A note on specializations of Grothendieck polynomials

Buch and Rimányi proved a formula for a specialization of double Grothendieck polynomials based on the Yang–Baxter equation related to the degenerate Hecke algebra. A geometric proof was found by Yong and Woo by constructing a Gröbner basis for the Kazhdan–Lusztig ideals. In this note, we give an elementary proof for this formula by using only divided difference operators.     

Deformed Kazhdan-Lusztig elements and Macdonald polynomials

We introduce deformations of Kazhdan-Lusztig elements and specialised non-symmetric Macdonald polynomials, both of which form a distinguished basis of the polynomial representation of a maximal parabolic subalgebra of the Hecke algebra. We give explicit integral formula for these polynomials, and explicitly describe the transition matrices between classes of polynomials. We further develop a combinatorial interpretation of homogeneous evaluations using an expansion in terms of Schubert polynomials in the deformation parameters.     

Applications of the Frobenius Formulas for the Characters of the Symmetric Group and the Hecke Algebras of Type A

We give a simple combinatorial proof of Ram's rule for computing the characters of the Hecke Algebra. We also establish a relationship between the characters of the Hecke algebra and the Kronecker product of two irreducible representations of the Symmetric Group which allows us to give new combinatorial interpretations to the Kronecker product of two Schur functions evaluated at a Schur function of hook shape or a two row shape. We also give a formula for the regular representation of the Hecke algebra.     

Geometric interpretation of Murphy bases and an application

In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of submodules of a free module over principal ideal local rings of length two with a finite residue field.     

Homological algebra for affine Hecke algebras

In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules.This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature, since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters, for all positive parameters (we will report on this application in a separate article).     

Schur elements for the Ariki–Koike algebra and applications

We study the Schur elements associated to the simple modules of the Ariki–Koike algebra. We first give a cancellation-free formula for them so that their factors can be easily read and programmed. We then study direct applications of this result. We also complete the determination of the canonical basic sets for cyclotomic Hecke algebras of type G(l,p,n) in characteristic 0.     

Construction of tame supercuspidal representations

We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of -adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.

     

A general framework for the polynomiality property of the structure coefficients of double-class algebras

In this paper, we build a general framework in which we give a formula that shows the form of the structure coefficients of double-class algebras and centers of group algebras. This formula allows us to give a polynomiality property for the structure coefficients of some important algebras. In particular, we re-obtain the polynomiality property of the structure coefficients in the cases of the center of the symmetric group algebra and the Hecke algebra of the pair \((\mathcal {S}_{2n},\mathcal {B}_{n}).\) We also assign a new polynomiality property for the structure coefficients of the center of the hyperoctahedral group algebra and the Hecke algebra of the pair \((\mathcal {S}_n\times \mathcal {S}_{n-1}^\mathrm{opp},{{\mathrm{diag}}}(\mathcal {S}_{n-1}))\).     

The symbolical and cancellation-free formulae for Schur elements

In this paper we give the symbolical formula and cancellation-free formula for the Schur elements associated to the simple modules of the degenerate cyclotomic Hecke algebras. As some applications, we show that the Schur elements are symmetric polynomials with rational integer coefficients and give a different proof of Ariki–Mathas–Rui’s criterion on the semisimplicity of the degenerate cyclotomic Hecke algebras.     

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.     

Singular parameters for the Birman–Murakami–Wenzl algebra

In this paper, we classify the singular parameters for the Birman–Murakami–Wenzl algebra over an arbitrary field. Equivalently, we give a criterion for the Birman–Murakami–Wenzl algebra being Morita equivalent to the direct sum of the Hecke algebras associated to certain symmetric groups.     

Combinatorial properties of the noncommutative Faà di Bruno algebra

We give a new combinatorial interpretation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder–Frabetti–Krattenthaler for the antipode of the noncommutative Faà di Bruno algebra.     

Derivees logarithmiques pour une s-derivation algebrique

We use the results on the minimal basis of the centre of an Iwahori–Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism for Iwahori–Hecke algebras defined by L. Jones (Jones, L. Centres of Generic Hecke Algebras; Ph.D. Thesis; University of Virginia, 1987.).

     

A super Frobenius formula for the characters of Iwahori–Hecke algebras

In this article, we establish a super Frobenius formula for the characters of Iwahori–Hecke algebras. We define Hall–Littlewood supersymmetric functions in a standard manner to make supersymmetric functions from symmetric functions, and give some properties of supersymmetric functions. Based on Schur–Weyl reciprocity between Iwahori–Hecke algebras and the general quantum super algebras, which was obtained in Mitsuhashi [H. Mitsuhashi, Schur–Weyl reciprocity between the quantum superalgebra and the Iwahori–Hecke algebra, Algeb. Represent. Theor. 9 (2006), pp. 309–322.], we derive that the Hall–Littlewood supersymmetric functions, 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 that was obtained in Ram [A. Ram, A Frobenius formula for the characters of the Hecke algebra, Invent. Math. 106 (1991), pp. 461–488.] and the classical super case.     

Combinatorial proof of the inversion formula on the Kazhdan–Lusztig R-polynomials

In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\) -polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equidistribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma’s equidistribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.     

Cyclic generators for irreducible representations of affine Hecke algebras

We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.     

Algèbre de Temperley–Lieb de type B

We give a dual presentation, in the sense of the dual presentation of Artin groups, of the Temperley–Lieb algebra of type B. In particular, we obtain a basis of this algebra by considering the homomorphic images of the simple elements of the dual monoid. This algebra is the largest quotient of the Hecke algebra whose irreducible representations are parametrized by pairs of Young diagrams with at most one column in each component. To cite this article: C. Vincenti, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Infinitesimal Hecke algebras

In this Note, we define infinitesimal analogues of the Iwahori–Hecke algebras associated with finite Coxeter groups. These are reductive Lie algebras for which we announce several decomposition results. These decompositions yield irreducibility results for representations of the corresponding (pure) generalized braid groups deduced from Hecke algebra representations through tensor constructions. To cite this article: I. Marin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).