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A local-global principle in group representation theory

(Damiano Rossi, Feb. 12, 2025)
In its broader definition, representation theory is the attempt to linearize algebraic structures by studying their actions on vector spaces. The general hope is to recover information about the algebraic structure by answering a (typically) easier question. In the case of (finite) group representation theory, one could tackle this problem one prime at a time. On one hand, we can describe the structure of a group by looking at its p-local structure (given by the set of p-subgroups and their embedding) for each prime p. On the other hand, we can describe the representations of such a group by analyzing their properties at each prime p. The local-global principle in group representation theory asserts that, for each fixed prime p, the p-local structure of a group is directly and intimately linked to the representation theory of the group looked at through the prime p. I will present several fundamental conjectures in the area and explain how these can all be recovered from a unifying statement known as Dade's Conjecture. I will then describe a research program I designed to prove Dade's Conjecture and explain its connections to algebraic topology, homotopy theory, and algebraic geometry.