A Random Walk on the Indecomposable Summands of Tensor Products of Modular Representations of SL2 $\left ({\mathbb {F}_p}\right )$

Abstract

In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$

“…Clebsch-Gordan rule. The decomposition of the tensor products of the form Sym i E ⊗ Sym j E with 0 ≤ i, j ≤ p − 1 into indecomposable summands, known as a Clebsch-Gordan rule, has been established, for instance, in [McD22,Theorem 3.7], [Glo78, (5.5) and (6.3)] and [Kou90a, Corollary 1.2(a) and Proposition 1.3(c)]. We will use the following version set in the stable module category of kG.…”

Modular plethystic isomorphisms for two-dimensional linear groups

“…Existing results. We emphasise that while much is known about tensor products of the symmetric powers Sym ℓ E of SL 2 (K) and the related projective and tilting modules when K has prime characteristic (see [EH02], [Kou90a], [McD21]), their modular behaviour under Schur functors is far less studied.…”