scholarly journals Generating Functions for Hecke Algebra Characters

2011 ◽  
Vol 63 (2) ◽  
pp. 413-435 ◽  
Author(s):  
Matjaž Konvalinka ◽  
Mark Skandera
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Generating Functions ◽  
Hecke Algebra ◽  
Close Connection ◽  
Group Characters ◽  
Quantum Polynomial ◽  
Quantum Immanants ◽  
Quantum Determinant ◽  
Master Theorem

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.

scholarly journals Combinatorial formulas for double parabolic R-polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Justin Lambright ◽  
Mark Skandera
Keyword(s):  
Polynomial Ring ◽  
Hecke Algebra ◽  
Combinatorial Interpretations ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Quantum Polynomial

International audience The well-known R-polynomials in ℤ[q], which appear in Hecke algebra computations, are closely related to certain modified R-polynomials in ℕ[q] whose coefficients have simple combinatorial interpretations. We generalize this second family of polynomials, providing combinatorial interpretations for expressions arising in a much broader class of computations. In particular, we extend results of Brenti, Deodhar, and Dyer to new settings which include parabolic Hecke algebra modules and the quantum polynomial ring. Les bien connues polynômes-R en ℤ[q], qui apparaissent dans les calcules d'algébre de Hecke, sont relationés à certaines polynômes-R modifiés en ℕ[q], dont les coefficients ont simples interprétations combinatoires. Nous généralisons cette deuxième famille de polynômes, fournissant des interprétations combinatoires pour les expressions qui se posent dans une catégorie beaucoup plus vaste de calculs. En particulier, nous étendons des résultats de Brenti, Deodhar, et Dyer à des situations nouvelles, qui comprennent modules paraboliques pour l'algébre de Hecke, et l'anneau des polynômes quantiques.

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 KAZHDAN–LUSZTIG CELLS AND THE MURPHY BASIS

2006 ◽  
Vol 93 (3) ◽  
pp. 635-665 ◽  
Author(s):  
MEINOLF GECK
Keyword(s):  
Symmetric Group ◽  
Hecke Algebra ◽  
Fundamental Properties ◽  
Algebraic Proofs ◽  
First Time

Let $H$ be the Iwahori–Hecke algebra associated with $S_n$, the symmetric group on $n$ symbols. This algebra has two important bases: the Kazhdan–Lusztig basis and the Murphy basis. We establish a precise connection between the two bases, allowing us to give, for the first time, purely algebraic proofs for a number of fundamental properties of the Kazhdan–Lusztig basis and Lusztig's results on the $a$-function.

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

Algorithms 307: Symmetric group characters

1967 ◽  
Vol 10 (7) ◽  
pp. 451-452 ◽  
Author(s):  
J. K. S. McKay
Keyword(s):  
Symmetric Group ◽  
Group Characters

scholarly journals A Diagrammatic Temperley-Lieb Categorification

2010 ◽  
Vol 2010 ◽  
pp. 1-47 ◽  
Author(s):  
Ben Elias
Keyword(s):  
Symmetric Group ◽  
Weyl Group ◽  
Planar Graphs ◽  
Monoidal Category ◽  
Hecke Algebra ◽  
Coxeter Complex ◽  
Quotient Category ◽  
Irreducible Modules ◽  
Soergel Bimodules

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1-skeleton of the Coxeter complex and to the topology of 2D cobordisms. We demonstrate how further subquotients of this category will categorify the irreducible modules of the Temperley-Lieb algebra.

COUNTING ELEMENTS IN THE SYMMETRIC GROUP

2009 ◽  
Vol 19 (03) ◽  
pp. 305-313 ◽  
Author(s):  
DAVID EL-CHAI BEN-EZRA
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Combinatorial Identities ◽  
Subgroup Growth

By using simple ideas from subgroup growth of pro-finite groups we deduce some combinatorial identities on generating functions counting various elements in symmetric groups.

Hidden structures of knot invariants

2014 ◽  
Vol 29 (29) ◽  
pp. 1430063 ◽  
Author(s):  
Alexey Sleptsov
Keyword(s):  
Symmetric Group ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Double Affine Hecke Algebra ◽  
Loop Expansion ◽  
Group Characters ◽  
The Family ◽  
Homfly Polynomials ◽  
Genus Expansion

We discuss a connection of HOMFLY polynomials with Hurwitz covers and represent a generating function for the HOMFLY polynomial of a given knot in all representations as Hurwitz partition function, i.e. the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and the loop expansion through Vassiliev invariants explicitly demonstrate this phenomenon. We study the genus expansion and discuss its properties. We also consider the loop expansion in details. In particular, we give an algorithm to calculate Vassiliev invariants, give some examples and discuss relations among Vassiliev invariants. Then we consider superpolynomials for torus knots defined via double affine Hecke algebra. We claim that the superpolynomials are not functions of Hurwitz type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are beta-deformed to Hamiltonians of the Calogero–Moser–Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials.

scholarly journals Generalized Pattern Avoidance Condition for the Wreath Product of Cyclic Groups with Symmetric Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-17
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel ◽  
Manda Riehl
Keyword(s):  
Symmetric Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Pattern Avoidance ◽  
Catalan Numbers ◽  
Cyclic Groups ◽  
Bijective Proof ◽  
Avoidance Condition

We continue the study of the generalized pattern avoidance condition for Ck≀Sn, the wreath product of the cyclic group Ck with the symmetric group Sn, initiated in the work by Kitaev et al., In press. Among our results, there are a number of (multivariable) generating functions both for consecutive and nonconsecutive patterns, as well as a bijective proof for a new sequence counted by the Catalan numbers.

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