# Spinorial Representations of Symmetric Groups

## TL;DR

This paper investigates when real representations of symmetric and related groups lift to their double covers, providing criteria based on character values and computing associated topological invariants.

## Contribution

It introduces new criteria for lifting representations to double covers of orthogonal groups and calculates the second Stiefel-Whitney classes for these cases.

## Key findings

- Criteria for lifting to double covers based on character values
- Explicit computation of second Stiefel-Whitney classes for symmetric groups
- Application to products of symmetric groups

## Abstract

A real representation $\pi$ of a finite group may be regarded as a homomorphism to an orthogonal group $\Or(V)$. For symmetric groups $S_n$, alternating groups $A_n$, and products $S_n \times S_{n'}$ of symmetric groups, we give criteria for whether $\pi$ lifts to the double cover $\Pin(V)$ of $\Or(V)$, in terms of character values. From these criteria, we compute the second Stiefel-Whitney classes of these representations.