Thomas Gobet
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ABSOLUTE ORDER AND INVOLUTIONS Open
We study the restriction of the absolute order on a Coxeter group W to an interval $[1,w]_T$ , where $w\in W$ is an involution. We characterise and classify those involutions w for which $[1,w]_T$ is a lattice, using the notion of involuti…
Absolute order and involutions Open
We study the restriction of the absolute order on a Coxeter group $W$ to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterize and classify those involutions $w$ for which $[1,w]_T$ is a lattice, using the notion of invol…
Faithful Burau-like representations of some rank two Garside groups and torus knot groups Open
We give a method to produce faithful representations of the groups $G(n,m)=\langle X, Y \ \vert \ X^m = Y^n \rangle$ in $\mathrm{GL}_2(\mathbb{C}[t^{\pm 1}, q^{\pm 1}])$. These groups are Garside groups and the Garside normal forms of elem…
Nonsymmetric Cauchy kernel, crystals and last passage percolation Open
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On maximal dihedral reflection subgroups and generalized noncrossing partitions Open
In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group , every pair of distinct reflections lie in a unique maximal dihedral reflection subgroup of . Our proof only relies on the combinator…
Elements of minimal length and Bruhat order on fixed point cosets of Coxeter groups Open
We study the restriction of the strong Bruhat order on an arbitrary Coxeter group $W$ to cosets $x W_L^θ$, where $x$ is an element of $W$ and $W_L^θ$ the subgroup of fixed points of an automorphism $θ$ of order at most two of a standard pa…
A new Garside structure on torus knot groups and some complex braid groups Open
Several distinct Garside monoids having torus knot groups as groups of fractions are known. For [Formula: see text] two coprime integers, we introduce a new Garside monoid [Formula: see text] having as Garside group the [Formula: see text]…
On maximal dihedral reflection subgroups and generalized noncrossing partitions Open
In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group $W$, every pair $t,t'$ of distinct reflections lie in a unique maximal dihedral reflection subgroup of $W$. Our proof only relies on t…
Odd and even Fibonacci lattices arising from a Garside monoid Open
We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid par…
Non symmetric Cauchy kernel, crystals and last passage percolation Open
We use non-symmetric Cauchy kernel identities to get the law of last passagepercolation models in terms of Demazure characters. The construction is basedon some restrictions of the RSK correspondence that we rephrase in a unifiedway which …
Toric reflection groups Open
Several finite complex reflection groups have a braid group which is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order $k$ for some $k\geq 2$, and meridians ar…
Parametrization, structure and Bruhat order of certain spherical quotients Open
Let be a reductive algebraic group and let be the stabilizer of a nilpotent element of the Lie algebra of . We consider the action of on the flag variety of , and we focus on the case where this action has a finite number of orbits (i.e., …
On torus knot groups and a submonoid of the braid group Open
The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $σ_1$ and $σ_1 σ_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of Garside monoids g…
On torus knot groups and a submonoid of the braid group Open
The submonoid of the $3$-strand braid group $\\mathcal{B}_3$ generated by\n$\\sigma_1$ and $\\sigma_1 \\sigma_2$ is known to yield an exotic Garside\nstructure on $\\mathcal{B}_3$. We introduce and study an infinite family\n$(M_n)_{n\\geq …
Dual Garside structures and Coxeter sortable elements Open
In Artin–Tits groups attached to spherical Coxeter groups, we give a combinatorial formula to express the simple elements of the dual braidmonoids in the classicalArtin generators. Every simple dual braid is obtained by lifting an S -reduc…
Hecke algebras of normalizers of parabolic subgroups Open
In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their defini…
A Soergel-like category for complex reflection groups of rank one Open
We introduce analogues of Soergel bimodules for complex reflection groups of rank one. We give an explicit parametrization of the indecomposable objects of the resulting category and give a presentation of its split Grothendieck ring by ge…
On generalized categories of Soergel bimodules in type $A_2$ Open
In this note, we compute the split Grothendieck ring of a generalized category of Soergel bimodules of type $A_2$, where we take one generator for each reflection. We give a presentation by generators and relations of it and a parametrizat…
Simple dual braids, noncrossing partitions and Mikado braids of type Dn Open
We show that the simple elements of the dual Garside structure of an Artin\ngroup of type $D_n$ are Mikado braids, giving a positive answer to a conjecture\nof Digne and the second author. To this end, we use an embedding of the Artin\ngro…
On cycle decompositions in Coxeter groups Open
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 paraboli…
Noncrossing partitions, fully commutative elements and bases of the Temperley–Lieb algebra Open
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 elemen…