Coinvariant algebras and fake degrees for spin Weyl groups of classical type

TL;DR

This paper introduces a spin coinvariant algebra for Weyl groups, computes the spin fake degrees for classical types, and extends the concept of fake degrees to the spin setting, enriching the representation theory of Weyl groups.

Contribution

It formulates the notion of spin coinvariant algebra for all Weyl groups and computes the spin fake degrees for classical types, a novel extension of classical fake degree theory.

Findings

Computed all spin fake degrees for classical Weyl groups.

Established the concept of spin coinvariant algebra for Weyl groups.

Extended fake degree computations to the spin setting.

Abstract

The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon-Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exceptional Weyl groups are given in a sequel.