  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.
