Related papers: Cells in quantum affine sl_n
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
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…
As a generalisation of Graham and Lehrer's cellular algebras, affine cellular algebras have been introduced in [12] in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley-Lieb…
We study the properties of the extended graphical calculus for categorified quantum $sl(n)$. The main results include proofs of Reidemeister 2 and Reidemeister 3-like moves involving strands corresponding to arbitrary thicknesses and…
Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic…
In \cite{FT19}, Finkelberg and Tsymbaliuk introduced the notion of shifted quantum affine algebras and described their role in the study of quantized Coulomb branches associated to certain 3D $N = 4$ quiver gauge theories. We describe a new…
In 1979, Kazhdan and Lusztig introduced the notion of "cells" (left, right and two-sided) for a Coxeter group $W$, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite…
We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an…
We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a…
We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new…
We define Kazhdan-Lusztig bases and study asymptotic forms for affine $q$-Schur algebras following Du and McGerty. We will show that the analogues of Lusztig's conjectures for Hecke algebras with unequal parameters hold for affine $q$-Schur…
In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Brou\'e,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups.…
We study factorization algebras on configuration spaces of points on the curved, colored by elements of the root lattice. We show that the factorization algebra attached to Lusztig's quantum group can be obtained as a direct image of a…
The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking.…
A quantum Frobenius map a la Lusztig for $\mathfrak{sl}_2$ is categorified at a prime root of unity.
We study the Hopf algebra structure of Lusztig's quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build…
We give a description of a certain induced module for a quantum group of type $A$. Together with our previous results this gives a proof of Lusztig's conjectural multiplicity formula for non-restricted modules over the De Concini-Kac type…
We study the explicit formula of Lusztig's integral forms of the level one quantum affine algebra $U_q(\hat{sl}_2)$ in the endomorphism ring of symmetric functions in infinitely many variables tensored with the group algebra of $\mathbb Z$.…
In 1983, Lusztig defined a map $\sigma$ from affine permutations of $n$ to partitions of $n$. He conjectured that for any partition $\lambda$ of $n$, $\sigma^{-1}(\lambda)$ is a two-sided cell. Shi, in 1986, proved part of this conjecture.…
The goal of this work is to provide an elementary construction of the canonical basis $\mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztig's symmetries. To that effect, we obtain a…