Alperin Weight Conjecture

Alperin's weight conjecture is a counting statement in modular representation theory. For a finite group $G$ and a prime $p$, a $p$-weight is a pair $(Q,\phi)$ with $Q$ a $p$-subgroup of $G$ and $\phi$ an irreducible character of $N_G(Q)/Q$ of $p$-defect zero. The conjecture, put forward by Alperin in the mid-1980s, asserts that the number of simple $kG$-modules in characteristic $p$ equals the number of $G$-conjugacy classes of weights [Alperin1987Weights]. A global invariant of the group algebra would thus be determined entirely by $p$-local data.

The modern approach runs through reduction theorems: the conjecture has been reduced to inductive conditions to be verified for the finite simple groups [Navarro2018Characters], placing it in the same local–global programme as the McKay conjecture [Spath2013AWCReduction]. These conditions have been checked in many families of simple groups, and recent work refines the reduction via character triples and the Glauberman correspondence [Rossi2026AWC].

The verification across all simple groups is incomplete, and the conjecture — along with its blockwise and Galois-equivariant refinements — remains open.