The purpose of this paper is to describe a general procedure for computing analogues of Young's seminormal representations of the symmetric groups. The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The combinatorics of these elements allows one to compute irreducible representations explicitly and often very easily. In this paper we do these computations for Weyl groups and Iwahori-Hecke algebras of types $A_n$, $B_n$, $D_n$, $G_2$. Although these computations are in reach for types $F_4$, $E_6$, and $E_7$, we shall, in view of the length of the current paper, postpone this to another work.
This paper studies the seminormal bases $\{f_{\mathfrak{s}\mathfrak{t}}\}$ and the dual seminormal bases $\{g_{\mathfrak{s}\mathfrak{t}}\}$ of the non-degenerate and the degenerate cyclotomic Hecke algebras ${H}_{\ell,n}$ of type $G(\ell,1,n)$. We present some explicit formulae for the constants $α_{\mathfrak{s}\mathfrak{t}}:=g_{\mathfrak{s}\mathfrak{t}}/f_{\mathfrak{s}\mathfrak{t}}\in K^\times$ in terms of the $γ$-coefficients $\{γ_{\mathfrak{u}}, γ'_{\mathfrak{u}}\}$ of $H_{\ell,n}$. In particular, we answer a question of Mathas on the rationality of square roots of some quotients of product…
We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vignéras.
We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vigneras, it completes the classification of irreducible modulo $p$ representations.
In this short note we give a new proof of the quantum generalization of Regev's theorems by applying the Murnaghan-Nakayama formula for skew characters of the generic Iwahori-Hecke algebra.
The Iwahori-Hecke algebra has a ``natural'' trace $τ$. This trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important commutative sub-algebra ${\bf C}[θ_x]$, that was described and studied by Bernstein, Zelevinski and Lusztig. In this note we compute the generating function for the value of $τ$ on the basis $θ_x$.
Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize Kato and Matsumoto irreducibility criteria.
In this paper we prove a quantum generalization of Regev's theorems in (Israel. J. Math. 195 (2013), 31--35) by applying the Schur-Weyl duality between the quantum superalgebra and Iwahori-Hecke algebra. We also present an alternative proof of the quantized generalizations using the skew character theory of Iwahori-Hecke algebras.
We study the seminormal basis ${f_t}$ for the Specht modules of the Iwahori-Hecke algebra ${\cal H}_n(q)$ of type $A_{n-1}$. We focus on the base change coefficients between the seminormal basis ${f_t}$ and Young's natural basis ${x_t}$ with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving radial lengths. Even for general tableaux we obtain new formulas. On the way we prove a new result about summands of the restricted Specht module at root of unity.
In this note, we show that the natural analogue of certain finiteness result of Barthel--Livn$\acute{\text{e}}$ on $GL_2$ fails for $GL_3$. More precisely, within the pro-$p$-Iwahori invariants of a maximal compact induction of $GL_3$, we show there exist non-zero Iwahori--Hecke submodules of \emph{infinite} codimension.
On overview of Coxeter groups $W$, their Iwahori-Hecke algebras $H$, Lusztig's asymptotic algebras $J$, cellular algebras in general and cellularity of $H$ in particular, as well as an introduction to Gyoja's $W$-graph algebra and the $W$-graph decomposition conjecture.
We state four open problems on Iwahori-Hecke algebras. The first one states a relation between some algebras appearing in Solleveld's work and some explicit Hecke algebras appearing in the study of unipotent representations. The second one is a boundedness conjecture. The third one is a conjecture of well-definedness of an asymptotic Hecke algebra. The fourth one concerns positive conjugacy classes in a Weyl group.
The reducibility of the Specht modules for the Iwahori--Hecke algebras in type $A$ is still open in the case where the defining parameter $q$ equals -1. We prove the reducibility of a large class of Specht modules for these algebras.
We give explicit formulae for certain elements occurring in the Bernstein presentation of an affine Hecke algebra, in terms of the usual Iwahori- Matsumoto generators. We utilize certain minimal expressions for said elements and we give a sheaf-theoretic interpretation for the existence of these minimal expressions.
