A Random Walk on the Indecomposable Summands of Tensor Products of Modular Representations of SL2left ({mathbb {F}_p}right )

Eoghan Mcdowell

doi:10.1007/s10468-021-10034-0

Eoghan Mcdowell

Open Access

https://doi.org/10.1007/s10468-021-10034-0

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Abstract

In this paper we introduce a novel family of Markov chains on the simple representations of SL2left ({mathbb {F}_p}right ) 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 SL2left ({mathbb {F}_p}right ), emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2left ({mathbb {F}_p}right )-representations.

Highlights

This paper introduces a novel family of Markov chains on the simple representations of SL2 Fp in defining characteristic
The chains are defined by, beginning at some simple module, tensoring by a fixed simple module and choosing a non-projective indecomposable summand with probability depending on a weighting given to each indecomposable module
We will see that the states are precisely the non-projective simple representations. The study of this random walk is inspired by Benkart–Diaconis–Liebeck–Tiep [3], who consider chains defined by choosing composition factors of tensor products

Summary

Introduction

This paper introduces a novel family of Markov chains on the simple representations of SL2 Fp in defining characteristic. We will see that the states are precisely the non-projective simple representations The study of this random walk is inspired by Benkart–Diaconis–Liebeck–Tiep [3], who consider chains defined by choosing composition factors of tensor products. If we were to replace SL2 Fp with an algebraic group SL2(k), this “reduced” tensor product would be the tensor product of a so-called fusion category of the kind studied in [2, Section 2], and our random walk can be described in terms of fusion rules In this setting, the useful symmetries of the table of multiplicities are given by [9, Axiom 3, p. In order to compute the transitions of our random walk, we require knowledge of the decomposition of tensor products of simple modules into indecomposable modules Such a decomposition is known as a Clebsch–Gordan rule; the statement of the rule in our case is given in Theorem 3.7.

Simple Representations

Clebsch–Gordan Rule

Tables of Multiplicities

Random Walks on Indecomposable Modules