Lumpings of Algebraic Markov Chains Arise from Subquotients

Abstract

A function on the state space of a Markov chain is a "lumping" if observing only the function values gives a Markov chain. We give very general conditions for lumpings of a large class of algebraically-defined Markov chains, which include random walks on groups and other common constructions. We specialise these criteria to the case of descent operator chains from combinatorial Hopf algebras, and, as an example, construct a "top-to-random-with-standardisation" chain on permutations that lumps to a popular restriction-then-induction chain on partitions, using the fact that the algebra of symmetric functions is a subquotient of the Malvenuto-Reutenauer algebra.2 Part I: General Theory 2.1 Matrix notation Given a matrix A , let A(x, y) denote its entry in row x, column y, and write A T for the transpose of A.Let V be a vector space (over R) with basis B, and T : V → V be a linear map. Write [T] B for the matrix of T with respect to B . In other words, the entries of [T] B satisfy T(x) = y∈B [T] B (y, x)y for each x ∈ B. 1 (The sections have both custom numbering and standard numerical numbering, to be consistent with the journal version.)

Markov Chains from Descent Operators on Combinatorial Hopf Algebras

“…Remarkably, the above formula also gives the distribution of the card of smallest value after t top-to-random shuffles of [AD86] -this is because the probability distribution of a deck under top-to-random shuffling, after t steps starting at the identity permutation, agrees with those of the to-do list chain [Pan18,Th. 3.14].…”

“…Because the Littlewood-Richardson coefficients involved in this case are particularly simple, this chain has a neat interpretation in terms of partition diagrams: remove a random box using the hook walk of [GNW79], then add a random box according to the complementary hook walk of [GNW84]. See [Pan18,Sec. 3.2.1] for a detailed derivation.…”

“…So the new to-do list chain opens up a whole new viewpoint to study the top-torandom shuffle. [Pan18,Th. 3.11] also relates the to-do list chain to the restriction-then-induction chains of Fulman [Ful04] (see below), namely that observing the RSK shape of the to-do list chain gives the restriction-then-induction chain.…”

Projections of the Aldous chain on binary trees: Intertwining and consistency

“…Our approach combines two well-known criteria for functions of Markov chains to be Markov: intertwining, which arises elsewhere in the down-up literature [5], and the Kemeny-Snell criterion [19], also known as Dynkin's criterion. See also [23]. The uniform n-tree down-up chain (T (j)) j≥0 is intertwined on top of an intermediate Markov chain (ρ 3, middle panel), which then relates to (ρ…”

Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

“…The Doob transform is a classical tool in Markov chain theory [37,Chapter 8]. For many applications and a literature review see [48].…”