Trivial source character tables of SL2(q)

Trivial source character tables of $\text{SL}_2(q)$, Part II

“…By definition the trivial source character table of the group G at the prime ℓ, denoted Triv ℓ (G), is the species table (or representation table) of the trivial source ring of kG in the sense of Benson and Parker; see [BP84]. However, as in [BFL22], we follow [LP10, Section 4.10] and consider Triv ℓ (G) as the block square matrix defined by the following notational convention.…”

“…We note that the group G acts by conjugation on the pairs (Q v , s) and the values of τ G Qv,s do not depend on the choice of (Q v , s) in its G-orbit. We refer the reader to §2 of our previous paper [BFL22] for details and further properties of trivial source modules and trivial source character tables.…”

“…Therefore, we recall the following essential notions about cyclic blocks. Further details can be found in the first part of our paper [BFL22] and also in [HL21].…”

“…The trivial source character table of G at the prime ℓ, denoted by Triv ℓ (G), is by definition the species table of the trivial source ring of kG in the sense of [BP84]. The present article is a sequel to [BFL22], in which Böhmler and the authors calculate the trivial source character tables for the special linear group SL 2 (q) over the finite field F q when q and ℓ are odd and ℓ ∤ q, and when q is odd, ℓ = 2 and SL 2 (q) has quaternion Sylow 2-subgroups. We refer the reader to the latter article for a complete introduction to trivial source character tables.…”

“…The paper is organised as follows. In Section 2 we recall the notation and definitions for trivial source character tables and for blocks with cyclic defect groups which were established in [BFL22]. Section 3 contains notation and preliminary results on the structure of SL 2 (2 f ).…”