Iwahori Hecke Algebra Research Articles

On the denominators of Young's seminormal basis

We study the seminormal basis {ft} for the Specht modules of the Iwahori-Hecke algebra Hn(q) of type An−1. We focus on the base change coefficients between the seminormal basis {ft} and Murphys' standard basis {xt} with emphasis on the denominators of these coefficients. In certain important cases we obtain simple formulas for these coefficients involving hook lengths. Even for general standard tableaux we obtain new formulas. On the way we prove a new result about submodules of the restricted Specht module at root of unity.

On irreducible characters of the Iwahori-Hecke algebra in type A

In this paper, we use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type A. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. We also formulate a determinant type Murnaghan-Nakayama formula and give another proof of the combinatorial Murnaghan-Nakayama rule. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Leduc-Ram formula.

Decomposable Specht Modules Indexed by Bihooks II

Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori–Hecke algebra of type B. In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.

A Hecke algebra on the double cover of a Chevalley group over ℚ2

We prove that a certain genuine Hecke algebra $\mathcal{H}$ on the non-linear double cover of a simple, simply-laced, simply-connected, Chevalley group $G$ over $\mathbb{Q}_{2}$ admits a Bernstein presentation. This presentation has two consequences. First, the Bernstein component containing the genuine unramified principal series is equivalent to $\mathcal{H}$-mod. Second, $\mathcal{H}$ is isomorphic to the Iwahori-Hecke algebra of the linear group $G/Z_{2}$, where $Z_{2}$ is the $2$-torsion of the center of $G$. This isomorphism of Hecke algebras provides a correspondence between certain genuine unramified principal series of the double cover of $G$ and the Iwahori-unramified representations of the group $G/Z_{2}$.

Flags and tangles

We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori–Hecke algebras of type A_n as endomorphism algebras of certain objects.

Transition Matrices Between Young's Natural and Seminormal Representations

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.

On structure constants of Iwahori–Hecke algebras for Kac–Moody groups

We consider the Iwahori-Hecke algebra associated to an almost split Kac-Moody group $G$ (affine or not) over a nonarchimedean local field $K$. It has a canonical double-coset basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ indexed by a sub-semigroup $W^+$ of the affine Weyl group $W$. The multiplication is given by structure constants $a^ {\mathbf u}_{\mathbf w,\mathbf v}\in N=Z_{\geq0}$ : $T_{\mathbf w}*T_{\mathbf v}=\sum_{\mathbf u\in P_{\mathbf w,\mathbf v}} a^ {\mathbf u}_{\mathbf w,\mathbf v} T_{\mathbf u}$. A conjecture, by Bravermann, Kazhdan, Patnaik, Gaussent and the authors, tells that $a^ {\mathbf u}_{\mathbf w,\mathbf v}$ is a polynomial, with coefficients in $N$, in the parameters $q_{i}-1,q'_{i}-1$ of $G$ over $K$. We prove this conjecture when $\mathbf w$ and $\mathbf v$ are spherical or, more generally, when they are said generic: this includes all cases of $\mathbf w,\mathbf v\in W^+$ if $G$ is of affine or strictly hyperbolic type. In the split affine case (where $q_{i}=q'_{i}=q$, $\forall i$) we get a universal Iwahori-Hecke algebra with the same basis $(T_{\mathbf w})_{\mathbf w\in W^+}$ over a polynomial ring $Z[Q]$; it specializes to our Iwahori-Hecke algebra when one sets $Q=q$.

Decompositions of principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields

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.

Adjustment matrices for the principal block of the Iwahori-Hecke algebra [formula omitted

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 H5e is the identity matrix whenever e≠4. When e=4, we are able to calculate all but two entries of the adjustment matrix.

Dual Garside structures and Coxeter sortable elements

In Artin-Tits groups attached to Coxeter groups of spherical type, we give a combinatorial formula to express the simple elements of the dual braid monoids in the classical Artin generators. Every simple dual braid is obtained by lifting an $S$-reduced expression of its image in the Coxeter group, in a way which involves Reading's $c$-sortable elements. It has as an immediate consequence that simple dual braids are Mikado braids (the known proofs of this result either require topological realizations of the Artin groups or categorification techniques), and hence that their images in the Iwahori-Hecke algebras have positivity properties. In the classical types, this requires to give an explicit description of the inverse of Reading's bijection from $c$-sortable elements to noncrossing partitions of a Coxeter element $c$, which might be of independent interest. The bijections are described in terms of the noncrossing partition models in these types. While the proof of the formula is case-by-case, it is entirely combinatorial and we develop an approach which reduces a uniform proof to uniformly proving a lemma about inversion sets of $c$-sortable elements.

On imprimitive representations of finite reductive groups in non-defining characteristic

In this paper, we begin with the classification of Harish-Chandra imprimitive representations in non-defining characteristic. We recall the connection of this problem to certain generalizations of Iwahori-Hecke algebras and show that Harish-Chandra induction is compatible with the Morita equivalence by Bonnafé and Rouquier, thus reducing the classification problem to quasi-isolated blocks. Afterwards, we consider imprimitivity of unipotent representations of certain classical groups. In the case of general linear and unitary groups, our reduction methods then lead to results for arbitrary Lusztig series.

