Quivers of monoids with basic algebras

Int. J. Algebra Comput.

et al. 2015

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We develop a general theory of Markov chains realizable as random walks on R-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by 169 170 A. Ayyer et al.the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop.

Section: Introductionmentioning

confidence: 99%

Markov chains, ${\mathscr R}$-trivial monoids and representation theory

Ayyer

Schilling

Int. J. Algebra Comput.

et al. 2015

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“…A detailed study of the representation theory of J -trivial monoids was undertaken in [12]. More generally, a number of authors have considered the representation of R-trivial monoids, monoids in which each principal right ideal has a unique generator, especially in connection with Markov chains [3,5,19]. Left regular bands are precisely the regular R-trivial monoids.…”

Section: Introductionmentioning

confidence: 99%

Projective indecomposable modules and quivers for monoid algebras

Margolis

Communications in Algebra

2018

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We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, J -trivial and R-trivial monoids and left regular bands of groups.

“…It is a monoid with respect to the operation A · B = {ab : a ∈ A, b ∈ B}. The following statement was observed by several people (in particular, S. Margolis and the second author mention this without proof in [19]):…”

Section: Subset Realizationmentioning

confidence: 90%

Double Catalan monoids

Mazorchuk

2011

J Algebr Comb

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In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.