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.

中文翻译：

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张量积马尔可夫链

我们分析了由分解不可约表示的张量积而产生的马尔可夫链族。这阐明了用于构建不可约表示的 Burnside-Brauer 定理、McKay 对应关系和 Pitman 的 2M-X 定理。这些链是明确可对角化的，我们使用特征值/特征向量来给出相关随机游走的收敛速度。对于模块化表示，链是不可逆的，分析细节出奇地复杂。在量子群的情况下，链不能对角化，但使用广义特征向量的新分析证明是成功的。