An equivariant bijection between irreducible Brauer characters and weights for Sp(2n, q)

TL;DR

This paper proves the blockwise Alperin weight conjecture for symplectic groups over finite fields of odd characteristic and constructs an equivariant bijection between irreducible Brauer characters and weights under certain conditions.

Contribution

It establishes the conjecture for Sp(2n, q) in odd non-defining characteristics and constructs an equivariant bijection assuming a unitriangular decomposition matrix.

Findings

Proved the blockwise Alperin weight conjecture for Sp(2n, q).

Constructed an equivariant bijection under specific assumptions.

Confirmed the conjecture for cases with unitriangular decomposition matrices.

Abstract

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Spaeth and Koshitani. Thus, to prove this conjecture, it suffices to verify the corresponding inductive condition for all finite simple groups. The first is to establish an equivariant bijection between irreducible Brauer characters and weights for the universal covering groups of simple groups. Assume q is a power of some odd prime p. We first prove the blockwise Alperin weight conjecture for Sp2n(q) and odd non-defining characteristics. If the decomposition matrix of Sp2n(q) is unitriangular with respect to an Aut(Sp2n(q))-stable basic set (this assumption holds for linear primes), we can establish an equivariant bijection between the irreducible Brauer characters and weights.