Representations of Symmetric groups $S_n$ can be considered as homomorphisms to the orthogonal group $\mathrm{O}(d,\mathbb{R})$, where $d$ is the degree of the representation. If the determinant of the representation is trivial, we call it achiral. In this case, its image lies in the special orthogonal group $\mathrm{SO}(V)$. It is called chiral otherwise. The group $\mathrm{O}(V)$ has a non-trivial topological double cover $\mathrm{Pin}(V)$. We say the representation is spinorial if it lifts to $\mathrm{Pin}(V)$. We obtained a criterion for whether the representation is spinorial in terms of its character. We found similar criteria for orthogonal representations of Alternating groups and products of symmetric groups. One can use these results to count the number of spinorial irreducible representations of $S_n$, which are parametrized by partitions of $n$. We say a partition is spinorial if the corresponding irreducible representation of $S_n$ is spinorial. In this talk, we shall present a summary of these results and count for the number of odd-dimensional, irreducible, achiral, spinorial partitions of $S_n$. We shall also prove that almost all the irreducible representations of $S_n$ are achiral and spinorial. This is joint work with my supervisor Dr. Steven Spallone.

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Last updated: 17 May 2024