We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding visual properties, but also from the fact that they represent a novel and intriguing class of supercharacters.

We announce our successful computation of a list of representatives of the conjugacy classes of subgroups of Sn for n ≤ 18, including the 7 274 651 classes of subgroups of S18.

We describe a set of subgroups isomorphic to S4 in PΓO(5, q), q ≡ 5(6), and prove that they belong to exactly two different conjugacy classes. As an application we use a representative group in each of the conjugacy classes to construct a number of BLT-sets of Q(4, q), some of which were previously found by computer in [5] (see also [4]), others of which are new.

Ordered chains (such as chains of amino acids) are ubiquitous in biological cells, and these chains perform specific functions contingent on the sequence of their components. Using the existence and general properties of such sequences as a theoretical motivation, we study the statistical physics of systems whose state space is defined by the possible permutations of an ordered list, i.e., the ...

For m a non-negative integer and G a Coxeter group, we denote by QIm(G) the ring of m-quasiinvariants of G, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with QI0(G) the whole polynomial ring, and the limit QI∞(G) the usual ring of invariants. Remarkably, the ring QIm(G) is freely generated over the ideal generated by the invariants of G without constant term,...

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K . The anti-automorphism g 7→ g of G can be extended linearly to an anti-automorphism a 7→ a∗ of KG . Let S∗(KG) = {x ∈ U(KG) | x∗ = x} be the set of all symmetric units of U(KG) . We consider the following question: for which groups G and commutative rings K it is true that S∗(KG) is a subgroup in U(KG...

In this paper we classify the irreducible representations of the symmetric group Sn and give a proof of the hook formula for the dimension of each irreducible. 1. The Irreducible Representations of Sn We construct the irreducible representations of the symmetric group. The number of irreducible representations of Sn is the number of conjugacy classes of Sn, which is the number of partitions of ...

We lift Cardelli, Ghelli and Gordon’s secrecy group creation operator [1] to a relative of the spicalculus that supports symmetric key cryptography, and show a natural extension of the associated type system. We then formulate a notion of secrecy preservation in the presence of cryptography, and prove that well-typed processes in the extended type system preserve secrecy of declared secrets, ev...

A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S(n), the symmetric group on any number, n, of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2p i...