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Exploiting Symmetries. Alternating Sign Matrices and the Weyl Character Formulas. David M. Bressoud Macalester College, St. Paul, MN Stanford University April 12, 2006. The Vandermonde determinant Weyl’s character formulae Alternating sign matrices - PowerPoint PPT Presentation
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David M. BressoudMacalester College, St. Paul, MNStanford UniversityApril 12, 2006
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
x1n−1 x2
n−1 L xnn−1
M M O Mx1 x2 L xn1 1 L 1
= −1( )I σ( ) xin−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)
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 by xi =xjxi −xj( )
i< j∏
x1n−1 x2
n−1 L xnn−1
M M O Mx1 x2 L xn1 1 L 1
= −1( )I σ( ) xin−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)
This function is 0 when so it is divisible by xi =xjxi −xj( )
i< j∏
But both polynomials have same degree, so ratio is constant, = 1.
= xi − x j( )i< j∏
x1n−1 x2
n−1 L xnn−1
M M O Mx1 x2 L xn1 1 L 1
= −1( )I σ( ) xin−σ i( )
i=1
n
∏σ∈Sn∑
Cauchy 1815
Any alternating function in divided by the Vandermonde determinant yields a symmetric function:
x1, x2 ,K , xn
x1λ1 +n−1 x2
λ1 +n−1 L xnλ1 +n−1
M M O Mx1
λn−1 +1 x2λn−1 +1 L xn
λn−1 +1
x1λn x2
λn L xnλn
x1n−1 x2
n−1 L xnn−1
M M O Mx1
1 x21 L xn
1
x10 x2
0 L xn0
=σλ x1, x2 ,K , xn( )
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.
x1λ1 +n−1 x2
λ1 +n−1 L xnλ1 +n−1
M M O Mx1
λn−1 +1 x2λn−1 +1 L xn
λn−1 +1
x1λn x2
λn L xnλn
x1n−1 x2
n−1 L xnn−1
M M O Mx1
1 x21 L xn
1
x10 x2
0 L xn0
=σλ x1, x2 ,K , xn( )
Issai Schur (1875–1941)
sλ 1,1,K ,1( ) is the dimension of the representation
sλ 1,1,K ,1( )=ρ + λ( )⋅ρρ ⋅ρρ∈An−1
+∏
wheρe ρ =n−12
,n −32
,K ,1−n2
⎛⎝⎜
⎞⎠⎟ =
12
ρρ∈An−1
+∑ ,
λ = λ1,λ2 ,K ,λn( ),
An−1+ = ei −ej 1≤i < j≤n{ } ,
ei iσ the unit vectoρ with 1 in the ith cooρdinate
Note that the symmetric group on n letters is the group of transformations of An−1 = ei −ej 1≤i ≠j≤n{ }
Weyl 1939 The Classical Groups: their invariants and representations
Sp2n λ; ρx( )=
x1λ1 +n −x1
−λ1−n L xnλ1 +n −xn
−λ1−n
M O Mx1
λn +1 −x1−λn −1 L xn
λn +1 −xn−λn −1
x1n −x1
−n L xnn −xn
−n
M O Mx1
1 −x1−1 L xn
1 −xn−1
Sp2n λ; ρx( ) 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)
Sp2n λ;1ρ
( )=ρ + λ( )⋅ρρ ⋅ρρ∈Cn
+∏
wheρe ρ = n,n−1,K ,1( )=12
ρρ∈Cn
+∑ ,
λ = λ1,λ2 ,K ,λn( ),
Cn+ = ei ±ej 1≤i < j≤n{ } U 2ei 1≤i ≤n{ } ,
ei iσ the unit vectoρ with 1 in the ith cooρdinate
The dimension of the representation is
Weyl 1939 The Classical Groups: their invariants and representations
x1n −x1
−n L xnn −xn
−n
M O Mx1
1 −x1−1 L xn
1 −xn−1
x1x2L xn( )n
xi −xj( )i< j∏
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at ±1, x j
−1 for 2 ≤ j ≤ n
Weyl 1939 The Classical Groups: their invariants and representations
x1n −x1
−n L xnn −xn
−n
M O Mx1
1 −x1−1 L xn
1 −xn−1
x1x2L xn( )n
xi −xj( )i< j∏
= xi2 −1( )
i∏ xixj−1( )
i< j∏
is a symmetric polynomial. As a polynomial in x1 it has degree n + 1 and roots at ±1, x j
−1 for 2 ≤ j ≤ n
Weyl 1939 The Classical Groups: their invariants and representations: The Denominator Formulas
x1n−12 −x1
−n+ 12 L xn
n−12 −xn−n+12
M O M
x112 −x1
−12 L xn12 −xn
−12
x1x2L xn( )n−12
xi −xj( )i< j∏
= xi −1( )i∏ xixj−1( )
i< j∏
x1n−1 + x1
−n+1 L xnn−1 + xn
−n+1
M O Mx1
0 + x1−0 L xn
0 + xn−0
x1x2L xn( )n−1
xi −xj( )i< j∏
=2 xixj−1( )i< j∏
Desnanot-Jacobi adjoint matrix thereom (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833
M ji is matrix M with row i and column j removed.
det M =detM1
1 ⋅detMnn −detMn
1 ⋅detM1n
detM1,n1,n
Given that the determinant of the empty matrix is 1 and the determinant of a 11 is the entry in that matrix, this uniquely defines the determinant for all square matrices.
