Traces on diagram algebras I: Free partition quantum groups, random lattice paths and random walks on trees Article Swipe
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· 2022
· Open Access
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· DOI: https://doi.org/10.1112/jlms.12562
· OA: W3034422231
We classify extremal traces on the seven direct limit algebras of noncrossing\npartitions arising from the classification of free partition quantum groups of\nBanica-Speicher (arXiv:0808.2628) and Weber (arXiv:1201.4723). For the\ninfinite-dimensional Temperley-Lieb-algebra (corresponding to the quantum group\n$O^+_N$) and the Motzkin algebra ($B^+_N$), the classification of extremal\ntraces implies a classification result for well-known types of central random\nlattice paths. For the $2$-Fuss-Catalan algebra ($H_N^+$) we solve the\nclassification problem by computing the \\emph{minimal or exit boundary} (also\nknown as the \\emph{absolute}) for central random walks on the Fibonacci tree,\nthereby solving a probabilistic problem of independent interest, and to our\nknowledge the first such result for a nonhomogeneous tree. In the course of\nthis article, we also discuss the branching graphs for all seven examples of\nfree partition quantum groups, compute those that were not already known, and\nprovide new formulas for the dimensions of their irreducible representations.\n
