Proofs & Confirmations The story of the

alternating sign matrix conjecture

David M. Bressoud

Macalester College

Arizona State University

Tempe, AZ

March 19, 2009 These slides are available at

www.macalester.edu/~bressoud/talks

Artwork by Greg Kuperberg

0 0 1 0 00 1 −1 0 11 −1 0 1 00 1 0 0 00 0 1 0 0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Square matrix:

•  Entries are 0, 1, –1

• Row and column sums are +1

•  Non-zero entries alternate in sign in each row

Bill Mills

Howard Rumsey

IDA-CCR

David Robbins (1942–2003)

MAA Robbins Prize in algebra, combinatorics, or discrete math

Charles L. Dodgson

aka Lewis Carroll

“Condensation of Determinants,” Proceedings of the Royal Society, London 1866

Desnanot-Jacobi adjoint matrix theorem (Desnanot for n ≤ 6 in 1819, Jacobi for general case in 1833

M ji is matrix M with row i and column j removed.

detM =detM1

1 ⋅detMnn − detMn

1 ⋅detM1n

detM1,n1,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)

detM =detM1

1 ⋅detMnn − detMn

1 ⋅detM1n

detM1,n1,n

detλ M =detλ M1

1 ⋅detλ Mnn + λ detλ Mn

1 ⋅detλ M1n

detλ M1,n1,n

det−1M = detM( )

detλ aji−1( )i, j=1

n= ai + λ aj( )1≤i< j≤n∏

detλa bc d

⎛⎝⎜

⎞⎠⎟= ad + λbc

detλ

a b cd e fg h j

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ = aej + λ bdj + afh( ) + λ2 bfg + cdh( ) + λ 3ceg

+ λ 1+ λ( )bde−1 fh

detM =detM1

1 ⋅detMnn − detMn

1 ⋅detM1n

detM1,n1,n

detλ M =detλ M1

1 ⋅detλ Mnn + λ detλ Mn

1 ⋅detλ M1n

detλ M1,n1,n

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( )+ sums over other permutations × λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +

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( )+ sums over other permutations × λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +

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 a1,4

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( )+ sums over other permutations × λ inversion number

+ λ 3 1+ λ−1( )a1,2a2,1a2,2−1 a2,3a3,4a4,2 +

+ λ 3 1+ λ−1( )2a1,2a2,1a2,2

−1 a2,3a3,2a3,3−1a3,4a4,3 +

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

n

1

2

3

4

5

6

7

8

9

An

1

2

7

42

429

7436

218348

10850216

911835460

0 0 1 0 00 1 −1 0 11 −1 0 1 00 1 0 0 00 0 1 0 0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

n

1

2

3

4

5

6

7

8

9

An

1

2

7

42

429

7436

218348

10850216

911835460

= 3 × 11 × 13

= 22 × 11 × 132

= 22 × 132 × 17 × 19

= 23 × 13 × 172 × 192

= 22 × 5 × 172 × 193 × 23

= 2 × 3 × 7

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

0 0 1 0 00 1 −1 0 11 −1 0 1 00 1 0 0 00 0 1 0 0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

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+…

0 0 1 0 00 1 −1 0 11 −1 0 1 00 1 0 0 00 0 1 0 0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

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

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

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,kAn,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,kAn,k+1

=

n − 2k −1

⎛⎝⎜

⎞⎠⎟+

n −1k −1

⎛⎝⎜

⎞⎠⎟

n − 2n − k −1

⎛⎝⎜

⎞⎠⎟+

n −1n − k −1

⎛⎝⎜

⎞⎠⎟

An =3 j +1( )!n + j( )!j=0

n−1

∏ =1!⋅4!⋅7! 3n − 2( )!n!⋅ n +1( )! 2n −1( )!

Richard Stanley

Richard Stanley

George Andrews

Andrews’ Theorem: the number of descending plane partitions of size n is

An =3 j +1( )!n + j( )!j=0

n−1

∏ =1!⋅4!⋅7! 3n − 2( )!n!⋅ n +1( )! 2n −1( )!

What is a descending

plane partition?

