2,647,618 research outputs found

    Groups with few pp'-character degrees

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    We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group GG contains only two distinct values not divisible by a given prime number p>3p>3, then Opppp(G)=1O^{pp'pp'}(G)=1. This is done by using the classification of finite simple groups

    pp-adic estimates for multiplicative character sums

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    This article is an expanded version of the talk given by the first author at the conference "Exponential sums over finite fields and applications" (ETH, Z\"urich, November, 2010). We state some conjectures on archimedian and pp-adic estimates for multiplicative character sums over smooth projective varieties. We also review some of the results of J. Dollarhide, which formed the basis for these conjectures. Applying his results, we prove one of the conjectures when the smooth projective variety is Pn{\mathbb P}^n itself.Comment: 9 page

    Character codegrees of maximal class p-groups

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    Let GG be a pp-group and let χ\chi be an irreducible character of GG. The codegree of χ\chi is given by G:ker(χ)/χ(1)|G:\text{ker}(\chi)|/\chi(1). If GG is a maximal class pp-group that is normally monomial or has at most three character degrees then the codegrees of GG are consecutive powers of pp. If G=pn|G|=p^n and GG has consecutive pp-power codegrees up to pn1p^{n-1} then the nilpotence class of GG is at most 2 or GG has maximal class

    Variations on average character degrees and pp-nilpotence

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    We prove that if pp is an odd prime, GG is a solvable group, and the average value of the irreducible characters of GG whose degrees are not divisible by pp is strictly less than 2(p+1)/(p+3)2(p+1)/(p+3), then GG is pp-nilpotent. We show that there are examples that are not pp-nilpotent where this bound is met for every prime pp. We then prove a number of variations of this result.Comment: 14 page

    Character sheaves on unipotent groups in characteristic p>0

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    These are slides for a talk given by the authors at the conference "Current developments and directions in the Langlands program" held in honor of Robert Langlands at the Northwestern University in May of 2008. The slides can be used as a short introduction to the theory of characters and character sheaves for unipotent groups in positive characteristic, developed by the authors in a series of articles written between 2006 and 2011. We give an overview of the main results of this theory along with a bit of motivation

    Characters of p'-degree and Thompson's character degree theorem

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    A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group GG is 1 or divisible by a prime pp, then GG has a normal pp-complement. We obtain a significant improvement of this result by considering the average of pp'-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.Comment: 23 page

    p-Blocks Relative to a Character of a Normal Subgroup

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    Let G be a finite group, let N be a normal subgroup of G, and let theta in Irr(N) be a G-invariant character. We fix a prime p, and we introduce a canonical partition of Irr(G|theta) relative to p. We call each member B_theta of this partition a theta-block, and to each theta-block B_theta we naturally associate a conjugacy class of p-subgroups of G/N, which we call the theta-defect groups of B_theta. If N is trivial, then the theta-blocks are the Brauer p-blocks. Using theta-blocks, we can unify the Gluck-Wolf-Navarro-Tiep theorem and Brauer's Height Zero conjecture in a single statement, which, after work of B. Sambale, turns out to be equivalent to the the Height Zero conjecture. We also prove that the k(B)-conjecture is true if and only if every theta-block B_theta has size less than or equal the size of any of its theta-defect groups, hence bringing normal subgroups to the k(B)-conjecture

    Structural and bonding character of potassium-doped p-terphenyl superconductors

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    Recently, there is a series of reports by Wang et al. on the superconductivity in K-doped p-terphenyl (KxC18H14) with the transition temperatures range from 7 to 123 Kelvin. Identifying the structural and bonding character is the key to understand the superconducting phases and the related properties. Therefore we carried out an extensive study on the crystal structures with different doping levels and investigate the thermodynamic stability, structural, electronic, and magnetic properties by the first-principles calculations. Our calculated structures capture most features of the experimentally observed X-ray diffraction patterns. The K doping concentration is constrained to within the range of 2 and 3. The obtained formation energy indicates that the system at x = 2.5 is more stable. The strong ionic bonding interaction is found in between K atoms and organic molecules. The charge transfer accounts for the metallic feature of the doped materials. For a small amount of charge transferred, the tilting force between the two successive benzenes drives the system to stabilize at the antiferromagnetic ground state, while the system exhibits non-magnetic behavior with increasing charge transfer. The multiformity of band structures near the Fermi level indicates that the driving force for superconductivity is complicated.Comment: 8 pages, 7 figure

    Short Character Sums and the P\'{o}lya-Vinogradov Inequality

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    We show in a quantitative way that any odd character χ\chi modulo qq of fixed order g2g \geq 2 satisfies the property that if the P\'{o}lya-Vinogradov inequality for χ\chi can be improved to max1tqntχ(n)=oq(qlogq)\max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q) then for any ϵ>0\epsilon > 0 one may exhibit cancellation in partial sums of χ\chi on the interval [1,t][1,t] whenever t>qϵt > q^{\epsilon}, i.e.,ntχ(n)=oq(t) for all t>qϵ.\sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t) \text{ for all $t > q^{\epsilon}$.} This generalizes and extends a result of Fromm and Goldmakher. We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing gg exhibit cancellation in short sums then the P\'{o}lya-Vinogradov inequality can be improved for all odd primitive characters of order gg. Some applications are also discussed.Comment: 25 page

    A ϕ1,3\phi_{1,3}-filtration of the Virasoro minimal series M(p,p') with 1<p'/p< 2

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    The filtration of the Virasoro minimal series representations M^{(p,p')}_{r,s} induced by the (1,3)-primary field ϕ1,3(z)\phi_{1,3}(z) is studied. For 1< p'/p< 2, a conjectural basis of M^{(p,p')}_{r,s} compatible with the filtration is given by using monomial vectors in terms of the Fourier coefficients of ϕ1,3(z)\phi_{1,3}(z). In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series (p'=p+1), we establish for each mm the equality between the character of the degree mm monomial basis and the character of the degree mm component in the associated graded module Gr(M^{(p,p+1)}_{r,s}) with respect to the filtration defined by ϕ1,3(z)\phi_{1,3}(z).Comment: 34 pages, no figur
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