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

Cohomology of idempotent braidings, with applications to factorizable monoids

We develop new methods for computing the Hochschild (co)homology of monoids which can be presented as the structure monoids of idempotent set-theoretic solutions to the Yang--Baxter equation. These include free and symmetric monoids; factorizable monoids, for which we find a generalization of the K{\"u}nneth formula for direct products; and plactic monoids. Our key result is an identification of the (co)homologies in question with those of the underlying YBE solutions, via the explicit quantum symmetrizer map. This partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We also obtain new structural results on the (co)homology of general YBE solutions

Growth alternative for Hecke-Kiselman monoids

The Gelfand-Kirillov dimension of Hecke-Kiselman algebras defined by oriented graphs is studied. It is shown that the dimension is infinite if and only if the underlying graph contains two cycles connected by an (oriented) path. Moreover, in this case, the Hecke-Kiselman monoid contains a free noncommutative submonoid. The dimension is finite if and only if the monoid algebra satisfies a polynomial identity

Littlewood-Richardson coefficients for reflection groups

In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies $a_{ij}a_{ji}\ge 4$ for all $i,j$, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott-Samelson varieties Gamma_\ii(G).Comment: 51 pages, AMSLaTeX, typos correcte

Factorable Monoids : Resolutions and Homology via Discrete Morse Theory

We study groups and monoids that are equipped with an extra structure called factorability. A factorable group can be thought of as a group G together with the choice of a generating set S and a particularly well-behaved normal form map G → S*, where S* denotes the free group over S. This is related to the theory of complete rewriting systems, collapsing schemes and discrete Morse theory. Given a factorable monoid M, we construct new resolutions of Z over the monoid ring ZM. These resolutions are often considerably smaller than the bar resolution E*M. As an example, we show that a large class of generalized Thompson groups and monoids fits into the framework of factorability and compute their homology groups. In particular, we provide a purely combinatorial way of computing the homology of Thompson's group F