Half-turn symmetric alternating sign matrices and Tokuyama type factorisation for orthogonal group characters

TL;DR

This paper derives a new Tokuyama type factorisation formula for orthogonal group characters using weighted half-turn symmetric alternating sign matrices, extending previous deformations of Weyl's denominator formula to this root system.

Contribution

It introduces a novel orthogonal group version of Tokuyama's factorisation formula based on combinatorial enumeration of HTSASMs and deformations of Weyl's denominator formula.

Findings

Derived a new orthogonal group Tokuyama formula

Connected HTSASMs with Weyl denominator deformations

Extended previous root system deformations to orthogonal groups

Abstract

Half turn symmetric alternating sign matrices (HTSASMs) are special variations of the well-known alternating sign matrices which have a long and fascinating history. HTSASMs are interesting combinatorial objects in their own right and have been the focus of recent study. Here we explore counting weighted HTSASMs with respect to a number of statistics to derive an orthogonal group version of Tokuyama's factorisation formula, which involves a deformation and expansion of Weyl's denominator formula multiplied by a general linear group character. Deformations of Weyl's original denominator formula to other root systems have been discovered by Okada and Simpson, and it is thus natural to ask for versions of Tokuyama's factorisation formula involving other root systems. Here we obtain such a formula involving a deformation of Weyl's denominator formula for the orthogonal group multiplied by a deformation of an orthogonal group character.