# A random walk on the indecomposable summands of tensor products of modular representations of $\mathrm{SL}_2(\mathbb{F}_p)$

## TL;DR

This paper introduces a new family of Markov chains based on tensoring simple representations of (\u200bF_p), analyzes their properties, and links these to the underlying representation theory, providing insights into the structure of tensor products.

## Contribution

It presents a novel Markov chain framework on simple modules of (F_p), establishing their reversibility, connected components, and stationary distributions, with new elementary proofs of tensor product decompositions.

## Key findings

- Chains are reversible and have explicitly determined stationary distributions.
- Connected components of the chains relate to symmetries in tensor products.
- Elementary proof provided for tensor product decomposition of simple representations.

## Abstract

In this paper we introduce a novel family of Markov chains on the simple representations of $\mathrm{SL}_2(\mathbb{F}_p)$ 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 $\mathrm{SL}_2(\mathbb{F}_p)$, emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple $\mathrm{SL}_2(\mathbb{F}_p)$-representations.