What question did this study set out to answer?

The aim is to establish criteria for orthogonal representations to lift to spinorial representations within compact Lie groups.

March 29, 2026Open Access

Spinorial Representations of Orthogonal Groups

Key Points

The aim is to establish criteria for orthogonal representations to lift to spinorial representations within compact Lie groups.
Analyzed the structure of G as a real compact Lie group composed of G0 and C2.
Developed criteria based on the highest weights and character values of representations.
Computed the first and second Stiefel-Whitney classes for the orthogonal group representations.
Identified conditions under which orthogonal representations lift to Pin(V).
Computed explicit forms of the first and second Stiefel-Whitney classes of the representations.
Demonstrated a relationship between highest weights and liftability of representations.

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 criteria for whether an orthogonal representation : G O(V ) lifts to Pin(V ) in terms of the highest weights of and also in terms of character values.From these criteria we compute the first and second Stiefel-Whitney classes of the representations of the orthogonal groups.

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