Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Behrend, Roger E. ORCID: https://orcid.org/0000-0002-6143-7439, 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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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)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:	08 Nov 2023 11:46
URI:	https://orca.cardiff.ac.uk/id/eprint/84214

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