Frobenius extensions and the exotic nilCoxeter algebra for $G(m,m,3)$

TL;DR

This paper introduces the exotic nilCoxeter algebra associated with the complex reflection group G(m,m,3), exploring its structure, new relations, and Frobenius extension properties in the context of q-deformed reflection representations.

Contribution

It defines a new algebra called the exotic nilCoxeter algebra for G(m,m,3), proves a new relation called the roundabout relation, and studies its Frobenius extension properties.

Findings

The exotic nilCoxeter algebra is finite-dimensional with surprising features.

A new relation called the roundabout relation is established.

The Frobenius trace is identified within the algebra for G(m,m,3).

Abstract

In a previous paper of the first author, the type A affine Cartan matrix was q-deformed to produce a deformation of the reflection representation of the affine Weyl group. This deformation plays a role in the quantum geometric Satake equivalence. In this paper we introduce the study of q-deformed divided difference operators. When q is specialized to a primitive 2m-th root of unity, this affine reflection representation factors through a quotient, the complex reflection group $G(m,m,n)$. The divided difference operators now generate a finite-dimensional algebra we call the exotic nilCoxeter algebra. This algebra is new and has surprising features. In addition to the usual braid relations, we prove a new relation called the roundabout relation. A classic result of Demazure, for Weyl groups, states that the polynomial ring of the reflection representation is a Frobenius extension over its subring of invariant polynomials, and describes how the Frobenius trace can be constructed within the nilCoxeter algebra. We study the analogous Frobenius extension for $G(m,m,n)$, and identify the Frobenius trace within the exotic nilCoxeter algebra for $G(m,m,3)$.