We consider the problem of classifying irreducible Specht modules for the Iwahori-Hecke algebra of type B with parameters Q,q. We solve this problem completely in the case where q is not a root of unity, and in the case q=-1 we reduce the problem to the corresponding problem in type A.
This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a non-Archimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein's presentation of the Iwahori-Hecke algebra.
We first present an Iwahori-Matsumoto presentation of affine Yokonuma-Hecke algebras $\widehat{Y}_{r,n}(q)$ to give a new proof of the fact, which was previously proved by Chlouveraki and Sécherre, that $\widehat{Y}_{r,n}(q)$ is a particular case of the pro-$p$-Iwahori-Hecke algebras defined by Vignéras, meanwhile, we give one application. Using the new presentation, we then give a third presentation of $\widehat{Y}_{r,n}(q),$ from which we immediately get an unexpected result, that is, the extended affine Hecke algebra of type $A$ is a subalgebra of the affine Yokonuma-Hecke algebra.
Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.
Let $p$ be a prime and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{H}_n$ denote the Iwahori--Hecke algebra of the symmetric group $\mathfrak{S}_n$ over $\mathbb{F}$ at $q=-1$. We prove that there are only finitely many partitions $λ$ such that both $λ$ and $λ'$ are 2-singular and the Specht module $S^λ$ for $\mathcal{H}_{|\la|}$ is irreducible.
We establish a connection between certain unique models, or equivalently unique functionals, for representations of p-adic groups and linear characters of their corresponding Hecke algebras. This allows us to give a uniform evaluation of the image of spherical and Iwahori-fixed vectors in the unramified principal series for this class of models. We provide an explicit alternator expressionfor the image of the spherical vectors under these functionals in terms of the representation theory of the dual group.
Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike, Broué and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible …
This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit "integral" closed formula for the Gram determinants of the Specht modules in terms of the combinatorics which utilizes the KLR gradings. We then use seminormal forms to give a deformation of the KLR algebras of type $A$. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As…
This article gives a fairly self-contained treatment of the basic facts about the Iwahori-Hecke algebra of a split p-adic group, including Bernstein's presentation, Macdonald's formula, the Casselman-Shalika formula, and the Lusztig-Kato formula.
Let $\mathscr{H}_n$ denote the Iwahori-Hecke algebra corresponding to the symmetric group $\mathfrak{S}_n$. We set up a Green correspondence for bimodules of these Hecke algebras, and a Brauer correspondence between their blocks. We examine Specht modules for $\mathscr{H}_n$ and compute the vertices of certain Specht modules, before using this to give a classification of the vertices of blocks of $\mathscr{H}_n$ in any characteristic. Finally, we apply this classification to resolve the Dipper-Du conjecture about the structure of vertices of indecomposable $\mathscr{H}_n$-modules.
James's Conjecture predicts that the adjustment matrix for blocks of the Iwahori-Hecke algebra of the symmetric group is the identity matrix when the weight of the block is strictly less than the characteristic of the field. In this paper, we consider the case when the characteristic of the field is greater than or equal to 5, and prove that the adjustment matrix for the principal block of $\mathcal{H}_{5e}$ is the identity matrix whenever $e\neq4$. When $e=4$, we are able to calculate all but two entries of the adjustment matrix.
Let w be an element of the Weyl group of a reductive group G defined and split over a finite field. We consider the variety of triples (g,B,B') where g is a unipotent element of G and B, B' are Borel subgroups of G such that B contains g and B',gB'g^{-1} are in relative position w. We show that the number of rational points of this variety can be expressed in terms of a trace on the Iwahori-Hecke algebra. We also show that this variety is smooth, irreducible, if w is elliptic, of minimal length in its conjugacy class.
