EL-labelings, Supersolvability and 0-Hecke Algebra Actions on Posets

We show that a finite graded lattice of rank n is supersolvable if and only if it has an EL-labeling where the labels along any maximal chain form a permutation. We call such a labeling an S_n EL-labeling and we consider finite graded posets of rank n with unique top and bottom elements that have an S_n EL-labeling. We describe a type A 0-Hecke algebra action on the maximal chains of such posets. This action is local and gives a representation of these Hecke algebras whose character has characteristic that is closely related to Ehrenborg's flag quasi-symmetric function. We ask what other classes of posets have such an action and in particular we show that finite graded lattices of rank n have such an action if and only if they have an S_n EL-labeling.Comment: 18 pages, 8 figures. Added JCTA reference and included some minor corrections suggested by refere

Graded left modular lattices are supersolvable

We provide a direct proof that a finite graded lattice with a maximal chain of left modular elements is supersolvable. This result was first established via a detour through EL-labellings in [McNamara-Thomas] by combining results of McNamara and Liu. As part of our proof, we show that the maximum graded quotient of the free product of a chain and a single-element lattice is finite and distributive.Comment: 7 pages; 2 figures. Version 2: typos and a small error corrected; diagrams prettier; exposition improved following referee's suggestions; version to appear in Algebra Universali

A new subgroup lattice characterization of finite solvable groups

We show that if G is a finite group then no chain of modular elements in its subgroup lattice L(G) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L(G) has length at least two more than that of the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L(G) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.Comment: 15 pages; v2 has minor changes for publication; v3 minor typos fixe

On Kiselman quotients of 0-Hecke monoids

Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify these monoids up to isomorphism, determine their idempotents and show that they are $\mathcal{J}$-trivial. For type $A$ we show that Catalan numbers appear as the maximal cardinality of our monoids, in which case the corresponding monoid is isomorphic to the monoid of all order-preserving and order-decreasing total transformations on a finite chain. We construct various representations of these monoids by matrices, total transformations and binary relations. Motivated by these results, with a mixed graph we associate a monoid, which we call a Hecke-Kiselman monoid, and classify such monoids up to isomorphism. Both Kiselman semigroups and Kiselman quotients of 0-Hecke monoids are natural examples of Hecke-Kiselman monoids.Comment: 14 pages; International Electronic Journal of Algebra, 201

Omitting parentheses from the cyclic notation

The purpose of this article is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form a bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes.Comment: new results adde

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

On Bruhat posets associated to compositions

The purpose of this work is to initiate a combinatorial study of the Bruhat-Chevalley ordering on certain sets of permutations obtained by omitting the parentheses from their standard cyclic notation. In particular, we show that these sets form bounded, graded, unimodal, rank-symmetric and EL-shellable posets. Moreover, we determine the homotopy types of the associated order complexes