On the Navarro-Willems conjecture for blocks of finite groups

We prove that a set of characters of a finite group can only be the set of characters for principal blocks of the group at two different primes when the primes do not divide the group order. This confirms a conjecture of Navarro and Willems in the case of principal blocks.     

Counting characters of small degree in upper unitriangular groups

Let U_n denote the group of upper n×n unitriangular matrices over a fixed finite field of order q. That is, U_n consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U_n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U_n of degree q^e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases.     

Generalized blocks for symmetric groups

We study, via character-theoretic methods, an ℓ-analogue of the modular representation theory of the symmetric group, for an arbitrary integer ℓ≥2. We find that many of the invariants of the usual block theory (ie. when ℓ is prime) generalize in a natural fashion to this new context. Oblatum 21-III-2002 & 5-VIII-2002?Published online: 8 November 2002     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Fields of values of Fong characters

If G is a p-solvable finite group with a p-complement H and φ ∈ IBr(G), then P. Fong showed that there exists α ∈ Irr(H) such that α^G=Φ_φ. In this note we prove that α can be chosen such that the field of values index divides φ (1)p. Received: 6 May 2005     

Correspondences between constituents of projective characters

Let G be a finite p-solvable group. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the irreducible constituents of p′-degree of the principal projective character of G and those of . Received: 2 May 2007, Revised: 17 September 2007     

Rouquier blocks

This paper investigates the Rouquier blocks of the Hecke algebras of the symmetric groups and the Rouquier blocks of the q-Schur algebras. We first give an algorithm for computing the decomposition numbers of these blocks in the ``abelian defect group case' and then use this algorithm to explicitly compute the decomposition numbers in a Rouquier block. For fields of characteristic zero, or when q=1 these results are known; significantly, our results also hold for fields of positive characteristic with q≠1. We also discuss the Rouquier blocks in the ``non–abelian defect group' case. Finally, we apply these results to show that certain Specht modules are irreducible.     

Finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels

In this paper, we study finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels. We prove several results concerning these groups and completely classify 2-groups with at most five nonlinear irreducible characters satisfying this property.     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

Growth of entire functions of exponential type defined by convolution

Based on the Borel transformation and the Hadamard multiplication theorem on singularities on the convolution of holomorphic functions, results on the growth of entire functions defined by convolution of an entire function of exponential type with a function holomorphic at the origin are obtained. Received: 29 August 2005     

Fong characters and chains of normal subgroups

In this note, we show that if is a π-partial character of the π-separable group is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then has a Fong character α Irr(H) such that for every subgroup , every irreducible constituent of α _H∩N is Fong for N. We also show that if is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of are Fong for M. Received: 21 July 2006 Revised: 17 January 2007     

Interlacing of zeros of shifted sequences of one-parameter orthogonal polynomials

We study the interlacing property of zeros of Laguerre polynomials of adjacent degree, where the free parameters differ by an integer, and of the same degree, where the free parameter is shifted continuously. Similar interlacing results are proven for the positive zeros of Gegenbauer polynomials. Research by Kathy Driver is supported by the National Research Foundation of South Africa under grant number 2053730. Research by Kerstin Jordaan is partially supported by the National Research Foundation of South Africa under grant number 2054423.     

Strongly connective degree sets I: nonabelian solvable quotients

When G is a finite nonabelian group, we associate the common-divisor graph (G) with G by letting nontrivial degrees in cd(G) be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of and every vertex outside of is adjacent to some member of When G has a nonabelian solvable quotient, we show that if (G) is connected and has a diameter of at most 2, then indeed cd(G) has a strongly connective subset.Received: 7 July 2004; revised: 5 October 2004     

Groups whose nonlinear irreducible characters separate element orders or conjugacy class sizes

A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C _G (x)| = |C _G (y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes. Received: 10 April 2007     

Comment on a paper of Rao et al., an entry of Ramanujan and a new 3F2(1)

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].     

Metaplectic operators for finite abelian groups and ?

The Segal-Shale-Weil representation associates to a symplectic transformation of the Heisenberg group an intertwining operator, called metaplectic operator. We develop an explicit construction of metaplectic operators for the Heisenberg group H(G) of a finite abelian group G, an important setting in finite time-frequency analysis. Our approach also yields a simple construction for the multivariate Euclidean case G = ?^d.     

Association schemes coming from minimal flats in classical polar spaces

Let denote a 2ν-dimensional vector space over a finite field F_q with a non-degenerate form of Witt index ν, and G denote the corresponding isometry group. Let W be a fixed maximal flat, and Θ be the set of all the minimal flats not contained in W. The set-wise stabilizer of W in G acts transitively on Θ and thus this action determines an association scheme. In this paper, we compute the intersection numbers and character tables for these schemes.     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.