A closed formula in the deformed affine nilHecke algebra

TL;DR

This paper derives explicit formulas for Demazure operators acting on a q-deformed affine symmetric group representation, with special cases at roots of unity, advancing understanding in quantum algebra and geometric representation theory.

Contribution

It provides the first explicit closed formula for scalar values of degree -k Demazure operators on monomials in the deformed affine nilHecke algebra.

Findings

Explicit closed formula for Demazure operators when n=3

Simplified formula at roots of unity

Enhanced understanding of q-deformed affine symmetric group representations

Abstract

There is a q-deformation of the reflection representation of the affine symmetric group, which arises in the quantum geometric Satake equivalence, and in the study of the complex reflection groups $G(m,m,n)$. Demazure operators (often called divided difference operators) act on the polynomial ring of this deformed representation. When $n=3$ we prove an explicit closed formula for the scalar one obtains when applying a degree $-k$ Demazure operator to a monomial of degree $k$. We also prove a simpler formula for the scalar obtained after specializing q to a root of unity.