Corpus ID: 140219540

# Tensor products of modular representations of $\operatorname{SL}_2(\mathbb{F}_p)$ and a random walk on their indecomposable summands

@article{McDowell2019TensorPO,
title={Tensor products of modular representations of \$\operatorname\{SL\}\_2(\mathbb\{F\}\_p)\$ and a random walk on their indecomposable summands},
author={Eoghan McDowell},
journal={arXiv: Representation Theory},
year={2019}
}
• Eoghan McDowell
• Published 30 April 2019
• Mathematics
• arXiv: Representation Theory
In this paper we give a novel, concise and elementary proof of the decomposition of tensor products of simple modular $\operatorname{SL}_2(\mathbb{F}_p)$-representations. This result is used to decompose tensor products involving their projective covers and to decompose symmetric squares. We define a Markov chain on the simple modular $\operatorname{SL}_2(\mathbb{F}_p)$-representations via tensoring with a fixed simple module and choosing an indecomposable summand according to a specified… Expand
2 Citations

#### Figures from this paper

Plethysms of symmetric functions and representations of $\mathrm{SL}_2(\mathbb{C})$
• Mathematics
• 2019
Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there isExpand
Plethysms of symmetric functions and representations of SL 2 (C)
• Mathematics
• 2021
Let ∇λ denote the Schur functor labelled by the partition λ and let E be the natural representation of SL2(C). We make a systematic study of when there is an isomorphism ∇λSymE ∼= ∇μSymE ofExpand

#### References

SHOWING 1-10 OF 12 REFERENCES
Tensor Product Markov Chains.
• Mathematics
• 2018
We analyze families of Markov chains that arise from decomposing tensor products of irreducible representations. This illuminates the Burnside-Brauer Theorem for building irreducible representations,Expand
A study of certain modular representations
Let p be a prime number, F the field of p elements, M the semigroup of all 2 x 2 matrices over F , G the group GL(2, p) of invertible elements of M , and S the normal subgroup SL(2, p) of GExpand
Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Preface Part I General Theory 1 Matrix Lie Groups 1.1 Definition of a Matrix Lie Group 1.2 Examples of Matrix Lie Groups 1.3 Compactness 1.4 Connectedness 1.5 Simple Connectedness 1.6 HomomorphismsExpand
The λstructure of the green ring of GL(2, (F p) in characteristic P, III
In this paper the reader is assumed to have taken notice of [I]. In [III] 1 we described the \$lambda;, and s-, structure of the Green ring of GL(2,F p), and Sl(2,F p). We shall now construct aExpand
Markov chains and mixing times
For our purposes, a Markov chain is a (finite or countable) collection of states S and transition probabilities pij, where i, j ∈ S. We write P = [pij] for the matrix of transition probabilities.Expand
Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups
Preface Part I. Semi-Simple Modules: Part II. Projective Modules: Part III. Modules and Subgroups: Part IV. Blocks: Part V. Cyclic Blocks.
The algebra of reversible Markov chains
• Mathematics
• 2010
For a Markov chain, both the detailed balance condition and the cycle Kolmogorov condition are algebraic binomials. This remark suggests to study reversible Markov chains with the tool of AlgebraicExpand