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

Abstract

The longstanding Alperin weight conjecture and its blockwise version have been reduced to simple groups recently by Navarro, Tiep, Späth 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 Sp 2n (q) and odd non-defining characteristics. If the decomposition matrix of Sp 2n (q) is unitriangular with respect to an Aut(Sp 2n (q))-stable basic set (this assumption holds for linear primes), we can establish an equivariant bijection between the irreducible Brauer characters and weights. , then (R, ϕ) is called a B-weight. Denote the set of all G-conjugacy classes of B-weights by W(B). In the sequel, the term "weight" will mean a single weight or a conjugacy class of weights depending on the context. J. L. Alperin gives the following conjecture in [1] called blockwise Alperin weight (BAW) conjecture.Conjecture (Alperin). Let G be a finite group, ℓ a prime and B an ℓ-block of G,This conjecture has been reduced to the simple groups, which means that the BAW conjecture holds for a finite group G if all non-abelian simple groups involved in G satisfy the inductive blockwise Alperin weight (iBAW) condition. See [31] for a definition of the iBAW condition and see [19] for another version. Basically speaking, the iBAW condition consists of two parts: the first one requires to establish an equivariant bijection between irreducible Brauer characters and weights; the second one (normally embedded conditions) requires to consider the extension of irreducible Brauer characters and weight characters. This inductive condition has been verified for several cases, such as: simple alternate groups, many sporadic simple groups, simple groups of Lie type and the defining characteristic, some few cases of simple groups of Lie type and primes different from the defining characteristic; see [7], [10], [13], [19], [21], [24], [28], [29], [31], etc.The purpose of this paper is to start the consideration of the iBAW condition for PSp 2n (q) with q a power of an odd prime p and an odd prime ℓ p. To do this, we first need to establish a blockwise bijection between the irreducible Brauer characters and weights of the universal covering group Sp 2n (q) of PSp 2n (q).Theorem 1. Let p be an odd prime, q = p f and ℓ an odd prime different from p, then the blockwise Alperin weight conjecture holds for Sp 2n (q) for ℓ.In [15], the classification of blocks and the partition of irreducible ordinary characters into blocks for CSp 2n (q) are obtained. Using the results of Fong and Srinivasan in [15] and the results of Lusztig in [20], we can classify the blocks of Sp 2n (q) and partition irreducible ordinary characters into blocks. To give a parametrization of irreducible Brauer characters, we use the basic set E(Sp 2n (q), ℓ...

On the Alperin-McKay conjecture for simple groups of type A

“…For unipotent blocks Feng has established in [Fen18] the inductive BAW condition for unipotent blocks and Li-Zhang have treated in [LZ18] other blocks under additional assumptions on the outer automorphism group. Also Li constructed in [Li18] an equivariant bijection for the inductive BAW condition in symplectic groups under some assumption on the ℓ-modular decomposition matrix. More particular cases of simple groups of small rank were checked in [FLL17,Sch16,SF14].…”

The Inductive Blockwise Alperin Weight Condition for PSp4(q)

“…2.D The version of basic subgroups given above is convenient when one considers the action of automorphisms on weights (see [22]) and the extension problem. Now, we will give another conjugate of the basic subgroup R m,α,γ,c , which is convenient when one considers the inclusion of some Brauer pairs in §3.B.…”

“…For Lusztig symbols, defects and ranks, cores and quotients, degenerate Lusztig symbols, etc., see [29, §5]. We remark that degenerate symbols are counted twice when one parametrizes characters, blocks and weights for conformal symplectic groups (see [17] and §4 in this paper), while degenerate symbols are counted only once when one parametrizes characters, blocks and weights for symplectic groups (see [22]). The two copies of the degenerate symbol λ are denoted as λ and λ ′ .…”

“…The two copies of the degenerate symbol λ are denoted as λ and λ ′ . As in [22], the empty Lusztig symbol is viewed as degenerate, thus a non-degenerate symbol is a fortiori not empty. But in many ocasions, the empty Lusztig symbol will be distinguished from the non-empty ones in this paper; the reason for this is that for conformal symplectic groups, the non-empty degenerate symbols are counted twice while the empty symbol is only counted once; see for example Table 1.…”

On the inductive blockwise Alperin weight condition for classical groups

“…We should mention that Conghui Li has proved independently Theorem 5.16 in [37] with different methods.…”