An n × n complex sign pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is f(λ). If S is a spectrally arbitrary complex sign pattern matrix, and no proper subpattern of S is spectrally arbitrary, then S is a minimal sp...

In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, NSym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur...

A result of Roquette 3] states that if D is an absolutely irreducible representation of a p-group G over the eld of complex numbers, then D can be realized in K((g) j g 2 G), where is the character of D and K = Q or K = Q(i) according to whether p 6 = 2 or p = 2. Based on Baum and Clausen's 1] algorithm for computing the irreducible representations of supersolvable groups, we give an elementary...

Let F ′/F be a finite normal extension of number fields with Galois group Gal(F ′/F ). Let χ be an irreducible character of Gal(F ′/F ) of degree greater than one and L(s, χ) the associated Artin L-function. Assuming the truth of Artin’s conjecture, we have explicitly determined a zero-free region about 1 for L(s, χ). As an application we show that, for a CM-field K of degree 2n with solvable n...

We prove that a finite group G $G$ has normal Sylow p $p$ -subgroup P $P$ if, and only every irreducible character of appearing in the permutation ( 1 ) $({\bf 1}_P)^G$ with multiplicity coprime to degree . This confirms prediction by Malle Navarro from 2012. Our proof above result depends on reduction simple groups ultimately combinatorial analysis properties branching coefficients for symmetr...

Abstract We study the fields of values irreducible characters a finite group degree not divisible by prime p . In case where $p=2$ , we fully characterise these fields. order to accomplish this, generalise main result [ILNT] higher irrationalities. do same for odd primes, except that in this analogous results hold modulo simple-to-state conjecture on character quasi-simple groups.

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph ∆(G) is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ∈ cd(G). We determine the graph ∆(G) for the finite simple groups of types A`(q) and A`(q ), that is, for the simple linear and unitary gr...

Let G be a finite group and let cd(G) be the set of irreducible ordinary character degrees of G. The degree graph of G is the graph ∆(G) whose set of vertices is the set of primes dividing degrees in cd(G), with an edge between primes p and q if pq divides some degree in cd(G). We determine the graph ∆(G) for the finite simple groups of types B`, C`, D` and D`; that is, for the simple orthogona...

Let G be a finite group. We denote by ρ(G) the set of primes which divide some character degrees of G and by σ(G) the largest number of distinct primes which divide a single character degree of G. We show that |ρ(G)| ≤ 2σ(G) + 1 when G is an almost simple group. For arbitrary finite groups G, we show that |ρ(G)| ≤ 2σ(G) + 1 provided that σ(G) ≤ 2.

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the gr...