Let p be a prime, G a locally finite p -group, K a commutative ring of characteristic p . The anti-automorphism g 7→ g of G extends linearly to an anti-automorphism a 7→ a∗ of KG . An element a of KG is called symmetric if a∗ = a . In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group. Let G be a group, K a commutative ring (with 1), an...

The geometry of moduli spaces of complex abelian varieties and of compactifications of those moduli spaces has been the object of much study in the last few years. Given such a compactified moduli space it is natural to ask about the fundamental group of a resolution of singularities. This problem has been studied in some special cases, for instance in [K], [HK] and [HS]. It follows from the re...

We consider sums of the form π∈Sn χ λ/µ (π) q inv(π) where χ λ/µ is a skew character of the symmetric group and inv(π) is the number of inversions of π. Our main result gives a lower bound on the number of factors of 1 + q and 1 − q which divide the above sum, and is shown to be sharp when λ/µ is a hook partition shape. Résumé Nous considérons des sommes de la forme

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diag S(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand ...

In this article we prove the following result: that for any two natural numbers k and `, and for all sufficiently large symmetric groups Sn, there are k disjoint sets of ` irreducible characters of Sn, such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó in [5]. The method...

We construct the Specht modules and prove that they completely characterize the irreducible representations of the symmetric group. We will prove certain properties of these representations using combinatorial tools (such as calculating the dimension using Hook’s length formula). Only an introductory knowledge of group theory and linear algebra will be assumed and representation theory concepts...