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Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend, Roger E., Fischer, Ilse and Konvalinka, Matjaz 2017. Diagonally and antidiagonally symmetric alternating sign matrices of odd order. Advances in Mathematics 315 , pp. 324-365. 10.1016/j.aim.2017.05.014

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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)x(2n+1) DASASMs is \prod_{i=0}^n (3i)!/(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)x(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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Uncontrolled Keywords: alternating sign matrices; six-vertex model
Publisher: Elsevier
ISSN: 0001-8708
Date of First Compliant Deposit: 28 July 2017
Date of Acceptance: 17 May 2017
Last Modified: 20 Jan 2021 21:43

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