Percy A. MacMahon

Plane Partition

Work begun in

1897

6 5 5 4 3 3

Plane partition of 75

# of pp’s of 75 = pp(75)

6 5 5 4 3 3

Plane partition of 75

# of pp’s of 75 = pp(75) = 37,745,732,428,153

Generating function:

1+ pp j( )j=1

∞

∑ q j = 1+ q + 3q2 + 6q3 +13q4 +…

=1

1− qk( )kk=1

∞

∏

=1

1− q( ) 1− q2( )2 1− q3( )31912 MacMahon proves that the generating function for plane partitions in an n × n × n box is 1− qi+ j+ k−1

1− qi+ j+ k−21≤i, j ,k≤n∏

Symmetric Plane Partition 4 3 2 1 1

3 2 2 1

2 2 1

1 1

1

At the same time, he conjectures that the generating function for symmetric plane partitions is

1− qi+ j+ k−1

1− qi+ j+ k−21≤i= j≤n1≤k≤n

∏ 1− q2 i+ j+ k−1( )

1− q2 i+ j+ k−2( )1≤i< j≤n1≤k≤n

∏

“The reader must be warned that, although there is little doubt that this result is correct, … the result has not been rigorously established. … Further investigations in regard to these matters would be sure to lead to valuable work.’’ (1916)

1971 Basil Gordon proves case for n = infinity

1971 Basil Gordon proves case for n = infinity

1977 George Andrews and Ian Macdonald independently prove general case

Cyclically Symmetric Plane Partition

Macdonald’s Conjecture (1979): The generating function for cyclically symmetric plane partitions in B(n,n,n) is

“If I had to single out the most interesting open problem in all of enumerative combinatorics, this would be it.” Richard Stanley, review of Symmetric Functions and Hall Polynomials, Bulletin of the AMS, March, 1981.

1− q η 1+ht η( )( )

1− q η ht η( )η∈B /C3∏

1979, Andrews counts cyclically symmetric plane partitions

1979, Andrews counts cyclically symmetric plane partitions

1979, Andrews counts cyclically symmetric plane partitions

1979, Andrews counts cyclically symmetric plane partitions

1979, Andrews counts cyclically symmetric plane partitions

length

width

L1 = W1 > L2 = W2 > L3 = W3 > …

1979, Andrews counts descending plane partitions

length

width

L1 > W1 ≥ L2 > W2 ≥ L3 > W3 ≥ …

6 6 6 4 3

3 3

2

length

width 6 6 6 4 3

3 3

2

Mills, Robbins, Rumsey Conjecture: # of n × n ASM’s with 1 at top of column j equals # of DPP’s ≤ n with exactly j–1 parts of size n.

Discovered an easier proof of Andrews’ formula, using induction on j and n.

Used this inductive argument to prove Macdonald’s conjecture

“Proof of the Macdonald Conjecture,” Inv. Math., 1982

But they still didn’t have a proof of their conjecture!

1983

Totally Symmetric Self-Complementary Plane Partitions

Vertical flip of ASM = complement of DPP ?

David Robbins

Totally Symmetric Self-Complementary Plane Partitions

Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is

3 j +1( )!n + j( )!j=0

n−1

∏ =1!⋅4!⋅7! 3n − 2( )!n!⋅ n +1( )! 2n −1( )!

Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is

1989: William Doran shows equivalent to counting lattice paths

1990: John Stembridge represents the counting function as a Pfaffian (built on insights of Gordon and Okada)

1992: George Andrews evaluates the Pfaffian, proves Robbins’ Conjecture

3 j +1( )!n + j( )!j=0

n−1

∏ =1!⋅4!⋅7! 3n − 2( )!n!⋅ n +1( )! 2n −1( )!

December, 1992

Doron Zeilberger announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n.

December, 1992

Doron Zeilberger announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n.

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,kAn,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 Davis

Physicists had been studying ASM’s for decades, only they called them the six-vertex model.

Horizontal = 1

Vertical = –1

0 0 1 0 00 1 −1 0 11 −1 0 1 00 1 0 0 00 0 1 0 0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

1960’s

Anatoli Izergin Vladimir Korepin

1980’s

Rodney Baxter’s

Triangle-to-triangle relation

det 1xi − yj( ) axi − yj( )

⎛

⎝⎜

⎞

⎠⎟

xi − yj( ) axi − yj( )i, j=1

n∏xi − x j( ) 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∏

a = z−4 , xi = z2 , yi = 1

RHS = z − z−1( )n n−1( )z + z−1( )2N A( )

A∈An

∑z = eπ i /3

RHS = −3( )n n−1( )/2 An

det 1xi − yj( ) axi − yj( )

⎛

⎝⎜

⎞

⎠⎟

xi − yj( ) axi − yj( )i, j=1

n∏xi − x j( ) 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∏

1996

Doron Zeilberger uses this determinant to prove the original conjecture

“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

The End (which is really just the beginning)

These slides can be downloaded from

www.macalester.edu/~bressoud/talks