Finite convergent presentations of plactic monoids for semisimple lie algebras

We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties

Artin group injection in the Hecke algebra for right-angled groups

For any Coxeter system we consider the algebra generated by the projections over the parabolic quotients. In the finite case it turn out that this algebra is isomorphic to the monoid algebra of the Coxeter monoid (0-Hecke algebra). In the infinite case it contains the Coxeter monoid algebra as a proper subalgebra. This construction provides a faithful integral representation of the Coxeter monoid algebra of any Coxeter system. As an application we will prove that a right-angled Artin group injects in Hecke algebra of the corresponding right-angled Coxeter group

Betti numbers of toric varieties and eulerian polynomials

It is well-known that the Eulerian polynomials, which count permutations in $S_n$ by their number of descents, give the $h$-polynomial/$h$-vector of the simple polytopes known as permutohedra, the convex hull of the $S_n$-orbit for a generic weight in the weight lattice of $S_n$. Therefore the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this paper we derive recurrences for the $h$-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a non-generic weight, namely a weight fixed by only the simple reflections $J=\{s_{n},s_{n-1},s_{n-2} \cdots,s_{n-k+2},s_{n-k+1}\}$ for some $k$ with respect to the $A_n$ root lattice. Furthermore, they give rise to certain rationally smooth toric varieties $X(J)$ that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincar\'e polynomial of $X(J)$ in terms of the Eulerian polynomials

Filtrations and Distortion in Infinite-Dimensional Algebras

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra

Coloring Complexes and Combinatorial Hopf Monoids

We generalize the notion of coloring complex of a graph to linearized combinatorial Hopf monoids. These are a generalization of the notion of coloring complex of a graph. We determine when a combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the quasisymmetric function invariants associated to the combinatorial Hopf monoid. We show that the collection of all such coloring complexes forms a combinatorial Hopf monoid, which is the terminal object in the category of combinatorial Hopf monoids with convex characters. We also study several examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure

Hopf Algebras in General and in Combinatorial Physics: a practical introduction

This tutorial is intended to give an accessible introduction to Hopf algebras. The mathematical context is that of representation theory, and we also illustrate the structures with examples taken from combinatorics and quantum physics, showing that in this latter case the axioms of Hopf algebra arise naturally. The text contains many exercises, some taken from physics, aimed at expanding and exemplifying the concepts introduced