scholarly journals Stable decompositions for some symmetric group characters arising in braid group cohomology

2011 ◽  
Vol 118 (3) ◽  
pp. 1136-1139 ◽  
Author(s):  
David J. Hemmer
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Group Cohomology ◽  
Group Characters

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

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 Quantum double finite group algebras and link polynomials

1994 ◽  
Vol 49 (2) ◽  
pp. 177-204 ◽  
Author(s):  
I. Tsohantjis ◽  
M.D. Gould
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
Braid Group ◽  
Unitary Representations ◽  
Group Algebras ◽  
Diagonal Form ◽  
Closed Formula ◽  
Quantum Double ◽  
Induced Representations

Unitary representations of the braid group and corresponding link polynomials are constructed corresponding to each irreducible representation of a quantum double finite group algebra. Moreover the diagonal form of the braid generator is derived from which a general closed formula is obtained for link polynomials. As an example, link polynomials corresponding to certain induced representations of the symmetric group and its subgroups are determined explicitly.

A toral configuration space and regular semisimple conjugacy classes

1995 ◽  
Vol 118 (1) ◽  
pp. 105-113 ◽  
Author(s):  
G. I. Lehrer
Keyword(s):  
Topological Space ◽  
Symmetric Group ◽  
Configuration Space ◽  
Braid Group ◽  
Cohomology Ring ◽  
General Setting ◽  
De Rham Cohomology ◽  
Pure Braid Group ◽  
De Rham ◽  
Image Position

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration spaceThe symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

THE BRAID GROUP $B_{n,m}(\mathbb{S}^{2})$ AND A GENERALISATION OF THE FADELL–NEUWIRTH SHORT EXACT SEQUENCE

2005 ◽  
Vol 14 (03) ◽  
pp. 375-403 ◽  
Author(s):  
DACIBERG LIMA GONÇALVES ◽  
JOHN GUASCHI
Keyword(s):  
Exact Sequence ◽  
Symmetric Group ◽  
Cross Section ◽  
Short Exact Sequence ◽  
Braid Group ◽  
Configuration Spaces ◽  
Group Homomorphism

Let m,n ∈ ℕ. We define [Formula: see text] to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup Sn × Sm of the symmetric group Sn+m on n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence: [Formula: see text] where [Formula: see text] is the group homomorphism (defined for all n ∈ ℕ) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration [Formula: see text] of the quotients of the corresponding configuration spaces. Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p* and the fibration p admit (respectively, do not admit) a section. If n = 3, then p* and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p* and p admit a section then m ≡ ε1(n - 1)(n - 2) - ε2n(n - 2) (mod n(n - 1)(n - 2)), where ε1,ε2 ∈ {0,1}. Finally, we show that [Formula: see text] is generated by two of its torsion elements.

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.

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