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

Symmetric groups and conjugacy classes

Let S_n be the symmetric group on n-letters. Fix n>5. Given any nontrivial $\alpha,\beta\in S_n$, we prove that the product $\alpha^{S_n}\beta^{S_n}$ of the conjugacy classes $\alpha^{S_n}$ and $\beta^{S_n}$ is never a conjugacy class. Furthermore, if n is not even and $n$ is not a multiple of three, then $\alpha^{S_n}\beta^{S_n}$ is the union of at least three distinct conjugacy classes. We also describe the elements $\alpha,\beta\in S_n$ in the case when $\alpha^{S_n}\beta^{S_n}$ is the union of exactly two distinct conjugacy classes.Comment: 7 page

Product of Conjugacy Classes of the Alternating Group An

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptional of I (I the identity conjugacy class), when n=5+4k , k>=0

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)

Product of Conjugacy Classes of the Alternating Group An

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptional of I (I the identity conjugacy class), when n=5+4k , k>=0

Asymptotics of characters of symmetric groups, genus expansion and free probability

The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles the genus expansion from the random matrix theory. As the main application we study irreducible representations of symmetric groups S_q for large q. We find the asymptotic behavior of characters when the corresponding Young diagram rescaled by a factor q^{-1/2} converge to a prescribed shape. The character formula (known as the Kerov polynomial) can be viewed as a power series, the terms of which correspond to two-dimensional surfaces with prescribed genus and we compute explicitly the first two terms, thus we prove a conjecture of Biane.Comment: version 2: change of title; the section on Gaussian fluctuations was moved to a subsequent paper [Piotr Sniady: "Gaussian fluctuations of characters of symmetric groups and of Young diagrams" math.CO/0501112

The Topological Symmetric Orbifold

We analyse topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the quantum cohomology of the Hilbert scheme on the boundary.Comment: 33 pages. v2: Remarks on proof adde