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How the Alternating Sign Matrix Conjecture Was Solved, Volume 46

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How the Alternating Sign Matrix Conjecture Was Solved, Volume 46 fea-bressoud.qxp 5/17/99 2:12 PM Page 637 How the Alternating Sign Matrix Conjecture Was Solved David Bressoud and James Propp Introduction These tools did not come from outside deter- Perusing the four volumes of Muir’s The Theory of minant theory; rather, the classical theory of de- Determinants in the Historical Order of Develop- terminants grew into nineteenth-century invari- ment, one might be tempted to conclude that the ant theory, a field whose twentieth-century progeny theory of determinants was well and truly beaten include partition theory and the q-calculus, rep- to death in the nineteenth century. In fact, the resentation theory and symmetric functions, and field is thriving, and it has continued to yield chal- statistical mechanics. The proofs of the Alternat- lenging problems of deceptive elegance and sim- ing Sign Matrix Theorem have served to strengthen plicity. The Alternating Sign Matrix Conjecture was ties between these fields and to suggest new av- one of the most notorious of these problems. For enues of research. fifteen years it defied assaults by some of the An alternating sign matrix (ASM) is a matrix of world’s best mathematicians; then in 1995 three 0’s, 1’s, and 1’s in which the entries in each row distinct proofs appeared. The first, by Doron Zeil- berger, drew on results and techniques from par- or column sum to 1 and the nonzero entries in each tition theory, symmetric functions, and constant row or column alternate in sign. An example is term identities, with a pivotal role played by the 00010 partial difference operator philosophy and by com- 01011 puter algebra. Greg Kuperberg found the second proof, which relied on the machinery of statistical 1 1010 . mechanics and in particular on the Yang-Baxter 00100 equation for the 6-vertex lattice model. The third 01000 proof, again by Zeilberger, expanded Kuperberg’s approach to prove a more general result. It com- bined the Yang-Baxter equation with the q-calcu- This generalization of the notion of permutation lus and its associated orthogonal polynomials, and matrices was discovered by David Robbins and it relied on the WZ-method of Herbert Wilf and Zeil- Howard Rumsey in the early 1980s, but to tell our berger. Wilf and Zeilberger would later receive the story properly, we should begin with Charles Steele Prize for this algorithmic approach to dis- Lutwidge Dodgson (better known as Lewis Car- covering and proving series identities (Notices, roll). April 1998). Dodgson devised a method of evaluating de- terminants called condensation that is eminently David Bressoud is professor of mathematics at Macalester suited to hand-calculations. Recall that the deter- College, St. Paul, MN. His e-mail address is bressoud@ macalester.edu. minant of an n-by-n matrix (ai,j) is defined as James Propp is associate professor of mathematics at the X Yn | | − I(π) University of Wisconsin. His e-mail address is ai,j = ( 1) ai,π(i), [email protected]. π i=1 JUNE/JULY 1999 NOTICES OF THE AMS 637 fea-bressoud.qxp 5/17/99 2:12 PM Page 638 1 11 If one applies Dodgson condensation to the 3-by-3 matrix 232 714147 abc 42 105 135 105 42 def , 429 1287 2002 2002 1287 429 ghi 7436 26026 47320 56784 47320 26026 7436 one first obtains the 2-by-2 matrix Figure 1. The counts of n-by-n ASMs with a 1 at the top of column k. ! ae bd bf ce , 2/2 dh eg ei fh 2/3 3/2 2/4 5/5 4/2 and from this one finds the 1-by-1 matrix whose 2/5 7/9 9/7 5/2 2/6 9/14 16/16 14/9 6/2 sole entry is 2 /7 11/20 25/30 30/25 20/11 7/ 2 ((ae2i aef h bdei + bdfh) (bdfh befg cdeh + ce2g))/e Figure 2. The ratios of adjacent terms from Figure 1. or, upon collection of terms, where ranges over all permutations of (1)aei +(1)af h +(1)bdi {1, 2,...,n} and I() is the inversion number of + (0)bde 1fh+ (1)bfg + (1)cdh +(1)ceg. , i.e., the minimal number of transpositions of ad- jacent columns needed to turn the matrix repre- Six of these terms correspond to the six permu- senting into the identity matrix. This formula is tation matrices. For example, (1)af h is associated practical for 3-by-3 and perhaps 4-by-4 matrices, with the matrix with 1 in the same positions as oc- but for large