Hecke Algebras Of Type Research Articles

Quantum supersymmetric pairs and ıSchur duality of type AIII

We construct quantum supersymmetric pairs (U,Uı) of type AIII and elucidate their fundamental properties. An ıSchur duality between the ıquantum supergroup Uı and the Hecke algebra of type B acting on a tensor space is established, providing a super generalization of the ıSchur duality of type AIII. Additionally, we construct a (quasi) K-matrix for arbitrary parameters, which facilitates the realization of the Hecke algebra action on the tensor space.

ıSchur duality and Kazhdan-Lusztig basis expanded

Expanding the classical works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an ıSchur duality between an ıquantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split ıSchur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the ıcanonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the ıSchur duality.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

Stability for Representations of Hecke Algebras of Type A

In this paper, we introduce the concept of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

Double centralisers and annihilator ideals of Young permutation modules

We study the double centraliser algebras and annihilator ideals of certain permutation modules for symmetric groups and their quantum analogues. Of particular interest is the behaviour of these algebras and ideals under base change. By exploiting connections with polynomial representation theory over a field we establish the double centraliser property and dimension invariance of annihilator ideals over fields. By elementary localisation techniques we obtain the corresponding results over arbitrary rings. The modules considered include the permutation modules relevant to partition algebras, as studied in the recent paper by Bowman, Doty and Martin, [3], and we obtain in particular their main results on the double centraliser property in this context. Much of our development is equally valid for the corresponding modules over Hecke algebras of type A and, for the most part, we give it at this level of generality.

Two Maps on Affine Type $A$ Crystals and Hecke Algebras

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$.

Two Dualities: Markov and Schur–Weyl

Abstract We show that quantum Schur–Weyl duality leads to Markov duality for a variety of asymmetric interacting particle systems. In particular, we consider the following three cases: (1) Using a Schur–Weyl duality between a two-parameter quantum group and a two-parameter Hecke algebra from [6], we recover the Markov self-duality of multi-species ASEP previously discovered in [23] and [3]. (2) From a Schur–Weyl duality between a co-ideal subalgebra of a quantum group and a Hecke algebra of type B [2], we find a Markov duality for a multi-species open ASEP on the semi-infinite line. The duality functional has not previously appeared in the literature. (3) A “fused” Hecke algebra from [15] leads to a new process, which we call braided ASEP. In braided ASEP, up to $m$ particles may occupy a site and up to $m$ particles may jump at a time. The Schur–Weyl duality between this Hecke algebra and a quantum group lead to a Markov duality. The duality function had previously appeared as the duality function of the multi-species ASEP$(q,m/2)$ [23] and the stochastic multi-species higher spin vertex model [24].

Affine Hecke algebras and generalizations of quiver Hecke algebras of type B

AbstractWe define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalization, for type B, of cyclotomic quiver Hecke algebras, which are a family of graded algebras closely related to algebras introduced by Varagnolo and Vasserot. Inspired by the work of Brundan and Kleshchev, we first give a family of isomorphisms for the corresponding result in type A which includes their original isomorphism. We then select a particular isomorphism from this family and use it to prove our result.

Decomposable Specht modules indexed by bihooks

We study the decomposability of Specht modules labelled by bihooks, bipartitions with a hook in each component, for the Iwahori--Hecke algebra of type $B$. In all characteristics, we determine a large family of decomposable Specht modules, and conjecture that these provide a complete list of decomposable Specht modules indexed by bihooks. We prove the conjecture for small $n$.

Signatures of representations of Hecke algebras and rational Cherednik algebras

Determining whether an irreducible representation of a group (or ⁎-algebra) admits a non-degenerate invariant, positive-definite Hermitian form is an important problem in representation theory. In this paper, we study the related notion of signatures. Let q∈C with |q|=1, κ=−1/c with c∈Q<0 with certain additional (mild) restrictions, and n∈Z≥2. We study representations Sλ(q) of Hn(q), the Hecke algebra of type An−1, and representations Mκ(λ) of Hκ, the rational Cherednik algebra of type An−1, which have unique (up to scaling) invariant Hermitian forms (here λ is a partition of n). The signature is the number of elements with positive norm minus the number of elements with negative norm, and we analogously define the signature character in the case that there is a natural grading on the module. We provide formulas for (1) signatures of modules over Hn(q) and (2) signature characters of modules over Hκ. We study the limit c→−∞, in which case the signature character has a simpler form in terms of inversions and descents of permutations in Sn. We provide examples corresponding to some special shapes, and small values of n. Finally, when q=e2πic, we show that the asymptotic signature character of the Hκ-module Mκ(τ) is the signature of the Hn(q)-module Sτ(q).

Uglov Bipartitions and Extended Young Diagrams

We study the class of Uglov bipartitions and prove a generalization of a conjecture by Dipper, James and Murphy. We give two consequences concerning the computation of canonical bases in affine type $A$ and the description of decomposition matrices for Hecke algebras of type $B_n$ in arbitrary characteristic.

Multiparameter quantum Schur duality of type B

We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

We develop several applications of the fact that the Yokonuma–Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori–Hecke algebras of type A . This includes a description of the semisimple and modular representation theory of the Yokonuma–Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism. In particular, for classical knots, a consequence of the construction is that the obtained set of invariants is topologically equivalent to the HOMFLYPT polynomial. We thus recover results of Chlouveraki et al. (2015, arXiv:1505.06666) about the Juyumaya–Lambropoulou invariants.

