All Robbins conjectures on the number of ASMs with symmetries are true

Algebra, Combinatorics 1 Minute

You need to know: Notation n!=1\cdot 2 \cdot \dots \cdot n, product notation \prod\limits_{i=1}^n, matrices.

Background: An n \times n matrix with elements a_{ij} is called an alternating-sign matrix (ASM) if (i) all a_{ij} are equal to -1, 0, or 1, (ii) the sum of elements in each row and column is 1, and (iii) the non-zero elements in each row and column alternate in sign. An ASM is called diagonally and antidiagonally symmetric (DASASM) if a_{ij} = a_{ji} = a_{n+1-j,n+1-i} for all 1\leq i,j \leq n.

The Theorem: On 18th December 2015, Roger Behrend, Ilse Fischer, and Matjaž Konvalinka submitted to arxiv a paper in which they proved that, for any positive integer n, the number of (2n+1)\times(2n+1) DASASMs is given by \prod\limits_{i=0}^{n} \frac{(3i)!}{(n+i)!}.

Short context: Alternating-sign matrices are important both in pure mathematics (determinant computation) and applications (so-called “six-vertex model” for ice modelling in statistical mechanics). The formula for the total number of ASMs was conjectured by Robbins and Rumsey in 1986 and proved by Zeilberger in 1996. In 1991, Robbins conjectured formulas for the number of ASMs with various symmetries. For many years, these conjecture served as a roadmap, stimulating progress in the area. Starting from work of Kuperberg, see here, several groups of researchers resolved by 2006 all these conjectures except one: the formula for the number of (2n+1)\times(2n+1) DASASMs. The Theorem resolves this remaining conjecture.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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