Lumpings of algebraic Markov chains arise from subquotients (Q2330411)

This paper studies lumping of Markov chains that arise from linear operators acting on a finite-dimensional vector space \(V\). More precisely, the author studies Markov chains that evolve on a fixed basis of \(V\), whose transition kernel corresponds to the matrix representing a linear operator \(T\colon V\to V\) with respect to the chosen basis. Examples of such processes include top-to-random and random transposition card shuffling, flipping random bits on binary words of fixed length, and random walks on groups. The main result of the paper gives algebraic conditions on \(T\), under which the image of the Markov process is itself again a Markov process. These conditions are of two types: (a) \(T\) descends to a well-defined map on some quotient space and the lumping map is the canonical projection (strong lumping), or (b) there is a subspace \(V'\) of \(V\), spanned by a subset of the basis, which is left invariant by \(T\), and the lumping map is the projection onto \(V'\). Finally the results are extend to the more general setting of descent operators on combinatorial Hopf algebras, building on previous work by the author [``Markov chains from descent operators on combinatorial Hopf algebras'', Preprint, \url{arXiv:1609.04312}]. These extended results are applied to the study of the probability distribution of the RSK shape [\textit{R. P. Stanley}, Enumerative combinatorics. Volume 2. Cambridge: Cambridge University Press (1999; Zbl 0928.05001)] and lumping of riffle-shuffles via descent [\textit{C. A. Athanasiadis} and \textit{P. Diaconis}, Adv. Appl. Math. 45, No. 3, 410--437 (2010; Zbl 1239.60071)].