Carl Jacobi (1804–1851)
det M =detM1
1 ⋅detMnn −detMn
1 ⋅detM1n
detM1,n1,n
detλ M =detM1
1 ⋅detMnn + λ detMn
1 ⋅detM1n
detM1,n1,n
det−1 M =detM( )
detλ aji−1( )i, j=1
n= ai + λaj( )
1≤i< j≤n∏
David Robbins (1942–2003)
det M =detM1
1 ⋅detMnn −detMn
1 ⋅detM1n
detM1,n1,n
detλ M =detM1
1 ⋅detMnn + λ detMn
1 ⋅detM1n
detM1,n1,n
detλa bc d
⎛⎝⎜
⎞⎠⎟ =ad + λbc
detλ
a b cd e fg h j
⎛
⎝⎜⎜
⎞
⎠⎟⎟ =aej+ λ bdj+ afh( )+ λ 2 bfg+ cdh( )+ λ 3ceg
+ λ 1+ λ( )bde−1 fh
detλ
a1,1 a1,2 a1,3 a1,4a2,1 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 σum σ oveρ otheρ peρm utationσ× λ inveρσion num beρ
+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L
+ λ 3 1+ λ−1( )2a1,2a2,1a2,2
−1 a2,3a3,2a3,3−1a3,4a4,3 +L
detλ
a1,1 a1,2 a1,3 a1,4a2,1 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 σum σ oveρ otheρ peρm utationσ× λ inveρσion num beρ
+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L
+ λ 3 1+ λ−1( )2a1,2a2,1a2,2
−1 a2,3a3,2a3,3−1a3,4a4,3 +L
0 1 0 01 −1 1 00 1 −1 10 0 1 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
0 1 0 01 −1 1 00 0 0 10 1 0 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
detλ
a1,1 a1,2 a1,3 a1,4a2,1 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 σum σ oveρ otheρ peρm utationσ× λ inveρσion num beρ
+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L
+ λ 3 1+ λ−1( )2a1,2a2,1a2,2
−1 a2,3a3,2a3,3−1a3,4a4,3 +L
0 1 0 01 −1 1 00 1 −1 10 0 1 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
0 1 0 01 −1 1 00 0 0 10 1 0 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
detλ
a1,1 a1,2 a1,3 a1,4a2,1 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 σum σ oveρ otheρ peρm utationσ× λ inveρσion num beρ
+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L
+ λ 3 1+ λ−1( )2a1,2a2,1a2,2
−1 a2,3a3,2a3,3−1a3,4a4,3 +L
detλ xi, j( )= λ Inv A( )
A= ai, j( )
∑ 1+ λ−1( )N A( )
xi, jai, j
i, j∏
Sum is over all alternating sign matrices, N(A) = # of –1’s
detλ
a1,1 a1,2 a1,3 a1,4a2,1 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 σum σ oveρ otheρ peρm utationσ× λ inveρσion num beρ
+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +L
+ λ 3 1+ λ−1( )2a1,2a2,1a2,2
−1 a2,3a3,2a3,3−1a3,4a4,3 +L
detλ xi, j( )= λ Inv A( )
A= ai, j( )
∑ 1+ λ−1( )N A( )
xi, jai, j
i, j∏
xi + λxj( )= λ Inv A( ) 1+ λ−1( )
N A( )xj
n−i( )ai, j
i, j∏
A∈An
∑1≤i< j≤n∏
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?
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
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
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
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
1
1 1
2 3 2
7 14 14 7
42 105 135 105 42
429 1287 2002 2002 1287 429
+ + +
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 00 ? ? ? ?0 ? ? ? ?0 ? ? ? ?0 ? ? ? ?
⎛
⎝
⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟
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
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
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
2
2 3
2 5 4
2 7 9 5
2 9 16 14 6
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:
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−2k−1
⎛⎝⎜
⎞⎠⎟ +
n−1k−1
⎛⎝⎜
⎞⎠⎟
n−2n−k−1
⎛⎝⎜
⎞⎠⎟ +
n−1n−k−1
⎛⎝⎜
⎞⎠⎟
Conjecture 1:
Conjecture 2 (corollary of Conjecture 1):
An,k
An,k+1
=
n−2k−1
⎛⎝⎜
⎞⎠⎟ +
n−1k−1
⎛⎝⎜
⎞⎠⎟
n−2n−k−1
⎛⎝⎜
⎞⎠⎟ +
n−1n−k−1
⎛⎝⎜
⎞⎠⎟
An =
3j+1( )!n + j( )!j=0
n−1
∏ =1!⋅4!⋅7!L 3n−2( )!n!⋅n +1( )!L 2n−1( )!
