Alperin weight conjecture

A symmetry can act by a matrix on vectors. In arithmetic modulo two, the matrix that sends (x,y) to (x+y,y) returns every vector to itself when applied twice, yet it preserves the line y=0. An irreducible action has no smaller nonzero proper subspace preserved by all the symmetries. For a finite group, such actions in prime characteristic are organized into families called blocks. Can their number in each block always be counted from local data associated with prime-power subgroups? That is Alperin’s weight conjecture.[1] [2]

Two matrices can obey the same symmetry rule while presenting different patterns of preserved subspaces. In characteristic two, both the identity and the matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ square to the identity; the latter preserves a line and is not a sum of two one-dimensional invariant actions. Modular representation theory studies such actions, and its irreducible ones are those with no nonzero proper invariant subspace. The conjecture asks whether their number in each block can always be counted through subgroup normalizers.

For a finite group G and prime p, $\operatorname{IBr}(B)$ lists the irreducible Brauer characters of block B. Use the Brauer-character convention of Martínez–Rizo–Rossi: $Q=O_p(N_G(Q))$, meaning Q is the largest normal p-subgroup of its normalizer. The character $\psi\in\operatorname{IBr}(N_G(Q))$ comes from a defect-zero character of $N_G(Q)/Q$, and it is a B-weight when $\operatorname{bl}(\psi)^G=B$. Write $\operatorname{Alp}(B)/G$ for their G-conjugacy classes. The target is $|\operatorname{IBr}(B)|=|\operatorname{Alp}(B)/G|$ for every G, p and B.[1] [2]

The count is proved in many families. A general method reduces work to strong conditions on finite simple groups, but those conditions still need verification. A reduction is a route to a solution, not the solution itself.[1] [2]

The strong inductive blockwise condition has additional automorphism and character-triple compatibility requirements. Proving its implication for arbitrary groups does not establish every simple-group input. Huang–Yılmaz’s equivalent global formulations and conditional descent consequences similarly leave the universal equality conjectural; refined versions are not extra counted pages.[1] [2]

Prime-characteristic arithmetic changes how symmetry actions break into their basic constituents: the two-step action above already has a preserved line without being a diagonal action. Local data would offer a way to count those constituents without recovering every global action separately. This connection between a group and its subgroup symmetries is the intrinsic reason the equality matters; it does not reconstruct all the actions or their interactions.[1] [2]

The normalizers preserve selected subgroup structure, while irreducible actions depend on the whole group. Relating them would explain why this local information can control a global blockwise count even when prime-characteristic actions have the nontrivial subspace behavior illustrated above. Automorphism and Galois refinements probe which correspondences survive additional symmetry or changes of coefficients. A numerical equality alone does not give an equivalence of entire module or derived categories; those compare substantially more information under additional hypotheses.[1] [2]