Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Abstract

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of $(2n+1)\times (2n+1)$ DASASMs is ${\prod }_{i=0}^n\frac{(3i)!}{(n+i)!}$, and a conjecture of Stroganov from 2008 that the ratio between the numbers of $(2n+1)\times (2n+1)$ DASASMs with central entry $-1$ and $1$ is $n/(n+1)$. Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.

33 pages; v3: published version

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Cited by in corpus (15)

• Bijective proofs of skew Schur polynomial factorizations
• Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order
• Factorization theorems for classical group characters, with applications to alternating sign matrices and plane partitions
• Vertex Models and Spin Chains in Formulas and Pictures
• Boundary emptiness formation probabilities in the six-vertex model at $\Delta =-\frac{1}{2}$
• Alternating sign trapezoids and a constant term approach
• Sum rules for the supersymmetric eight-vertex model
• A Fourfold Refined Enumeration of Alternating Sign Trapezoids
• Off-diagonally symmetric domino tilings of the Aztec diamond
• Refined Enumeration of Halved Monotone Triangles and Applications to Vertically Symmetric Alternating Sign Trapezoids
• Further results for classical and universal characters twisted by roots of unity
• Factorization of rational six vertex model partition functions
• Weight-preserving bijections between integer partitions and a class of alternating sign trapezoids
• Alternating sign matrices with reflective symmetry and plane partitions: $n+3$ pairs of equivalent statistics and a Cauchy-type identity
• Elliptic free-fermion model with OS boundary and elliptic Pfaffians