Tensor Product Markov Chains

TL;DR

This paper studies Markov chains derived from tensor product decompositions of representations, providing explicit diagonalization, convergence rates, and novel analysis techniques for both classical and quantum cases.

Contribution

It introduces a comprehensive analysis of tensor product Markov chains, including non-reversible and non-diagonalizable cases, with applications to representation theory and quantum groups.

Findings

Chains are explicitly diagonalizable with sharp convergence rates.

Chains for modular representations are non-reversible and analytically complex.

Quantum group chains are non-diagonalizable but analyzable via generalized eigenvectors.

Abstract

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, the McKay Correspondence, and Pitman's 2M-X Theorem. The chains are explicitly diagonalizable, and we use the eigenvalues/eigenvectors to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analytical details are surprisingly intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful.