Decomposition of certain products of conjugacy classes of Sn

AbstractUsing the character theory of the symmetric group Sn, we study the decomposition of the product of two conjugacy classes Kλ ∗ Kμ in the basis of conjugacy classes. This product takes place in the group algebra of the symmetric group, and the coefficient of the class Kγ in the decomposition, called structure constant, is a positive integer that counts the number of ways of writing a given permutation of type γ as a product of two permutations of type λ and μ. In this paper, we present new formulas for the decomposition of the products K1rn−r ∗ K1sn−s and K(r, n−r) ∗ K(s, n−s) over a restricted set of conjugacy classes Kγ. These formulas generalize the formula for the decomposition of the product of the class of full cycles with itself K(n) ∗ K(n)

On products of conjugacy classes of the symmetric group

AbstractThe product of conjugacy classes of the symmetric group in its group algebra is found as a linear combination of conjugacy classes with integer coefficients. The purpose of this paper is to give a partial answer to the problem of finding simple combinatorial rules to obtain these coefficients. In particular, we will show that the product C(n)∗C(n) of the class of circular permutations with itself can be decomposed in a simple manner

Kronecher powers and character polynomials

In this talk, I will present joint works with Cedric Chauve and Adriano Garsia. With C. Chauve, we studied Kronecker powers of the irreducible representation of Sn indexed with (n-1,1). We gave a combinatorial interpretation and a generating function for the coefficients of any irreducible representation in a k-th Kronecker power ( χ(n-1,1) )⊗k. With A. Garsia, we studied character polynomials qλ(x1,…,xn) which are polynomials in several variables with the fundamental property that their evaluation on the multiplicities (m1,m2, …,mn) of a partition µ of n gives the value of the irreducible character χ( n- | λ | , λ ) of the symmetric group Sn on the conjugacy class Cµ . Character polynomials are closely related to the problem of decomposition of Kronecker product of representations of Sn. They were defined by Specht in 1960. Since then they received little attention from the combinatorics community. I will show how character polyomials are related to Kronecker products, how to produce them, their algebraic structure and show some applications

Characters and conjugacy classes of the symmetric group

Rapport interne.This article addresses several conjectures due to Jacob Katriel concerning conjugacy classes of $\S{n}$ viewed as operators acting by multiplication. The first conjecture expresses, for a fixed partition \r of the form $r1^{n-r}$, the eigenvalues (or central characters) \eo\r\l in terms of contents of \l. While Katriel conjectured a generic form and an algorithm to compute missing coefficients, we provide an explicit expression. The second conjecture (presented at FPSAC'98 in Toronto) gives a general form for the expression of a conjugacy class in terms of \emph{elementary} operators. We prove it using a convenient description by differential operators acting on symmetric polynomials. To conclude, we partially extend our results on \eo\r\l to arbitrary partitions \r. || Nous démontrons plusieurs conjectures dues à Jacob Katriel qui portent sur les classes de conjugaisons de $\S{n}$ vues comme opérateurs agissant par multiplication. La première conjecture exprime, pour une partition fixée $\rho$ de la forme $r1^{n-r}$, l

Content evaluation and class symmetric functions

AbstractIn this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over Q(n) that we call the basis of class symmetric functions.By definition this basis provides an algebra isomorphism between Λ and the Farahat–Higman algebra FH governing for all n the products of conjugacy classes in the center Zn of the group algebra of the symmetric group Sn. We thus obtain a calculus of all connexion coefficients of Zn inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients.We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras

Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes