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Jeremie Guilhot

Publications and source records attributed to Jeremie Guilhot.

11 recordsLinked to original sources
arXiv2026

Kazhdan-Lusztig bases of parabolic Hecke algebras and applications to Schur-Weyl duality

With an eye to applications to type A and Schur-Weyl duality, we study Kazhdan-Lusztig bases for a general parabolic Hecke algebra. Parabolic Hecke algebras are idempotent subalgebras of Hecke algebras corresponding to parabolic subgroups, and for type A they coincide with the fused Hecke algebras appearing in a generalisation of the Schur-Weyl duality with the quantum group of GL(N). In this paper we investigate two different Kazhdan-Lusztig bases for parabolic Hecke algebras, together with the associated cells and the corresponding representations. We quickly specialise to type A, for which we describe the cells in terms of the RSK correspondence generalising thus the well-known description for the symmetric group. As a first application we recover the classification of irreducible representations of parabolic Hecke algebras of type A and provide a new construction of these representations. Next we turn to the Schur-Weyl duality and describe the kernel in terms of one the basis studied precedently. Moreover, we formulate some conjectures about a generator of these kernels in terms of Kazhdan-Lusztig basis elements, give some evidence and prove these conjectures in some special cases.

math.RT↗
arXiv2021

Generalised Howe dualityand injectivity of induction: the symplectic case

We study the symplectic Howe duality using two new and independent combinatorial methods: via determinantal formulae on the one hand, and via (bi)crystals on the other hand. The first approach allows us to establish a generalised version where weight multiplicities are replaced by branching coefficients. In turn, this generalised Howe duality is used to prove the injectivity of induction for Levi branchings as previously conjectured by the last two authors.

math.CO↗
arXiv2016

Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group $W\_0$. The set of Weyl characters ${\sf s}\_\la$ forms a basis of the center and Lusztig showed in [Lus15] that these characters act as translations on the Kazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest element of $W\_0$, that is we have $C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}$. As a consequence, the coefficients that appear when decomposing~$C\_{w\_0t\_{\la}}{\sf s}\_\tau$ in the Kazhdan-Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group $W\_0$. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition.

math.RT↗
arXiv2013

Cellularity of the lowest two-sided ideal of an affine Hecke algebra

In this paper we show that the lowest two-sided ideal of an affine Hecke algebra is affine cellular for all choices of parameters. We explicitely describe the cellular basis and we show that the basis elements have a nice decomposition when expressed in the Kazhdan-Lusztig basis. In type $A$ we provide a combinatorial description of this decomposition in term of number of paths.

math.RT↗
arXiv2013

Ordering Families using Lusztig's symbols in type B: the integer case

Let $\Irr(W)$ be the set of irreducible representations of a finite Weyl group $W$. Following an idea from Spaltenstein, Geck has recently introduced a preorder $\leq_L$ on $\Irr(W)$ in connection with the notion of Lusztig families. In a later paper with Iancu, they have shown that in type $B$ (in the asymptotic case and in the equal parameter case) this order coincides with the order on Lusztig symbols as defined by Geck and the second author in \cite{GJ}. In this paper, we show that this caracterisation extends to the so-called integer case, that is when the ratio of the parameters is an integer.

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arXiv2009

Kazhdan-Lusztig cells in the affine Weyl groups of rank 2

In this paper we determine the partition into Kazhdan-Lusztig cells of the affine Weyl groups of type $\tB_{2}$ and $\tG_{2}$ for any choice of parameters. Using these partitions we show that the semicontinuity conjecture of Bonnafé holds for these groups.

math.GR↗
arXiv2009

Some computations about Kazhdan-Lusztig cells in affine Weyl groups of rank 2

In the last section of the paper "Generalized induction of Kazhdan-Lusztig cells" and in "Kazhdan-Lusztig cells in affine Weyl groups of rank 2" the author described the partition into Kazhdan-Lusztig cells of the affine Weyl groups of rank 2 for all choices of parameters. The proof of these results relies on some explicit computations with GAP. In these notes we give some details of these computations.

math.RT↗
arXiv2008

Generalized induction of Kazhdan-Lusztig cells

Following Lusztig, we consider a Coxeter group $W$ together with a weight function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of $W$ which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of $W$ are cells in the whole group, and we decompose the affine Weyl group $\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions.

math.RT↗
arXiv2008

On the lowest two-sided cell in affine Weyl groups

Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most |W_{0}| left cells where W_{0} is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.

math.RT↗
arXiv2007

On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine Weyl group of type $\tilde{G_{2}}$. Using explicit computation with \textsf{CHEVIE}, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. For the proof, we show some equalities on the Kazhdan-Lusztig polynomials which hold for any affine Weyl groups.

math.RT↗