Alperin weight conjecture

This article is about a conjecture in the following area in/related to group theory: representation theory. View all conjectures and open problems

This is an equal cardinalities conjecture, viz a conjecture that the cardinalities of two sets, obtained in different but possibly related ways, are equal

This conjecture was made by: Alperin

This conjecture is believed to be true

Statement

Let $G$ be a finite group, $p$ a prime, and $B$ a $p$-block (viz an indecomposable summand of the group algebra of $G$ over the algebraic closure of $F_p$). Then, the following two numbers are equal:

• The number of irreducible modular characters with block $B$
• The number of $B$-weights

Relation with other conjectures

Here is a list of closely related conjectures:

• Alperin-McKay conjecture
• Dade invariant conjecture

Progress towards the conjecture

Symmetric groups and general linear groups

Alperin and Fong, in their paper Weights for symmetric and general linear groups, proved the conjecture for the symmetric groups and for the general linear groups.

Covering groups

Michler and Olsson have verified the conjecture for covering groups of symmetric and alternating groups, in their paper Weights for covering groups of symmetric and alternating groups.

References

• Weights for symmetric and general linear groups by Alperin and Fong, Journal of Algebra, 131, No. 1, 2-22
• Weights for covering groups of symmetric and alternating groups by Michler and Olsson