We prove that every finite-dimensional representation of a finite group over field characteristic p admits resolution by p-permutation modules. The proof involves reformulation in terms derived categories.

We present a succinct data structure for permutation graphs, and their superclass of circular i.e., structures using optimal space up to lower order terms. Unlike concurrent work on circle graphs (Acan et al. 2022), our also supports distance shortest-path queries, as well adjacency neighborhood all in time. particular the first exact oracle (circular) graphs. A second degree queries time indep...

Quantum theory stipulates that if two particles are identical in all physical aspects, the allowed states describing system either symmetric or antisymmetric with respect to permutations of single-particle states1–5. Experimentally, symmetry can be inferred indirectly from fact neglecting correct exchange theoretical analysis leads dramatic discrepancies observations6–13. The only way directly ...