1. Proofs & Confirmations The story of the
alternating sign matrix conjecture
David M. Bressoud
Macalester College
MAA Northeastern Section
November 20, 2004
Worcester, MA

2. Institute for Defense Analysis
Bill Mills
Institute for Defense Analysis 1 –1
Howard Rumsey
Dave Robbins

3. Charles L. Dodgson aka Lewis Carroll1 –1
“Condensation of Determinants,” Proceedings of the Royal Society, London 1866

4. 1 –1 n 1 2 3 4 5 6 7 8 9 An 42
429
7436
218348

5. How many n  n alternating sign matrices?1 –1 n 1 2 3 4 5 6 7 8 9 An 42
429
7436
218348
= 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?

6. 1 –1 n 1 2 3 4 5 6 7 8 9 An 42
429
7436
218348
very suspicious
= 2  3  7
= 3  11  13
= 22  11  132
= 22  132  17  19
= 23  13  172  192
= 22  5  172  193  23

7. There is exactly one 1 in the first row2 3 4 5 6 7 8 9 An 42
429
7436
218348
There is exactly one 1 in the first row 1 –1

8. There is exactly one 1 in the first row2 3 4 5 6 7 8 9 An
1+1
2+3+2 …
There is exactly one 1 in the first row 1 –1

9. 1 1 –1

10. 1 1 –1 + + +

11. 1 1 –1 + + +

12. 1
1 2/2 1
2 2/ /2 2
7 2/ /2 7
42 2/ /2 42
429 2/ /2 429 1 –1

13. 1
1 2/2 1
2 2/ /2 2
7 2/ / /2 7
42 2/ / / /2 42
429 2/ / / / /2 429 1 –1

14. 2/2
2/ /2
2/ / /2
2/ / / /2
2/ / / / /2 1 –1

15. 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

16. Numerators: 1+1 1+1 1+2 1+1 2+3 1+3 1+1 3+4 3+6 1+4
Numerators: 1 –1
Conjecture 1:

17. Conjecture 2 (corollary of Conjecture 1):–1
Conjecture 2 (corollary of Conjecture 1):

18. 1 –1
Richard Stanley

19. Richard Stanley George Andrews1 –1
Richard Stanley
Andrews’ Theorem: the number of descending plane partitions of size n is
George Andrews

20. All you have to do is find a 1-to-1 correspondence between n by n alternating sign matrices and descending plane partitions of size n, and conjecture 2 will be proven! 1 –1

21. What is a descending plane partition?
All you have to do is find a 1-to-1 correspondence between n by n alternating sign matrices and descending plane partitions of size n, and conjecture 2 will be proven! 1 –1
What is a descending plane partition?

22. Percy A. MacMahon
Plane Partition 1 –1
Work begun in
1897

23. Plane partition of 75 1 –1
# of pp’s of 75 = pp(75)

24. Plane partition of 75 1 –1
# of pp’s of 75 = pp(75) = 37,745,732,428,153

25. Generating function: 1 –1

26. 1 –1

27. 1 –1

28. 1 –1

29. 1912 MacMahon proves that the generating function for plane partitions in an n  n  n box is–1
At the same time, he conjectures that the generating function for symmetric plane partitions is

30. Symmetric Plane Partition
1 1 1 1 –1
“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)

31. 1971 Basil Gordon proves case for n = infinity–1

32. 1971 Basil Gordon proves case for n = infinity–1
1977 George Andrews and Ian Macdonald independently prove general case

33. Macdonald’s observation: both generating functions are special cases of the following1 –1
where G is a group acting on the points in B and B/G is the set of orbits. If G consists of only the identity, this gives all plane partitions in B. If G is the identity and (i,j,k)  (j,i,k), then get generating function for symmetric plane partitions.

34. Does this work for other groups of symmetries?1 –1
G = S3 ? No
G = C3 ? (i,j,k)  (j,k,i)  (k,i,j)
It seems to work.

35. Cyclically Symmetric Plane Partition1 –1

36. Cyclically Symmetric Plane Partition1 –1

37. Cyclically Symmetric Plane Partition1 –1

38. Cyclically Symmetric Plane Partition1 –1

39. Macdonald’s Conjecture (1979): The generating function for cyclically symmetric plane partitions in B(n,n,n) is 1 –1
“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.

