Subgroup actions and Brauer induction for equivariant cosheaves

Investigate the actions of subgroups H ≤ G on equivariant cellular cosheaves (G-cosheaves) and characterize the relations between representations and characters under subgroup restriction and Brauer induction in this setting. Determine how irreducible decompositions, character counts, and associated invariants behave when passing from a finite group G to a subgroup H, and under Brauer induction, in the context of the G-cosheaf framework developed for symmetric graphic statics.

Background

The paper introduces equivariant cellular cosheaves and shows how finite group representations act on chain complexes encoding structural data (forces, positions) of symmetric frameworks. It develops character-based tools (including a symmetric Euler characteristic) and decompositions into irreducible components to analyze self-stresses and reciprocal diagrams by symmetry.

While the work focuses on actions of a finite group G, it does not analyze how subgroup actions H ≤ G interact with the cosheaf framework. The authors explicitly note that understanding representation-theoretic relations under subgroup restriction and Brauer induction remains open, pointing toward connections with identities for equivariant coefficient systems (e.g., Boltje, 1994).