arXiv subjects
Publications and source records attributed to Jérémie Guilhot.
The aim of this paper is to give a new explicit construction of Lusztig's asymptotic algebra in affine type $\mathsf{A}$. To do so, we construct a balanced system of cell modules, prove an asymptotic version of the Plancherel Theorem and develop a relative version of the Satake Isomorphism for each two-sided Kazhdan-Lusztig cell.
An oriented graph is said positively multiplicative when its adjacency matrix $A$ embeds in a matrix algebra admitting a basis $\mathsf{B}$ with nonnegative structure constants in which the matrix of the multiplication by $A$ coincides with $A$. The goal of this paper is to present basic properties of this notion and explain, through various simple examples, how it relates to highly non trivial problems like the combinatorial description of fusion rules, the description of the minimal boundary of graded graphs or the study of random walks on alcove tilings.
We introduce the combinatorial model of $J$-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam's Plancherel Theorem.
Let $\mathfrak{g}$ be an untwisted affine Lie algebra with associated Weyl group $W_a$. To any level 0 weight $γ$ we associate a weighted graph $Γ_γ$ that encodes the orbit of $γ$ under the action $W_a$. We show that the graph $Γ_γ$ encodes the periodic orientation of certain subsets of alcoves in $W_a$ and therefore can be interpreted as an automaton determining the reduced expressions in these subsets. Then, by using some relevant quotients of the homology ring of affine Grassmannians, we show that $Γ_γ$ is positively multiplicative. This allows us in particular to compute the structure constants of the homology rings using elementary linear algebra on multiplicative graphs. In another direction, the positivity of $Γ_γ$ yields the key ingredients to study a large class of central random walks on alcoves.
We study a class of commuting Markov kernels whose simplest element describes the movement of $k$ particles on a discrete circle of size $n$ conditioned to not intersect each other. Such Markov kernels are related to the quantum cohomology ring of the Grassmannian, which is an algebraic object counting analytic maps from $\mathbb{P}^1(\mathbb{C})$ to the Grassmannian space of k-dimensional vector subspaces of $\mathbb{C}^n$ with prescribed constraints at some points of $\mathbb{P}^1(\mathbb{C})$. We obtain a Berry-Esseen theorem and a local limit theorem for an arbitrary product of approximately $n^2$ Markov kernels belonging to the above class, when k is fixed. As a byproduct of those results, we derive asymptotic formulas for the quantum cohomology ring of the Grassmannian in terms of the heat kernel on $SU (k)$.
Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}% \subset \mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{% \mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group $W$ of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $% \overline{\mathfrak{g}}$. In this paper we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight $μ$ and $ν$ whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that $μ$ and $ν$ are conjugate under the action of an element of $W$ which is also a Dynkin diagram automorphism of $\overline{\mathfrak{g}% }$ ? We conjecture this is true in general. We prove this conjecture in type $A$ and, for the other root systems, in various situations providing $μ$ and $ν$ satisfy additional hypotheses. Our result can be interpreted as an analogue for branching coefficient of the main result of \cite{Raj} on tensor product multiplicities.
To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$ is associated a partition of $W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^\circ = \{s \in S | L(s)=0\}$, $S^+=\{s \in S | L(s) > 0\}$ and let $C$ be a Kazhdan-Lusztig (left, right or two-sided) $L$-cell. According to the semicontinuity conjecture of the first author, there should exist a positive natural number $m$ such that, for any weight function $L' : W \to \NM$ such that $L(s^+)=L'(s^+) > m L'(s^\circ)$ for all $s^+ \in S^+$ and $s^\circ \in S^\circ$, $C$ is a union of Kazhdan-Lusztig (left, right or two-sided) $L'$-cells. The aim of this paper is to prove this conjecture whenever $(W,S)$ is an affine Weyl group and $C$ is contained in the lowest two-sided $L$-cell.