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Thomas Gobet

Publications and source records attributed to Thomas Gobet.

27 records · Page 2Linked to original sources
arXiv2016

On cycle decompositions in Coxeter groups

The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called parabolic quasi-Coxeter elements of Coxeter groups (in the symmetric group every element is a parabolic quasi-Coxeter element). We show that such an element admits an analogue of the cycle decomposition. Elements which are not in this family still admit a generalized cycle decomposition, but it is not unique in general.

math.GR↗
arXiv2016

Coxeter-Catalan combinatorics and Temperley-Lieb algebras

We introduce bijections between generalized type $A_n$ noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley-Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley-Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on $n+1$ strands. We then show that our bijections come from triangular base changes between the diagram basis and the various bases indexed by noncrossing partitions, by explicitly describing the orders giving triangularity. These orders were introduced in a joint paper with Williams and provide exotic lattice structures on noncrossing partitions. Several combinatorial objects are introduced along the way, including an involution on the set of noncrossing partitions.

math.CO↗
arXiv2016

Dual braid monoids, Mikado braids and positivity in Hecke algebras

We study the rational permutation braids, that is the elements of an Artin-Tits group of spherical type which can be written $x^{-1} y$ where $x$ and $y$ are prefixes of the Garside element of the braid monoid. We give a geometric characterization of these braids in type $A_n$ and $B_n$ and then show that in spherical types different from $D_n$ the simple elements of the dual braid monoid (for arbitrary choice of Coxeter element) embedded in the braid group are rational permutation braids (we conjecture this to hold also in type $D_n$).This property implies positivity properties of the polynomials arising in the linear expansion of their images in the Iwahori-Hecke algebra when expressed in the Kazhdan-Lusztig basis. In type $A_n$, it implies positivity properties of their images in the Temperley-Lieb algebra when expressed in the diagram basis.

math.GR↗
arXiv2016

Twisted filtrations of Soergel bimodules and linear Rouquier complexes

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in case the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin-Tits group. These lifts turn out to be generalizations of mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups.

math.RT↗
arXiv2015

On the Hurwitz action in finite Coxeter groups

We provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element, that is, if and only if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup.

math.GR↗
arXiv2015

Noncrossing partitions, fully commutative elements and bases of the Temperley-Lieb algebra

We introduce a new basis of the Temperley-Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple elements of the Birman-Ko-Lee braid monoid to the Temperley-Lieb algebra. The combinatorics of the new basis involve the Bruhat order restricted to noncrossing partitions. As an application we can derive properties of the coefficients of the base change matrix between Zinno's basis and the well-known diagram or Kazhdan-Lusztig basis of the Temperley-Lieb algebra. In particular, we give closed formulas for some of the coefficients of the expansion of an element of the diagram basis in the Zinno basis.

math.QA↗
arXiv2015

Noncrossing partitions and Bruhat order

We give a criterion for Bruhat order on noncrossing partitions corresponding to the Coxeter element $c=s_1 s_2\cdots s_n$. Using it we prove that the Bruhat order endows noncrossing partitions with a lattice structure. We then explain what happens if we change the Coxeter element; in that case the lattice property fails and we explain which order to consider to get the same lattice structure as for the Coxeter element $c$. In particular we get bijections between noncrossing partitions associated to distinct Coxeter elements, which fix the set of reflections.

math.CO↗
arXiv2015

Noncrossing partitions and Bruhat order

We prove that the restriction of Bruhat order to noncrossing partitions in type $A_n$ for the Coxeter element $c=s_1s_2 ...s_n$ forms a distributive lattice isomorphic to the order ideals of the root poset ordered by inclusion. Motivated by the change-of-basis from the graphical basis of the Temperley-Lieb algebra to the image of the simple elements of the dual braid monoid, we extend this bijection to other Coxeter elements using certain canonical factorizations. In particular, we give new bijections---fixing the set of reflections---between noncrossing partitions associated to distinct Coxeter elements.

math.CO↗