Unitriangular Shape of Decomposition Matrices of Unipotent Blocks

Abstract

We show that the decomposition matrix of unipotent ℓ-blocks of a finite reductive group G(F q ) has a unitriangular shape, assuming q is a power of a good prime and ℓ is very good for G. This was conjectured by Geck [23] in 1990. We establish this result by constructing projective modules using a modification of generalised Gelfand-Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka, see [42, 2.4.5].

On modular Harish-Chandra series of finite unitary groups

“…A similar result to Theorem 3.1 should hold for finite classical groups of types B and C. In that case the weak Harish-Chandra branching rule is given by the sl e -crystal on a sum of level 2 Fock spaces [16], but there is a little more work to do as the analogue of [16,Theorem 5.10] identifying cuspidals has not been checked. The validity of any of these results relies on unitriangularity of the decomposition matrix, which was conjectured by Geck for all types and very recently proven in [2].…”

“…. for ι ∈ {0, 1} and thus to Levi subgroups that are of the same Dynkin type 2 A n [29]. However, the group GL e (q 2 ) also admits a (unique) unipotent cuspidal and can appear as a factor in a Levi subgroup.…”

“…In work by Gerber [27], the sl ∞ -crystal on level 2 Fock spaces was called the Heisenberg crystal because of its relationship with the action of the infinite-dimensional "Heisenberg Lie algebra." 2 We warn the reader that our preferred nomenclature of sl ∞ -crystal is a linguistic misdemeanor because we are only using "crystal" to mean "a simple directed graph carrying representation theoretic meaning which happens to be isomorphic as a graph to the crystal graph of a Lie algebra." We do not find it using any quantum group or limit as q goes to 0.…”

“…and call |λ| the size of λ. Let P denote the set of all partitions, including the empty partition ∅ and let P(n) denote the set of all partitions of size n. A bipartition (λ 1 , λ 2 ) is an ordered pair of partitions λ 1 , λ 2 ∈ P. We denote by P (2) the set of all bipartitions and by P (2) (n) the set of all bipartitions of size n, meaning that…”

“…In upgrading the definition of content to bipartitions there is some flexibility, as we might want to fix some notion of distance between the two partitions λ 1 and λ 2 ; this is done by shifting the contents of the boxes in the two partitions using a "charge." A charged bipartition is a pair |λ, s where λ = (λ 1 , λ 2 ) ∈ P (2) and s…”