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
Contents
Statement
Let be a finite group, a prime, and a -block (viz an indecomposable summand of the group algebra of over the algebraic closure of ). Then, the following two numbers are equal:
- The number of irreducible modular characters with block
- The number of -weights
Relation with other conjectures
Here is a list of closely related conjectures:
Justification/appeal of the 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