Tensor product Markov chains

Georgia Benkart, Persi Diaconis, Martin W. Liebeck, Pham Huu Tiep

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)17-83
Number of pages67
JournalJournal of Algebra
Volume561
DOIs
StatePublished - Nov 1 2020

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Keywords

  • Brauer character
  • Markov chain
  • McKay correspondence
  • Modular representation
  • Quantum group
  • Tensor product

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