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
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
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 + 29400x + 63308x2 + 66158x3 + 38390x4 +13037x5
+ 2660x6 + 328x7 + 26x8 + x9
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 + 29400x + 63308x2 + 66158x3 + 38390x4 +13037x5
+ 2660x6 + 328x7 + 26x8 + x9
xi + λxj( )= λ Inv 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
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 + 29400x + 63308x2 + 66158x3 + 38390x4 +13037x5
+ 2660x6 + 328x7 + 26x8 + x9
A n; 3( )=3n n−1( )
2n n−1( )
3 j−i( )+13 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
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+12 j
∏⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟A n( )
AHT 2n( )= −3( )n n−1( )/2 3 j−i( )+ 2j−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
December, 1992
Zeilberger announces a proof that # of ASM’s equals
3 j +1( )!n + j( )!j=0
n−1
∏
Doron Zeilberger
Rutgers University
December, 1992
Zeilberger announces a proof that # of ASM’s equals
3 j +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.
Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:
An,k
An,k+1
=
n−2k−1
⎛⎝⎜
⎞⎠⎟ +
n−1k−1
⎛⎝⎜
⎞⎠⎟
n−2n−k−1
⎛⎝⎜
⎞⎠⎟ +
n−1n−k−1
⎛⎝⎜
⎞⎠⎟
1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis
1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC DavisPhysicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).
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
Horizontal 1
Vertical –1
southwest
northeast
northwest
southeast
N = # of verticalI = inversion number = N + # of SW
x2, y3
Anatoli IzerginVladimir Korepin
SUNY Stony Brook
1980’s
det1
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
∑
× xiveρt∏ 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
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
a =z−4 , xi =z2 , yi =1
RHS= z−z−1( )n n−1( )
z+ z−1( )2N A( )
A∈An
∑
det1
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
∑
× xiveρt∏ yj axi −yj( )
SW , NE∏ xi −yj( )
NW , SE∏
det1
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
∑
× xiveρt∏ yj axi −yj( )
SW , NE∏ xi −yj( )
NW , SE∏
z =epi/3 : RHS= −3( )n n−1( )/2 An ,
z=epi/4 : RHS= −2( )n n−1( )/2 2N A( )
A∈An
∑ ,
z=epi/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
∑
1996
Doron Zeilberger uses this determinant to prove the original conjecture
“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics
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+12 j
∏⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟A n( )
AHT 2n( )= −3( )n n−1( )/2 3 j−i( )+ 2j−i + ni, j
∏⎛⎝⎜
⎞⎠⎟A n( )
AQT 4n( )=AHT 2n( )⋅A n( )2
Kuperberg, 2001: proved formulas for counting some new six-vertex models:
AUU 2n( )= −3( )n2
22n 3 j−i( )+ 2j−i + 2n +11≤i, j≤2n+1
2 j
∏
1 −1 0 10 1 −1 00 0 0 00 0 1 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
Kuperberg, 2001: proved formulas for many symmetry classes of ASM’s and some new ones
1 −1 0 10 1 −1 00 0 0 00 0 1 0
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
AUU 2n( )= −3( )n2
22n 3 j−i( )+ 2j−i + 2n +11≤i, j≤2n+1
2 j
∏
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.
sλ 1,1,K ,1( )=ρ + λ( )⋅ρρ ⋅ρρ∈A2n−1
+∏ =
3j+1( )2
⎢⎣⎢
⎥⎦⎥−
3i +1( )2
⎢⎣⎢
⎥⎦⎥
j−i1≤i< j≤2n∏
3−n(n−1)/2σλ 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−12,n −
32,K ,−n + 1
2( )
dim Sp4n λ( )=ρC + λ( )⋅ρρC ⋅ρρ∈C2n
+∏
=6n + 2 −
3i +12
⎢⎣⎢
⎥⎦⎥−
3j+12
⎢⎣⎢
⎥⎦⎥
4n + 2 −i−j
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟1≤i< j≤2n
∏3j+12
⎢⎣⎢
⎥⎦⎥−
3i +12
⎢⎣⎢
⎥⎦⎥
j−i
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
3n +1−3i +12
⎢⎣⎢
⎥⎦⎥
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 =12
ρ=ρ∈C2n
+∑ 2n,2n−1,K ,1( )
NEW for 2006 (preprint 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 − 12,n − 3
2,n − 32,K , 3
2, 32, 1
2( )
dim Sp4n λ( )× dim %O4n m( )=
ρC + λ( )⋅ρρC ⋅ρρ∈C2n
+∏
⎛⎝⎜
⎞⎠⎟
ρD + m( )⋅ρρD ⋅ρρ∈D2n
+∏
⎛⎝⎜
⎞⎠⎟
C2n+ = ei ±ej 1≤i < j≤2n{ } U 2ei 1≤i ≤2n{ }
ρC = 2n,2n−1,K ,1( )
D 2n+ = ei ±ej 1≤i < j≤2n{ }
ρD = 2n−1,2n−2,K ,1,0( )
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, J. Algebraic Combinatorics, 23 (2006), 43–69.