Algorithms 307: Symmetric group characters

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

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 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 Unitary Matrix Integrals, Primitive Factorizations, and Jucys-Murphy Elements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sho Matsumoto ◽  
Jonathan Novak
Keyword(s):  
Symmetric Group ◽  
Matrix Models ◽  
Unitary Matrix ◽  
Group Characters ◽  
Matrix Integrals ◽  
International Audience

International audience A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described. Une factorisation en transpositions d'une permutation est dite "primitive'' si ses facteurs sont ordonnés. Nous discutons du problème de l'énumération des factorisations primitives de permutations, et de sa place dans la hiérarchie des problèmes de factorisation précédemment étudiés. Nous présentons plusieurs formules énumérant certaines classes de factorisations primitives,et nous soulignons des connexions intéressantes avec les éléments Jucys-Murphy, les caractères des groupes symétriques, et les modèles de matrices.

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.

The symmetric group: Characters, products and plethysms

1973 ◽  
Vol 14 (9) ◽  
pp. 1176-1183 ◽  
Author(s):  
P. H. Butler ◽  
R. C. King
Keyword(s):  
Symmetric Group ◽  
Group Characters

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

scholarly journals Another Proof of the Harer-Zagier Formula

10.37236/5420 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Boris Pittel
Keyword(s):  
Fourier Transform ◽  
Probability Measure ◽  
Symmetric Group ◽  
Generating Function ◽  
Recurrence Equation ◽  
Young Diagrams ◽  
Group Characters ◽  
Bivariate Generating Function ◽  
The Fourier Transform

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on $S_{2n}$. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.

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