# Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

## TL;DR

This paper provides an explicit description of genuine projective representations of the symmetric group using Dirac cohomology, connecting representation theory, combinatorics, and algebraic geometry.

## Contribution

It introduces a new explicit model for projective representations via Dirac cohomology and describes Vogan's morphism for Hecke algebras in type A.

## Key findings

- Derived the branching graph for projective representations using Dirac theory.
- Constructed explicit models for projective representations.
- Connected Dirac cohomology with combinatorial and geometric data.

## Abstract

In this paper we will derive an explicit description of the genuine projective representations of the symmetric group $S_n$ using Dirac cohomology and the branching graph for the irreducible genuine projective representations of $S_n$. In 2015 Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of $S_n$ are related to the characters of elliptic graded modules. We derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac cohomology to construct an explicit model for the projective representations. We also described Vogan's morphism for Hecke algebras in type A using spectrum data of the Jucys-Murphy elements.