matrices it is inefficient. Most math- cupied by a, f, and h above, with 0’s elsewhere. In ematicians are familiar with Gaussian elimination addition, there is an extra (vanishing) term as a more practical method of evaluating deter- (0)bde 1fh that can be associated with the matrix minants by hand, but condensation is also useful with 1’s in the positions of b, d, f, and h and 1 and deserves to be better known. One starts with in the position of e: an n-by-n matrix and then successively computes 010 an (n 1) -by-(n 1) matrix, an (n 2) -by-(n 2) matrix, etc., until one arrives at a 1-by-1 matrix 1 11 . whose sole entry is the determinant of the origi- 010 nal n-by-n matrix. The rule for computing the k- yyr by-k matrix (n 1 k 1) is to take the k2 If one does the same thing for the general 4-by-4 2-by-2 connected subdeterminants of the (k +1)- matrix, one finds that, in addition to the 24 mono- by-(k +1) matrix and divide them by the corre- mials that make nonzero contributions to the de- sponding k2 central entries of the (k +2)-by-(k +2) terminant, there are also 18 monomials with van- matrix. (In the case k = n 1, no divisions are per- ishing coefficient. Each of these 42 monomials is associated with a 4-by-4 matrix of 0’s, 1’s, and formed.) Although the use of division may seem 1’s. In general, when Dodgson condensation is ap- like a liability, it actually provides a useful form plied to an n-by-n matrix and all like monomials of error checking for hand calculations with inte- are gathered together, the terms in the final ex- ger matrices: when the algorithm is performed pression (taking the vanishing terms along with the properly (with extra provisos for avoiding divi- nonvanishing ones) are associated with the n-by- sion by 0), all the entries of all the intervening ma- n matrices of 0’s, 1’s, and 1’s in which the trices are integers, so that when a division fails to nonzero entries in each row and column alternate come out evenly, one can be sure that a mistake in sign, beginning and ending with a +1. These are has been made somewhere. The method is also use- the alternating sign matrices (or ASMs) of order n, ful for computer calculations, especially since it can invented by Robbins and Rumsey in their study of be executed in parallel by many processors. The Dodgson condensation. k-by-k matrix that one computes by this procedure It was simple curiosity that led Robbins and has a natural interpretation: it is the matrix of de- Rumsey, now joined by William Mills, to investigate 2 terminants of the k (n k +1)-by- (n k +1)con- the number of ASMs. Letting An denote the set of nected submatrices of the original matrix. The n -by-n ASMs and An the cardinality of An, the proof of this assertion makes use of one of Jacobi’s three investigators found by computer calculation matrix identities. that the sequence An went 638 NOTICES OF THE AMS VOLUME 46, NUMBER 6 fea-bressoud.qxp 5/17/99 2:12 PM Page 639 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460,.... This was not a sequence any of them had seen before. The growth rate of the sequence and the absence of large prime divisors (e.g., 4 3+1 2+2 911835460 = 22 5 172 193 23) suggested to Mills, Robbins, and Rumsey that there was a formula for An as a ratio of products of fac- torials. To find this formula, they divided the set of n-by-n ASMs into classes according to the position of the 1 in the first row. Their tal- lies yielded a triangular array in which the kth entry of the nth row is the number of n-by-n ASMs with a 1 in row 1, column k, as shown in Figure 1. 2+1+1 1+1+1+1 Clearly the sum of the entries in each row is An, and it is not difficult to see as well that Figure 3. Young diagrams corresponding to partitions of 4. the first entry in each row must equal An1. When Mills, Robbins, and Rumsey looked at ratios ber of ways of representing the positive integer n of horizontally adjacent entries, they discovered as a sum of positive integers (without regard to the remarkable pattern shown in Figure 2. order) equals the coefficient of qn in the power- The nth row starts with 2/(n +1)and ends with series expansion of the infinite product (n +1)/2. The striking observation is that each Y∞ 1 ratio appears to arise from the two ratios diago- =1+q +2q2 +3q3 nally above by adding numerators and adding de- 1 − qk k=1 nominators.
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