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ℋ = ℋℂ, −1(𝔖 n ) be the Hecke algebra of the symmetric group 𝔖 n . For partitions λ and ν with ν 2 − regular, define the Specht module S(λ) and the irreducible module D(ν). Define d λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d λν for all partitions of the form λ = (a, c, 1 b ) and ν 2 − regular.

The Whitehead conjecture, the tower ofS1conjecture, and Hecke algebras of type A

In the early 1980s, the author proved G. W. Whitehead's conjecture about stable homotopy groups and symmetric products. In the mid 1990s, Arone and Mahowald showed that the Goodwillie tower of the identity had remarkably good properties when specialized to odd-dimensional spheres. In this paper, we prove that these results are linked, as has been long suspected. We give a state-of-the-art proof of the Whitehead conjecture valid for all primes, and simultaneously show that the identity tower specialized to the circle collapses in the expected sense. Key to our work is that Steenrod algebra module maps between the primitives in the mod p homology of certain infinite loopspaces are determined by elements in the mod p Hecke algebras of type A. Certain maps between spaces are shown to be chain homotopy contractions by using identities in these Hecke algebras.

$0$-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra

We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics. Nous définissons une action de l’algèbre de Hecke-$0$ de type A sur l’anneau Stanley-Reisner de l’algèbre de Boole. En étudiant cette action, on obtient une famille de fonctions symétriques non commutatives multivariées, qui se spécialisent pour les non commutatives fonctions de Hall-Littlewood symétriques et leur $(q,t)$-analogues introduits par Bergeron et Zabrocki. Nous obtenons également des identités de fonction quasisymmetrique multivariées, qui se spécialisent à la suite de Garsia et Gessel sur la fonction génératrice de la distribution conjointe de cinq statistiques de permutation.

SYMMETRIZERS AND ANTISYMMETRIZERS FOR THE BMW-ALGEBRA

Let n ∈ ℕ and Bn(r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that Bn(r, q) has two distinct linear representations generated by two central elements of Bn(r, q) called the symmetrizer and antisymmetrizer of Bn(r, q). These generate for n ≥ 3 the only one-dimensional two sided ideals of Bn(r, q) and generalize the corresponding notion for Hecke algebras of type A. The main result, Theorem 3.1, in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of Bn(r, q).

Fusion symmetric spaces and subfactors

We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when these categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors. This paper is a sequel of our previous paper [W4], where we introduced a q-deformation of Brauer’s centralizer algebra for orthogonal and symplectic groups; this algebra had already appeared more or less before in [Mo], see also discussion in [W4]. It is motivated by finding a deformation of orthogonal or symplectic subgroups of a unitary group which is compatible with the standard quantum deformation of the big group. This has been done before on the level of coideal subalgebras of Hopf algebras by Letzter. However, our categorical approach also allows us to extend this to the level of fusion tensor categories, where we find finite analogs of symmetric spaces related to the already mentioned groups. Moreover, we can establish C∗ structures, necessary for the construction of subfactors, in this categorical setting; this is not so obvious to see in the setting of co-ideal algebras. It is well-known how one can use a subgroup H of a (for simplicity here) finite group G to construct a module category of the representation category Rep G of G. This module category also appears in the context of subfactors of II1 von Neumann factors as follows: Let R be the hyperfinite II1 factor, and let N = RG ⊂M = RH be the fixed points under outer actions of G and H. Then the category of N − N bimodules is equivalent to Rep G, and the module category is given via the M−N bimodules of the inclusion N ⊂ M; its simple objects are labeled by the irreducible representations of H. In particular, an important invariant called the principal graph of the subfactor is determined by the restriction rules for representations from G to H. Important examples of subfactors were constructed from fusion categories whose Grothendieck semirings are quotients of the ones of semisimple Lie groups. So a natural question to ask is whether one can perform a similar construction in this context. More precisely, can we find restriction rules for type A fusion categories which describe a subfactor as before, and which will approach in the classical limit the usual restriction rules from U(N) to O(N). We answer this question in the positive in this paper via a fairly elementary construction. We show that certain semisimple quotients of the q-Brauer algebras have a C∗ structure and contain C∗-quotients of Hecke algebras of type A. The subfactor is then obtained as the closure of inductive limits of such algebras. Due to its close connection to Lie groups, we can

A q-Brauer algebra

We define a new q-deformation of Brauerʼs centralizer algebra which contains Hecke algebras of type A as unital subalgebras. We determine its generic structure as well as the structure of certain semisimple quotients. This is expected to have applications for constructions of subfactors of type II1 factors and for module categories of fusion categories of type A corresponding to certain symmetric spaces.

On Quantum Immanants and the Cycle Basis of the Quantum Permutation Space

There are many combinatorial expressions for evaluating characters of the Hecke algebra of type A. However, with rare exceptions, they give simple results only for permutations that have minimal length in their conjugacy class. For other permutations, a recursive formula has to be applied. Consequently, quantum immanants are complicated objects when expressed in the standard basis of the quantum permutation space. In this paper, we introduce another natural basis of the quantum permutation space, and we prove that coefficients of quantum immanants in this basis are class functions.