Hecke Algebra Hn Research Articles

Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

We define a unified trace form τ on the cyclotomic Hecke algebras Hn,K of type A, which generalize both Malle-Mathas' trace form on the non-degenerate version (with Hecke parameter ξ≠1) and Brundan-Kleshchev's trace form on the degenerate version. We use seminormal basis theory to construct an explicit pair of dual bases for Hn,K with respect to the form. We also construct an explicit basis for the cocenter of the corresponding cyclotomic Schur algebra, which shows that the cocenter has dimension independent of the ground field K, the Hecke parameter ξ and the cyclotomic parameters Q1,⋯,Qℓ.

Combinatorial interpretation of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations in 𝔖6

Abstract Deodhar [Geom. Dedicata 36, no. 1 (1990)] introduced the defect statistic on subexpressions of reduced expressions in the symmetric group 𝔖 n to construct an algorithmic description of the Kazhdan–Lusztig basis of the Hecke algebra H n (q). This led Billey–Warrington [J. Algebraic Combin. 13, no. 2 (2001)] and the second author [J. Pure Appl. Algebra 212 (2008)] to state very explicit combinatorial descriptions of the basis elements indexed by permutations avoiding certain patterns. We extend the above work by producing an exhaustive list of graphical representations of Kazhdan–Lusztig basis elements indexed by 45312-avoiding permutations w ∈𝔖5, 𝔖6.

Crystal of affine type [formula omitted] and Hecke algebras at a primitive 2ℓth root of unity

Let ℓ∈N with ℓ>1. In this paper we give a new realization of the crystal of affine type Aˆℓ−1 using the modular representation theory of the affine Hecke algebras Hn of type A and their level two cyclotomic quotients with Hecke parameter being a primitive 2ℓth root of unity. We construct “hat” versions of i-induction and i-restriction functors on the category RepI(Hn) of finite dimensional integral modules over Hn, which induce Kashiwara operators on a suitable subgroup of the Grothendieck groups of RepI(Hn). For any simple module M∈RepI(Hn), we prove that the simple submodules of resHn−2HnM which belong to Bˆ(∞) (Definition 5.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of slˆℓ and the affine Hecke algebras of type A at a primitive ℓth root of unity.

Sur une construction algébrique d'isomorphismes génériques d'algèbres de Hecke de type A

We construct explicit homomorphisms between the generic Hecke algebra Hn(q) of type A and the degenerate affine Hecke algebra DH(Sn) by means of a generic element Rqν(X,S) which satisfies the quadratic Hecke relation. For the specialization of DH(Sn) by means of the Jucys-Murphy's elements associated with the symmetric group Sn, we obtain explicit isomorphisms, after extension of the scalars to C(q), between Hn(q) and the group algebra of the symmetric group defined over C(q). Moreover, this isomorphism is compatible with the Hoefsmit's model of irreducible representations of Hn(q) and Sn.

Decomposable Specht modules for the Iwahori–Hecke algebra [formula omitted

Let Sλ denote the Specht module defined by Dipper and James for the Iwahori–Hecke algebra Hn of the symmetric group Sn. When e=2 we determine the decomposability of all Specht modules corresponding to hook partitions (a,1b). We do so by utilising the Brundan–Kleshchev isomorphism between 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 Sλ is indecomposable. When n is odd, we find an endomorphism of Sλ and use it to obtain a generalised eigenspace decomposition of Sλ.

A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

POSITIVE PERMUTATION BRAIDS AND PERMUTATION INVERSIONS WITH SOME APPLICATIONS

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra Hn+1from such basis of Hnis given.

An algorithm for semistandardising homomorphisms

Suppose μ is a partition of n and λ a composition of n, and let Sμ, Mλ denote the Specht module and permutation module defined by Dipper and James for the Iwahori–Hecke algebra Hn of the symmetric group Wn. We give an explicit fast algorithm for expressing a tableau homomorphism ϕˆA:Sμ→Mλ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.

Generating Functions for Hecke Algebra Characters

Abstract Certain polynomials in n2 variables that serve as generating functions for symmetric group characters are sometimes called (Sn) character immanants. We point out a close connection between the identities of Littlewood–Merris–Watkins and Goulden–Jackson, which relate Sn character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra Hn(q), we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood–Merris–Watkins identities and selected Goulden–Jackson identities that relate Hn(q) character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis–Lê–Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Cartan invariants of symmetric groups and Iwahori-Hecke algebras

Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the ℓ-Cartan matrix for Sn (equivalently, the invariant factors of the Cartan matrix for the Iwahori–Hecke algebra ℋn(q), where q is a primitive ℓth root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when ℓ=pr, p is prime and r⩽p, and went on to conjecture that the formulae should hold for all r. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition ℓ = p 1 r 1 ⋅ … ⋅ p k r k , the Cartan matrix of an ℓ-block of Sn is a product of Cartan matrices associated to p i r i -blocks of Sn. In particular, the invariant factors of the Cartan matrix associated to an ℓ-block of Sn can be recovered from the Cartan matrices associated to the p i r i -blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of Sn, not only for the full Cartan matrix, but also for an individual block. We collect evidence for this conjecture by showing that the formulae predict the correct determinant of the ℓ-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of Külshammer, Olsson and Robinson.

