On zeros of characters of finite groups

For a finite group G and a non-linear irreducible complex character χ of G write υ(χ) = {g ∈ G | χ(g) = 0}. In this paper, we study the finite non-solvable groups G such that υ(χ) consists of at most two conjugacy classes for all but one of the non-linear irreducible characters χ of G. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable φ-groups. As a corollary, we answer Research Problem 2 in [Y.Berkovich and L.Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math. 24 (1998), 619–630.] posed by Y.Berkovich and L.Kazarin.     

On zeros of characters of finite groups

We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if is a finite group such that , then has an element , such that , where is the maximal number of zeros in a row of the character table of . Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let be a finite group such that ; then has an irreducible character such that , where is the maximal number of zeros in a column of the character table of .

     

On the mean value of the two-term exponential sums with Dirichlet characters

The main purpose of this paper is using the analytic method to study the mean value properties of the two-term exponential sums with Dirichlet characters, and give an explicit formula for its fourth power mean.     

On weakly quasi-primitive characters of solvable groups

We define weakly quasi-primitive characters of solvable groups as a generalization of quasi-primitive characters, and present three main results about the zeros and the values for these characters, which in turn strengthen the corresponding theorems for quasi-primitive characters due to G. Navarro and T. Wilde.     

On bideterminantal formulas for characters of classical groups

New bideterminantal formulas for the irreducible symplectic and orthogonal characters are given that generalize the classical bideterminantal formulas. These formulas are analogous to Regev’s (Israel J. Math. 80 (1992), 155–160) bideterminantal formulas for Schur functions, the irreducible general linear characters. Also, new bideterminantal formulas for Proctor’s intermediate symplectic characters are derived.     

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acting on a regular rooted tree acts absolutely non-freely on the boundary of the tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II₁-factor representations, and infinitely many continuous ergodic invariant random subgroups.     

On a certain application of the theory of characters for commutative groups

We study a certain application of the theory of characters for commutative compact topological groups and weakly linearly compact groups.     

Block characters of the symmetric groups

A block character of a finite symmetric group is a positive definite function which depends only on the number of cycles in a permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of ${\mathfrak{S}}_{n}$ . The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group ${\mathfrak{S}}_{\infty}$ , along with their connection to the Thoma characters of the infinite linear group GL _∞(q) over a Galois field.     

On the mean value of the mixed exponential sums with Dirichlet characters and general gauss sum

The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers m, n, k, q, with k ? 1 and q ? 3, and Dirichlet characters χ, χ? modulo q we define a mixed exponential sum $$C(m,n;k;\chi ;\overline \chi ;q) = \sum\limits_{a = 1}^q {\chi (a){G_k}(a,\overline \chi )e\left( {{{m{a^k} + n\overline {{a^k}} } \over q}} \right)} ,$$ , with Dirichlet character χ and general Gauss sum G _k (a, χ?) as coefficient, where Σ′ denotes the summation over all a such that (a, q) = 1, aā ≡ 1 mod q and e(y) = e^2πiy . We mean value of $$\sum\limits_m {\sum\limits_\chi {\sum\limits_{\overline \chi } {C{{\left| {m,n;k;\chi ;\overline \chi ;q} \right|}^4}} } } ,$$ , and give an exact computational formula for it.     

On functional equations associated with characters of unitary representations of groups

This is mainly concerned with the following functional equation: