Spinorial representations of symmetric groups

Abstract

Let G be a real compact Lie group, such that G = G 0 ⋊ C 2 , with G 0 simple. Here G 0 is the connected component of G containing the identity and C 2 is the cyclic group of order 2. We give a criterion whether an orthogonal representation π : G → O(V ) lifts to Pin(V ) in terms of the highest weights of π. We also calculate the first and second Stiefel-Whitney classes of the representations of the Orthogonal groups.

Spinoriality of Orthogonal Representations of GL(n,q)

“…Any transposition in S n is conjugate in G to a 1 . According to Theorem 1.1 of [9], we have m π ≡ 0 or 3 mod 4. Therefore π is spinorial.…”

“…This is one of a series of papers investigating the spinoriality question for well-known groups. Please see [12] for criteria when G is a connected reductive Lie group, [8] for orthogonal groups, and [9] for criteria when G is a symmetric or alternating group. This paper covers the group G = GL n (F q ), our foray into finite groups of Lie type.…”

Stiefel-Whitney Classes of Representations of Dihedral Groups

“…The paper [3] of Ganguly and Spallone computed the second SWC to characterize spinorial representations of symmetric groups. This led to a program of calculating the total SWCs of representations in terms of character values for various groups.…”

Total Stiefel Whitney classes for real representations of $\mathrm{GL}_n$ over $\mathbb{F}_q, \mathbb{R}$ and $\mathbb{C}$

“…We also compute the total SWC of an orthogonal representation of GL n (C) and GL n (R). The computation of second SWC for representations of S n and related groups can be found in [GS20]. Explicit expressions for the first and second SWC of the representations of O n (R) were found in [GJ21] in terms of highest weights.…”