Markov chains from linear operators and Hopf algebras

Abstract: If you study a linear operator that expands positively in some basis, then your results may be applicable to a Markov chain, whose transition probabilities are given by the matrix of the operator. This is the idea behind the theory of random walks on groups and monoids, where the eigen-data of the operator informs the long-term behaviour of the chain. We point out a lesser-known advantage of this framework: if the linear operator descends to a specific subquotient of its domain, then the corresponding Markov chain admits a projection / lumping. We apply this to a coproduct-then-product operator on Hopf algebras, to explain Jason Fulman's observation regarding the RSK-shape under card-shuffling. I hope this talk will enable and inspire you to explore new examples.

commutative algebracombinatoricsoperator algebrasprobabilityrings and algebras

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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