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Tensor product Markov chains. (English) Zbl 1467.60054

The authors 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 the use of eigenvalues/eigenvectors allows them to give sharp rates of convergence for the associated random walks. For modular representations, the chains are not reversible, and the analysis becomes intricate. In the quantum group case, the chains fail to be diagonalizable, but a novel analysis using generalized eigenvectors proves successful. Section 2 gives a review on the available literature; Section 3 contains some basic results for tensor product Markov chains in the modular case and gives sharp rates of convergence for the groups \(\mathrm{SL}_2(p)\) with respect to tensoring with the natural two-dimensional module and with the Steinberg module. Section 4 treats \(\mathrm{SL}_2(p^2)\), Section 5 features \(\mathrm{SL}_2(2^n)\), and Section 6 considers \(\mathrm{SL}_3(p)\). Section 7 examines the case of quantum \(\mathrm{SL}_2\) at a root of unity. Two appendices provide introductory information about Markov chains and modular representations.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
20C20 Modular representations and characters
20G42 Quantum groups (quantized function algebras) and their representations
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