Non-orientable representation spheres

Abstract

For any finite group, we pose the following question: given an irreducible real representation, is the associated representation sphere non-orientable if and only if the representation is nontrivial of real type? We prove that the answer to this question is yes if the group has a normal Sylow 2-subgroup, but exhibit a 2-nilpotent group of order 112 for which the answer is no via elementary arguments. We also link the question to the unit group of the Burnside ring, where we recover a basis discovered by Bouc for 2-groups, and pose a conjecture about detection of non-orientability from solvable subgroups. The question is motivated by issues arising from the construction of permutation twisted cohomology for finite groups.

Disclosure

“rity functions The following GAP code checks whether the parity functions of all irreducible real representations are linearly independent for all groups up to a fixed order, thus providing a proof of Theorem 2.13. This was written with generative AI assistance. We omit groups of odd order and 2-groups, since these cases are already known to be strong nirrorno. ################################################################## ## ## orientation_parity.g”

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Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:22:25.818303Z

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