Group Characters, Symmetric Functions, and the Hecke Algebra

2002 ◽  
Author(s):  
David Goldschmidt
Keyword(s):  
Symmetric Functions ◽  
Hecke Algebra ◽  
Group Characters

scholarly journals The Hopf structure of symmetric group characters as symmetric functions

2021 ◽  
Vol 4 (3) ◽  
pp. 551-574
Author(s):  
Rosa Orellana ◽  
Mike Zabrocki
Keyword(s):  
Symmetric Group ◽  
Symmetric Functions ◽  
Group Characters

scholarly journals Hecke algebra and quantum chromatic symmetric functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
International Audience

International audience We evaluate induced sign characters of $H_n(q)$ at certain elements of $H_n(q)$ and conjecture an interpretation for the resulting polynomials as generating functions for $P$-tableaux by a certain statistic. Our conjecture relates the quantum chromatic symmetric functions of Shareshian and Wachs to $H_n(q)$ characters. Nous évaluons les caractères de signe induits de $H_n(q)$ à certains éléments de $H_n(q)$ et nous conjecturons une interprétation des polynômes résultants comme fonctions génératrices pour les tableaux-$P$ par une certaine statistique. Cette conjecture établit un lien entre les fonctions chromatiques symétriques de Shareshian et Wachs et les caractères de $H_n(q)$.

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

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jia Huang
Keyword(s):  
Boolean Algebra ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
Type A ◽  
Quasisymmetric Function ◽  
Permutation Statistics ◽  
Nous Obtenons ◽  
Noncommutative Symmetric Functions ◽  
International Audience ◽  
Cette Action

International audience 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.

scholarly journals Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Sam Clearman ◽  
Matthew Hyatt ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
Irreducible Characters ◽  
Algebraic Interpretation ◽  
International Audience

International audience For irreducible characters $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ and induced sign characters $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the pattern 312, we combinatorially interpret the polynomials $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ and $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$. This gives a new algebraic interpretation of $q$-chromatic symmetric functions of Shareshian and Wachs. We conjecture similar interpretations and generating functions corresponding to other $H_n(q)$-traces. Pour les caractères irréductibles $\{ \chi_q^{\lambda} | \lambda \vdash n\}$ et les caractères induits du signe $\{\epsilon_q^{\lambda} | \lambda \vdash n\}$ du algèbre de Hecke, et les éléments $C'_w(q)$ du base Kazhdan-Lusztig avec $w$ qui évite le motif 312, nous interprétons les polynômes $\chi_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ et $\epsilon_q^{\lambda}(q^{\frac{\ell(w)}{2}} C'_w(q))$ de manière combinatoire. Cette donne une nouvelle interprétation aux fonctions symétriques $q$-chromatiques de Shareshian et Wachs. Nous conjecturons des interprétations semblables et des fonctions génératrices qui correspondent aux autres applications centrales de $H_n(q)$.

scholarly journals Group characters and algebra

1934 ◽  
Vol 233 (721-730) ◽  
pp. 99-141 ◽  
Keyword(s):  
Symmetric Group ◽  
Symmetric Functions ◽  
Irreducible Representations ◽  
Close Analogy ◽  
Group Characters ◽  
And Algebra

It has been known for some time* that the elements of a matrix of degree n may be arranged in sets which correspond to cycles of the symmetric group of order n !, and that there are relations connecting permanents and determinants, e.g. , /a* p y S a |3 y 8/ ( Further, MACMAHON and BRIOSCHI have pointed out the close analogy which exists between the threefold algebra of the symmetric functions an, hn and sn, and the theory of determinants, permanents, and the cycles of substitutions of the symmetric group. Here we trace the analogy to its source by fixing attention on the characters of the irreducible representations of the symmetric group of linear substitutions, as the centre of the whole theory. By this means divers theories of combinatory analysis and algebra are seen to be merely different aspects of the same theory. For the symmetric group of order n ! the characters are all integers, and we associate with each partition of n both a character of the group and a cycle of substitutions.

scholarly journals On the square root of the centre of the Hecke algebra of type A

2007 ◽  
Vol 82 (2) ◽  
pp. 209-220
Author(s):  
Andrew Francis ◽  
Lenny Jones
Keyword(s):  
Symmetric Group ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
Square Root ◽  
Minimal Basis ◽  
Elementary Symmetric Functions ◽  
Square Roots ◽  
Commutative Subalgebra ◽  
Central Elements ◽  
Generic Existence

AbstractIn this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.

scholarly journals SKEIN THEORY AND THE MURPHY OPERATORS

2002 ◽  
Vol 11 (04) ◽  
pp. 475-492 ◽  
Author(s):  
HUGH R. MORTON
Keyword(s):  
Generating Function ◽  
Symmetric Functions ◽  
Hecke Algebra ◽  
Type A ◽  
Natural Basis ◽  
Power Sum ◽  
Skein Theory

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.

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 Symmetric group characters as symmetric functions

2021 ◽  
Vol 390 ◽  
pp. 107943
Author(s):  
Rosa Orellana ◽  
Mike Zabrocki
Keyword(s):  
Symmetric Group ◽  
Symmetric Functions ◽  
Group Characters

CLASSES OF WEIGHTED SYMMETRIC FUNCTIONS

1988 ◽  
Vol 14 (2) ◽  
pp. 429
Author(s):  
Tran
Keyword(s):  
Symmetric Functions
Sign in / Sign up

Export Citation Format

Share Document