Let $H_k(W,q)$ be the Iwahori--Hecke algebra associated with a finite Weyl group $W$, where $k$ is a field and $0 \neq q \in k$. Assume that the characteristic of $k$ is not ``bad'' for $W$ and let $e$ be the smallest $i \geq 2$ such that $1+q+q^2+... +q^{i-1}=0$. We show that the number of simple $H_k(H,q)$-modules is ``generic'', i.e., it only depends on $e$. The proof uses some computations in the {\sf CHEVIE} package of {\sf GAP} and known results due to Dipper--James, Ariki--Mathas, Rouquier and the author.
We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over a local non-archimedean field. We use the hovel associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer K of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define the Iwahori-Hecke algebra as the algebra of some K-bi-invariant functions on the group with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup o…
Let F be a locally compact nonarchimedean field with residue characteristic p and G the group of F-rational points of a connected split reductive group over F. For k an arbitrary field, we study the homological properties of the Iwahori-Hecke k-algebra H' and of the pro-p Iwahori-Hecke k-algebra H of G. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of G. If G is semisimple, we also show that this upper bound is sharp, that both H and H' are Auslander-Gorenstein and that there is a duality functor on the finitely generated modu…
The pro-$p$-Iwahori Hecke algebra has an involution $ι$ defined in terms of Iwahori-Matsumoto basis. Then for a module $π$ of pro-$p$-Iwahori Hecke, $π^ι= π\circ ι$ is also a module. We calculate $π^ι$ for simple modules $π$. We also calculate the dual of $π$.
Let $F$ be a non-Archimedean local field of characteristic not equal to 2, let $E/F$ be a finite unramified extension field, and let $σ$ be a generator of $\text{Gal}(E, F )$. Let $f$ be an element of $Z(H_{I_E})$, the center of the Iwahori-Hecke algebra for $GL_2(E)$, and let $b$ be the Iwahori base change homomorphism from $Z(H_{I_E})$ to $Z(H_{I_F})$, the center of the Iwahori-Hecke algebra for $GL_2(F)$ [8]. This paper proves the matching of the $σ$-twisted orbital integral over $GL_2(E)$ of $f$ with the orbital integral over $GL_2(F)$ of $bf$. To do so, we compute the orbital and $σ$-twis…
In this paper we construct an analogue of Iwahori-Hecke algebras of SL_2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL_2, and prove that the product is well-defined. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori-Matsumoto type relations.
Let $S_λ$ denote the Specht module defined by Dipper and James for the Iwahori-Hecke algebra $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. When $e=2$ we determine the decomposability of all Specht modules corresponding to hook partitions $(a,1^b)$. We do so by utilising the Brundan-Kleshchev isomorphism between $\mathscr{H}$ and a Khovanov-Lauda-Rouquier algebra and working with the relevant KLR algebra, using the set-up of Kleshchev-Mathas-Ram. When $n$ is even, we easily arrive at the conclusion that $S_λ$ is indecomposable. When $n$ is odd, we find an endomorphism of $S_λ$ and us…
Let $\H_n$ be the Iwahori-Hecke algebra of the symmetric group $S_n$, and let $Z(\H_n)$ denote its centre. Let $B={b_1,b_2,...,b_t}$ be a basis for $Z(\H_n)$ over $R=\Z[q,q^{-1}]$. Then $B$ is called \emph{multiplicative} if, for every $i$ and $j$, there exists $k$ such that $b_ib_j= b_k$. In this article we prove that there are no multiplicative bases for $Z(\Z S_n)$ and $Z(\H_n)$ when $n\ge 3$. In addition, we prove that there exist exactly two multiplicative bases for $Z(\Z S_2)$ and none for $Z(\H_2)$.
