scholarly journals Hecke Algebra Actions on the Coinvariant Algebra

2000 ◽  
Vol 233 (2) ◽  
pp. 594-613 ◽  
Author(s):  
Ron M. Adin ◽  
Alexander Postnikov ◽  
Yuval Roichman
Keyword(s):  
Hecke Algebra ◽  
Coinvariant Algebra

scholarly journals 0-Hecke algebra actions on coinvariants and flags

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Generating Functions ◽  
Hecke Algebra ◽  
Flag Variety ◽  
Major Index ◽  
Inversion Number ◽  
International Audience ◽  
Descent Set ◽  
Complete Flag ◽  
Coinvariant Algebra

International audience By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. En étudiant l'action de l'algèbre de 0-Hecke sur l'algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions génératrices qui comptent les permutations avec un ensemble inverse de descentes fixé, selon leur nombre d'inversions et leur "major index''.

scholarly journals Tail Positive Words and Generalized Coinvariant Algebras

10.37236/6970 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Brendon Rhoades ◽  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Hecke Algebra ◽  
Macdonald Polynomials ◽  
Isomorphism Type ◽  
Monomial Basis ◽  
Combinatorial Properties ◽  
Algebraic Properties ◽  
Standard Monomial ◽  
Coinvariant Algebra

Let $n,k,$ and $r$ be nonnegative integers and let $S_n$ be the symmetric group. We introduce a quotient $R_{n,k,r}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which carries the structure of a graded $S_n$-module.  When $r \ge n$ or $k = 0$ the quotient $R_{n,k,r}$ reduces to the classical coinvariant algebra $R_n$ attached to the symmetric group. Just as algebraic properties of $R_n$ are controlled by combinatorial properties of permutations in $S_n$, the algebra of $R_{n,k,r}$ is controlled by the combinatorics of objects called tail positive words. We calculate the standard monomial basis of $R_{n,k,r}$ and its graded $S_n$-isomorphism type. We also view $R_{n,k,r}$ as a module over the 0-Hecke algebra $H_n(0)$, prove that $R_{n,k,r}$ is a projective 0-Hecke module, and calculate its quasisymmetric and nonsymmetric 0-Hecke characteristics. We conjecture a relationship between our quotient $R_{n,k,r}$ and the delta operators of the theory of Macdonald polynomials.

scholarly journals Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 779
Author(s):  
Charles F. Dunkl
Keyword(s):  
Preceding Paper ◽  
Hecke Algebra ◽  
Type A ◽  
Macdonald Polynomials ◽  
Special Points ◽  
Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

scholarly journals Representation Theory of a Hecke Algebra of G(r, p, n)

1995 ◽  
Vol 177 (1) ◽  
pp. 164-185 ◽  
Author(s):  
S Ariki
Keyword(s):  
Representation Theory ◽  
Hecke Algebra

REPRESENTATIONS OF THE HECKE ALGEBRA FOR GL4(q)

2005 ◽  
Vol 04 (06) ◽  
pp. 631-644
Author(s):  
KENICHI SHINODA ◽  
ILKNUR TULUNAY
Keyword(s):  
Hecke Algebra ◽  
Standard Basis

In this article, we explicitly calculated the values of the representations of the Hecke algebra [Formula: see text], associated with a Gelfand–Graev character of GL 4(q), at some of the standard basis elements.

Integrable models through representations of the hecke algebra

1991 ◽  
Vol 360 (2-3) ◽  
pp. 613-640 ◽  
Author(s):  
Nir Sochen
Keyword(s):  
Hecke Algebra ◽  
Integrable Models

The Decomposition Numbers of the Hecke Algebra of Type E ∗ 6

1993 ◽  
Vol 61 (204) ◽  
pp. 889 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Hecke Algebra ◽  
Decomposition Numbers

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

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