n
∏
xi − xj( )yi − yj( )1≤i< j≤n∏
= 1− a( )2N A( )
an(n−1)/2− Inv A( )
A∈An
∑
× xi
vert
∏ yj axi − yj( )SW, NE
∏ xi − yj( )NW, SE
∏
Proof:
LHS is symmetric polynomial in x’s and in y’s
Degree n – 1 in x1
By induction, LHS = RHS when x1 = y1
Sufficient to show that RHS is symmetric polynomial in x’s and
in y’s

55. LHS is symmetric polynomial in x’s and in y’s
Degree n – 1 in x1
By induction, LHS = RHS when x1 = –y1
Sufficient to show that RHS is symmetric polynomial in x’s and
in y’s — follows from Baxter’s triangle-to-triangle relation
Proof:
Rodney J. Baxter
Australian
National
University

56. a = z−4
, xi = z2
, yi = 1
RHS = z − z−1
( )
n n−1( )
z + z−1
( )
2N A( )
A∈An
∑
det
1
xi − yj( )axi − yj( )






xi − yj( )axi − yj( )i, j=1
n
∏
xi − xj( )yi − yj( )1≤i< j≤n∏
= 1− a( )2N A( )
an(n−1)/2−Inv A( )
A∈An
∑
× xi
vert
∏ yj axi − yj( )SW, NE
∏ xi − yj( )NW, SE
∏

57. det
1
xi − yj( )axi − yj( )






xi − yj( )axi + yj( )i, j=1
n
∏
xi − xj( )yi − yj( )1≤i< j≤n∏
= 1− a( )2N A( )
an(n−1)/2−Inv A( )
A∈An
∑
× xi
vert
∏ yj axi − yj( )SW, NE
∏ xi − yj( )NW, SE
∏
z = eπi/3
: RHS = −3( )n n−1( )/2
An ,
z = eπi/4
: RHS = −2( )n n−1( )/2
2N A( )
A∈An
∑ ,
z = eπi/6
: RHS = −1( )n n−1( )/2
3N A( )
A∈An
∑ .
a = z−4
, xi = z2
, yi = 1
RHS = z − z−1
( )
n n−1( )
z + z−1
( )
2N A( )
A∈An
∑

58. 1996
Doron Zeilberger
uses this
determinant to
prove the original
conjecture
“Proof of the refined alternating sign matrix
conjecture,” New York Journal of Mathematics

59. 2001, Kuperberg uses the power of the triangle-to-triangle
relation to prove some of the conjectured formulas:
AV 2n +1( )= −3( )n2 3 j − i( )+1
j − i + 2n +1i, j≤2n+1
2 j
∏










A n( )
AHT 2n( )= −3( )n n−1( )/2 3 j − i( )+ 2
j − i + ni, j
∏





 A n( )
AQT 4n( )= AHT 2n( )⋅ A n( )2

60. Kuperberg, 2001: proved formulas for counting
some new six-vertex models:
AUU 2n( )= −3( )n2
22n 3 j − i( )+ 2
j − i + 2n +11≤i, j≤2n+1
2 j
∏
1 −1 0 1
0 1 −1 0
0 0 0 0
0 0 1 0













61. Kuperberg, 2001: proved formulas for many
symmetry classes of ASM’s and some new ones
1 −1 0 1
0 1 −1 0
0 0 0 0
0 0 1 0












AUU 2n( )= −3( )n2
22n 3 j − i( )+ 2
j − i + 2n +11≤i, j≤2n+1
2 j
∏

62. Soichi Okada,
Nagoya University
1993, Okada finds the equivalent of the
λ-determinant for the other Weyl
Denominator Formulas.
2004, Okada shows that the formulas for
counting ASM’s, including those subject
to symmetry conditions, are simply the
dimensions of certain irreducible
representations, i.e. specializations of
Weyl Character formulas.

63. sλ 1,1,K ,1( )=
ρ + λ( )⋅r
ρ ⋅rr∈A2 n−1
+
∏ =
3j +1( )
2



 −
3i +1( )
2




j − i1≤i< j≤2n
∏
3−n(n−1)/2
sλ 1,1,K ,1( )=
3i +1( )!
n + i( )!i=0
n−1
∏
Number of n × n ASM’s is 3–n(n–1)/2
times the
dimension of the irreducible representation of
GL2n indexed by
λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )
A2n−1
+
= ei − ej 1 ≤ i < j ≤ 2n{ }
ρ = n − 1
2,n − 3
2,K ,−n + 1
2( )

64. dim Sp4n λ( )=
ρC + λ( )⋅r
ρC ⋅rr∈C2n
+
∏
=
6n + 2 −
3i +1
2



 −
3j +1
2




4n + 2 − i − j









1≤i< j≤2n
∏
3j +1
2



 −
3i +1
2




j − i










3n +1−
3i +1
2




2n +1− ii=1
2n
∏
Number of (2n+1) × (2n+1) vertically symmetric
ASM’s is 3–n(n–1)
times the dimension of the
irreducible representation of Sp4n indexed by
λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )
C2n
+
= ei ± ej 1 ≤ i < j ≤ 2n{ }U 2ei 1 ≤ i ≤ 2n{ }
ρC =
1
2
r =
r∈C2n
+
∑ 2n,2n −1,K ,1( )

65. NEW for 2004:
Number of (4n+1) × (4n+1) vertically and
horizontally symmetric ASM’s is 2–2n
3–n(2n–1)
times
λ = n −1,n −1,n − 2,n − 2,K ,1,1,0,0( )
µ = n − 1
2,n − 3
2,n − 3
2,K , 3
2, 3
2, 1
2( )
dim Sp4n λ( )× dim %O4n µ( )=
ρC + λ( )⋅r
ρC ⋅rr∈C2n
+
∏






ρD + µ( )⋅r
ρD ⋅rr∈D2n
+
∏






C2n
+
= ei ± ej 1 ≤ i < j ≤ 2n{ }U 2ei 1 ≤ i ≤ 2n{ }
ρC = 2n,2n −1,K ,1( )
D2n
+
= ei ± ej 1 ≤ i < j ≤ 2n{ }
ρD = 2n −1,2n − 2,K ,1,0( )

66. Proofs and Confirmations: The Story
of the Alternating Sign Matrix
Conjecture
Cambridge University Press & MAA, 1999
OKADA, Enumeration of Symmetry Classes of
Alternating Sign Matrices and Characters of Classical
Groups, arXiv:math.CO/0408234 v1 18 Aug 2004