Let $G$ be a connected reductive group over a non-archimedean local field $F$ and $I$ be an Iwahori subgroup of $G(F)$. Let $I_n$ is the $n$-th Moy-Prasad filtration subgroup of $I$. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with $I_n$-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups $G$ that split over an unramified extension of $F$. The first main result of this paper is a presentation of the Hecke algebra $\mathcal H(G(F),I_n)$, generalizing the previous work of Iwahori-Matsu…
We establish a super Frobenius formula for the characters of Iwahori-Hecke algebras. We show that the Hall-Littlewood sypersymmetric function, up to constant, generates the values of the irreducible characters of Iwahori-Hecke algebras at the elements corresponding to cycle permutations. Our formula in this article includes both the ordinary quantum case and the classical super case.
In this paper, we introduce the sign q-permutation representation of the Iwahori-Hecke algebra on the tensor space of the graded vector space. We establish Schur-Weyl reciprocity between the quantum general super Lie algebra and the Iwahori-Hecke algebra. Our result includes Schur-Weyl reciprocity of both the super case and the quantum case, and unifies those two important cases.
The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup ${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of $\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution algebra associated to $e_{\mathcal{P}}$ is called a Peter-Weyl Iwahori algebra. We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algbera have a natural $\mathbb{C}^\star$-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebra…
We establish branching rules between some Iwahori-Hecke algebra of type B and their subalgebras which are defined as fixed subalgebras by involutions including Goldman involution. The Iwahori-Hecke algebra of type D is one of such fixed subalgebras. We also obtain branching rules between those fixed subalgebras and their intersection subalgebra. We determine basic sets of irreducible representations of those fixed subalgebras and their intersection subalgebra by making use of generalized Clifford theory.
We introduce the computer algebra package {\sf PyCox}, written entirely in the {\sf Python} language. It implements a set of algorithms - in a spirit similar to the older {\sf CHEVIE} system - for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan--Lusztig cells and $W$-graphs, which works efficiently for all finite groups of rank $\leq 8$ (except $E_8$). We also discuss the computation of Lusztig's leading coefficients of character values and distinguished involutions (which works for $E_8$ as well). Our experiments…
We show that the center of the Iwahori--Hecke algebra of the symmetric group $S_n$ carries a natural filtered algebra structure, and that the structure constants of the associated graded algebra are independent of $n$. A series of conjectures and open problems are also included.
A $q$-analogue of combinatorics concerning the Cartan matrix for the Iwahori-Hecke algebra of type $A$ is investigated. We give several descriptions for the determinant of the graded Cartan matrix, which imply some combinatorial identities. A conjectural expression for the elementary divisors is also presented.
We determine the image of the Artin groups of types B and D inside the Iwahori-Hecke algebras, when defined over finite fields, in the semisimple case. This generalizes earlier work on type A by Brunat, Magaard and Marin. In this multi-parameter case, this image depends heavily on the parameters.
Let p be a prime number, and F a nonarchimedean local field of residual characteristic p. We explore the interaction between the pro-p-Iwahori-Hecke algebras of the group GL_n(F) and its derived subgroup SL_n(F). Using the interplay between these two algebras, we deduce two main results. The first is an equivalence of categories between Hecke modules in characteristic p over the pro-p-Iwahori-Hecke algebra of SL_2(Q_p) and smooth mod-p representations of SL_2(Q_p) generated by their pro-p-Iwahori-invariants. The second is a "numerical correspondence" between packets of supersingular Hecke modu…
We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.
In this paper we give Iwahori-Hecke type algebras H_q(g) associated with the Lie superalgebras g=A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of H_q(g) for generic q.
A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced earlier by the first author. The representations of the algebras of types $A_n$, $B_n$, $D_n$ and $G_2$ were computed and it is the purpose of this paper to extend these computations to $F_4$. The main goal here is to compute irreducible representations of the Iwahori-Hecke algebra of type $F_4$ by only using information in the character table of the Weyl group.
In this paper, we compute the Hecke action of a certain test function on the space of an unramified principal series of a connected reductive group over a non-archimedean local field by using the theory of Iwahori--Hecke algebra. As an application, we obtain a new expression of the local L-functions of unramified representations.
For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $τ$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $τ$ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In earlier work of Anandavardhanan-Borisagar, an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $π$ of $τ$, making use o…