40. 1 –1
Macdonald’s Second Conjecture (1979): For every subgroup G of S3, the product on the right counts the total number of plane partitions contained in B and invariant under G.
G = C3: proved by Andrews, 1979

41. 1 –1
Macdonald’s Second Conjecture (1979): For every subgroup G of S3, the product on the right counts the total number of plane partitions contained in B and invariant under G.
G = S3: proved by John Stembridge, 1995

42. 1979, Andrews counts cyclically symmetric plane partitions–1

43. 1979, Andrews counts cyclically symmetric plane partitions–1

44. 1979, Andrews counts cyclically symmetric plane partitions–1

45. 1979, Andrews counts cyclically symmetric plane partitions–1

46. 1979, Andrews counts cyclically symmetric plane partitions–1
L1 = W1 > L2 = W2 > L3 = W3 > …
width
length

47. 6 6 6 4 3 3 3 2 1979, Andrews counts descending plane partitions–1
L1 > W1 ≥ L2 > W2 ≥ L3 > W3 ≥ …
3 3 2
width
length

48. 6 6 6 4 3 3 3 2 What are the corresponding 6 subsets of DPP’s?
6 X 6 ASM  DPP with largest part ≤ 6
What are the corresponding 6 subsets of DPP’s? 1 –1
3 3 2
width
length

49. ASM with 1 at top of first column DPP with no parts of size n.
ASM with 1 at top of last column DPP with n–1 parts of size n. 1 –1
3 3 2
width
length

50. 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. 1 –1
3 3 2
width
length

51. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 –1

52. Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 –1
Discovered an easier proof of Andrews’ formula, using induction on j and n.

53. Used this inductive argument to prove Macdonald’s conjecture
Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 –1
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

54. But they still didn’t have a proof of their conjecture!
Mills, Robbins, & Rumsey proved that # of DPP’s ≤ n with j parts of size n was given by their conjectured formula for ASM’s. 1 –1
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!

55. 1983 Totally Symmetric Self-Complementary Plane Partitions–1
1983
Vertical flip of ASM  complement of DPP ?

56. Totally Symmetric Self-Complementary Plane Partitions1 –1

57. 1 –1

58. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is1 –1

59. Robbins’ Conjecture: The number of TSSCPP’s in a 2n X 2n X 2n box is1 –1
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

60. December, 1992
Doron Zeilberger announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 –1

61. December, 1992
Doron Zeilberger announces a proof that # of ASM’s of size n equals of TSSCPP’s in box of size 2n. 1 –1
1995 all gaps removed, published as “Proof of the Alternating Sign Matrix Conjecture,” Elect. J. of Combinatorics, 1996.

62. 1 –1
Zeilberger’s proof is an 84-page tour de force, but it still left open the original conjecture:

63. 1996 Kuperberg announces a simple proof–1
1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis

64. 1996 Kuperberg announces a simple proof–1
1996 Kuperberg announces a simple proof
“Another proof of the alternating sign matrix conjecture,” International Mathematics Research Notices
Greg Kuperberg
UC Davis
Physicists have been studying ASM’s for decades, only they call them square ice (aka the six-vertex model ).

65. H O H O H O H O H O H
H H H H H 1 –1

66. 1 –1

67. Horizontal  1 1 –1
Vertical  –1

68. southwest 1 –1
northwest
southeast
northeast

69. 1 –1
N = # of vertical
I = inversion
number
= N + # of NE
x2, y3

70. Triangle-to-triangle relation
1960’s
Rodney Baxter’s
Triangle-to-triangle relation 1 –1

71. Triangle-to-triangle relation
1960’s
Rodney Baxter’s
Triangle-to-triangle relation 1 –1
1980’s
Vladimir Korepin
Anatoli Izergin

72. 1 –1

73. 1 –1

74. 1996
Doron Zeilberger uses this determinant to prove the original conjecture 1 –1
“Proof of the refined alternating sign matrix conjecture,” New York Journal of Mathematics

75. The End 1 –1
(which is really just the beginning)

76. The End (which is really just the beginning)1 –1
(which is really just the beginning)
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