Spin invariant theory for the symmetric group

TL;DR

This paper develops a theory of invariants for the spin symmetric group using polynomial and exterior algebras, solving multiplicity problems with Schur Q-functions and providing combinatorial proofs.

Contribution

It introduces a new invariant theory for the spin symmetric group and connects it with Schur Q-functions and shifted q-hook formulas.

Findings

Solved the graded multiplicity problem using Schur Q-functions

Provided a bijective proof for the principal specialization of Schur Q-functions

Established a framework linking spin invariants with algebraic combinatorics

Abstract

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.