Spin chains and totally symmetric alternating sign matrices

by Christian Walmsley Hagendorf (UCLouvain/IRMP/GPP)

Thursday 16 Apr 2026, 14:00 → 15:00 Europe/Brussels

B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

Alternating sign matrices are central objects in enumerative combinatorics and closely related to the six-vertex model of statistical mechanics. Totally symmetric alternating sign matrices (TSASMs) form a highly constrained symmetry class whose enumeration is only partially understood. In this talk, I describe a connection between TSASM enumeration and a distinguished eigenvector of an integrable lattice model called the open XXZ spin chain. 

This eigenvector arises from a Laurent-polynomial solution of the boundary quantum Knizhnik-Zamolodchikov (bqKZ) equations. It leads to a multivariate Laurent-polynomial generalisation of the sum of the eigenvector's entries, which is uniquely characterised by a set of algebraic properties.

We show that a suitable partition function of a six-vertex model associated with TSASMs satisfies the same properties. By unique characterisation, this partition function coincides with the generalised sum of the eigenvector's entries. As a consequence, TSASM enumeration can be studied using the bqKZ solution. In particular, this connection yields a constant term formula for the number of TSASMs of arbitrary order.