Martin Gallauer

Some things I learned about permutation representations

Say I pick a field k and a finite group G, and I ask you to exhibit k-linear G-representations. The catch is: I don’t tell you which k nor G that I picked. There is one family you can always exhibit: the permutation representations k(G/H) for subgroups H of G. Arguably it is this universality that makes permutation representations so ubiquitous in mathematics. In this talk I will explain some of the things I learned about them over the last few years, notably in collaboration with Paul Balmer. These facts will be drawn from algebra (modular representation theory), (tensor-triangular) geometry, and combinatorics. Naturally, I’ll also mention things I don’t but would like to know.