The Dunkl weight function for rational Cherednik algebras

In this paper we prove the existence of the Dunkl weight function $$K_{c, \lambda }$$ for any irreducible representation $$\lambda $$ of any finite Coxeter group W, generalizing previous results of Dunkl. In particular, $$K_{c, \lambda }$$ is a family of tempered distributions on the real reflection representation of W taking values in $$\text {End}_\mathbb {C}(\lambda )$$, with holomorphic dependence on the complex multi-parameter c. When the parameter c is real, the distribution $$K_{c, \lambda }$$ provides an integral formula for Cherednik’s Gaussian inner product $$\gamma _{c, \lambda }$$ on the Verma module $$\Delta _c(\lambda )$$ for the rational Cherednik algebra $$H_c(W, \mathfrak {h})$$.queryPlease check and confirm the inserted city name ‘Stanford’ for the affiliation is correct. In this case, the restriction of $$K_{c, \lambda }$$ to the hyperplane arrangement complement $$\mathfrak {h}_{\mathbb {R}, reg}$$ is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on $$KZ(\Delta _c(\lambda ))$$, where KZ denotes the Knizhnik–Zamolodchikov functor introduced by Ginzburg–Guay–Opdam–Rouquier. This provides a concrete connection between invariant Hermitian forms on representations of rational Cherednik algebras and invariant Hermitian forms on representations of Iwahori–Hecke algebras, and we exploit this connection to show that the KZ functor preserves signatures, and in particular unitarizability, in an appropriate sense.

Iwahori component of Bessel model spaces

Let $k_0$ be a $p$-adic field of odd residual characteristic, and $G$ a special orthogonal group defined as acting on a split $2n+1$-dimensional orthogonal space $V$ over $k_0$. Let $H$ be the Iwahori Hecke algebra of $G$. A purpose of this short article is to compute the Iwahori component of a Bessel model space and identify it with an explicit projective $H$-module.

Relationship between Mullineux involution and the generalized regularization

The Mullineux involution is an important map on p-regular partitions that originates from the modular representation theory of Sn. In this paper we study the Mullineux transpose map and the generalized column regularization and prove a condition under which the two maps are exactly the same. Our results generalize the work of Bessenrodt, Olsson and Xu, and the combinatorial constructions is related to the Iwahori–Hecke algebra and the global crystal basis of the basic Uq(sl̂b)-module. In the conclusion, we provide several conjectures regarding the q-decomposition numbers and generalizations of results due to Fayers.

A comparison between pro-p Iwahori–Hecke modules and mod p representations

We give an equivalence of categories between certain subcategories of modules of pro-$p$-Iwahori Hecke algebras and modulo $p$ representations.

Specht modules labelled by hook bipartitions II

We continue the study of Specht modules labelled by hook bipartitions for the Iwahori–Hecke algebra of type B with e∈{3,4,…} via the cyclotomic Khovanov–Lauda–Rouquier algebra HnΛ. Over an arbitrary field, we explicitly determine the graded decomposition submatrices for HnΛ comprising rows corresponding to hook bipartitions.

Parameters for generalized Hecke algebras in type B

The irreducible representations of full support in the rational Cherednik category [Formula: see text] attached to a Coxeter group [Formula: see text] are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in [Formula: see text] of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup [Formula: see text] in [Formula: see text] for [Formula: see text] and [Formula: see text].

Peter–Weyl Iwahori Algebras

AbstractThe Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

Integral u-deformed involution modules

Let (W,S) be a Coxeter system and ⁎ an automorphism of W with order ≤2 and S⁎=S. Lusztig and Vogan ([20], [23]) have introduced a u-deformed version Mu of Kottwitz's involution module over the Iwahori–Hecke algebra Hu(W) with Hecke parameter u2, where u is an indeterminate. Lusztig has proved that Mu is isomorphic to the left Hu(W)-submodule of Hˆu generated by X∅:=∑w⁎=w∈Wu−ℓ(w)Tw, where Hˆu is the vector space consisting of all formal (possibly infinite) sums ∑x∈WcxTx (cx∈Q(u) for each x). He speculated that one can extend this by replacing u with any λ∈C∖{0,1,−1}. In this paper, we give a positive answer to his speculation for any λ∈K∖{0,1,−1} and any W, where K is an arbitrary ground field.

Self-dual representations of SL(2,F): an approach using the Iwahori–Hecke algebra

Let F be a non-Archimedean local field and Let be an irreducible smooth Iwahori-spherical representation of G. It is easy to see that such representations are always self-dual. The space V of π admits a non-degenerate G-invariant bilinear form which is unique up to scaling. It can be shown that the form is either symmetric or skew-symmetric and we set accordingly. In this article, we use the Bernstein–Lusztig presentation of the Iwahori–Hecke algebra of G and show that