There exist conjugate simple braids whose associated permutations are not strongly conjugate

AbstractIf two permutations are strongly conjugate, then their corresponding positive permutation braids (also called simple braids) are conjugate. In this paper we exhibit two conjugate simple braids whose associated permutations are not strongly conjugate. In terms of the grey and black graphs with vertices in the ultra summit set defined in [1], this result can be reformulated by saying that there are ultra summit sets with simple braids (hence with canonical length k = 1) in which the grey graph is not connected. Birman, Gebhardt and González Meneses have given similar examples, but with k ≥ 2 [1]. Recall that the set of simple braids on n strings is a basis of the Hecke algebra Hn. If two simple braids on n strings are conjugate, the associated permutations are centrally conjugate, which means that the coefficients of any central element of Hn corresponding to these simple braids are equal. Working on topological dynamics, Hall and Carvalho [8] have discovered two braids on 12 strings (one and its reverse) which are not conjugate. Since one braid is the reverse of the other, their corresponding permutations are centrally conjugate. For n ≤ 6 we checked that central conjugacy implies conjugacy of the corresponding simple braids.

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

The Hecke algebra Hn can be described as the skein R n of (n, n)-tangle diagrams with respect to the framed Homfly relations. This algebra R n contains well-known idempotents Eλ which are indexed by Young diagrams λ with n cells. The geometric closures Qλ of the Eλ in the skein of the annulus are known to satisfy a combinatorial multiplication rule which is identical to the Littlewood–Richardson rule for Schur functions in the ring of symmetric functions. This fact is derived from two skein theoretic lemmas by using elementary determinantal arguments. Previously known proofs depended on results for quantum groups.

Robinson–Schensted–Knuth insertion and characters of cyclotomic Hecke algebras

The cyclotomic Hecke algebras Hn,r = Hn(u1, . . . , ur; q) were defined by Ariki and Koike in [AK] as Iwahori-Hecke algebras of the complex reflection group Gn,r = Sn o (Z/rZ) where Sn is the symmetric group. If ζ is a primitive complex rth root of unity, then when q → 1 and ui → ζ, the algebra Hn,r specializes to the group algebra C[Gn,r]. The irreducible representations of Hn,r are constructed in [AK]. They are indexed by the set of all r-tuples of partitions with a total of n boxes, called r-partitions. For each r-partition μ, T. Shoji [Sho] defines a symmetric function qμ and proves that qμ = ∑

SKEIN THEORY AND THE MURPHY OPERATORS

The Murphy operators in the Hecke algebra Hn of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of Hn. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, [Formula: see text]. Symmetric functions of the Murphy operators are also central in Hn. I define geometrically a homomorphism from [Formula: see text] to the centre of each algebra Hn, and find an element in [Formula: see text], independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of [Formula: see text] in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

Markov Traces and Generic Degrees in Type Bn

Motivated by the theory of knots, Geck and Lambropoulou studied so-called Markov traces on the Iwahori–Hecke algebra Hn of type Bn. These traces depend on two parameters and are linear combinations of the irreducible characters of Hn, where the coefficients are called weights. Orellana determined the weights explicitly for certain special choices of the parameters. In this article, we derive from Orellana's result a formula for the weights in general. As an application, we obtain a new proof of Hoefsmit's formulas for the generic degrees of Hn. Finally, we present a conjectural formula for weights of Markov traces on Ariki–Koike algebras.

A note on the irreducible characters of the Hecke algebra {$H\sb n(q)$}

In this note we present a formula for the irreducible characters of the Hecke algebras Hn(q) of type An−1, in terms of irreducible characters of the symmetric group corresponding to classes with a single cycle of length greater than one.

The Structure of Affine Hecke Algebra HN for the XXZ Model with Boundary

A new affine Hecke algebra is found in studying the XXZ model with a diagonal boundary condition. In the special case of r = 0, we give the double structure of this new affine Hecke algebra, which is similar to the Bn type affine Hecke algebra.

The structure of n-variable polynomial rings as Hecke algebra modules

We determine the structure of the ring of n-variable polynomials as a module for the Hecke algebra Hn(q) (q an invertible parameter). We show how to construct polynomial bases for the q analogues of symmetric group Specht modules, fully generalizing the relevant result of Davies et al (1993, on two representations in the Temperley-Lieb quotient).

Frobenius formula for the characters of hecke algebras of type An−1at roots of unity

A Frobenius formula is found for the (m, k)-standard characters φσ ρ, of the Hecke algebras Hn(q) of type An-1, where q is a primitive pth root of unity, with p=m+k.

An algoritm for calculating characters of hecke algebrasHn(q) of typeAn-1whenqis a root of unity

Recently a Frobenius formula has been derived for characters of Hecke algebras Hn(q) of type An-1 when q is a root of unity (Cummins and King 1992,1993). This result is used lo derive a combinatorial procedure for calculating these characters. The procedure serves to generalise both that of Murnaghan and Nakayama for characters of Sn and that of Ram and Van der Jeugt for characters of Hn(q) when q w not a root of unity.