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## On Representations of Classical Groups over Finite Local Rings of Length Two

Singla, Pooja
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
We study the complex irreducible representations of special linear, symplectic, orthogonal and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by author.; Comment: 10 pages

## Brauer pairs of Camina p-groups of nilpotence class 2

Lewis, Mark L.
Tipo: Artigo de Revista Científica
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16.45%
In this paper, we find a condition that characterizes when two Camina $p$-groups of nilpotence class 2 form a Brauer pair.

## Lifts of partial characters with respect to a chain of normal subgroups

Lewis, Mark L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
In this paper, we consider lifts of $\pi$-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is a such a lift. With this condition, we will get a lower bound on the number of such lifts.

## Groups where all the irreducible characters are super-monomial

Lewis, Mark L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
Isaacs has defined a character to be super monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $G$ is an $M$-group with odd order, then every irreducible character is super monomial. We prove that the conjecture is true if $G$ is an $M$-group of odd order where every irreducible character is a $\{p \}$-lift for some prime $p$. We say that a group where irreducible character is super monomial is a super $M$-group. We use our results to find an example of a super $M$-group that has a subgroup that is not a super $M$-group.

## Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

Thiem, Nathaniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.; Comment: 24 pages

## Extreme lattices and vexillar designs

Meyer, Bertrand
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
We define a notion of vexillar design for the flag variety in the spirit of the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite subgroup of the orthogonal group, we explain how conditions on the group have the orbits of any flag under the group action be a design and point out why the minima of a lattice in the sense of the general Hermite constant forming a 4-design implies being extreme. The reasoning proves useful to show the extremality of many new expected examples ($E_8$, $\La_{24}$, Barnes-Wall lattices, Thompson-Smith lattice for instance) that were out of reach until now.; Comment: 16pages

## Variables separated equations: Strikingly different roles for the Branch Cycle Lemma and the Finite Simple Group Classification

Fried, Michael d.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
H. Davenport's Problem asks: What can we expect of two polynomials, over the integers, with the same ranges on almost all residue class fields? This stood out among many separated variable problems posed by Davenport, D.J. Lewis and A. Schinzel. By bounding the degrees, but expanding the maps and variables in Davenport's Problem, Galois stratification enhanced the separated variable theme, solving an Ax and Kochen problem from their Artin Conjecture work. J. Denef and F. Loeser applied this to add Chow motive coefficients to previously introduced zeta functions on a diophantine statement. By restricting the variables, but leaving the degrees unbounded, we found the striking distinction between Davenport's problem over the rationals, solved by applying the Branch Cycle Lemma, and its generalization over any number field, solved using the simple group classification. This encouraged J. Thompson to formulate the genus 0 problem on rational function monodromy groups. R. Guralnick and Thompson led its solution in stages. We look at at two developments since the solution of Davenport's problem. * Stemming from C. MacCluer's 1967 thesis, identifying a general class of problems, including Davenport's, as monodromy precise. * R(iemann) E(xistence) T(heorem)'s role as a converse to problems generalizing Davenport's...

## Products of conjugacy classes and fixed point spaces

Guralnick, Robert; Malle, Gunter
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth powers for any prime power r (in particular, a product of two squares).; Comment: 44 pages

## Microlocal analysis on wonderful varieties. Regularized traces and global characters

Cupit-Foutou, Stephanie; Parthasarathy, Aprameyan; Ramacher, Pablo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
Let $\mathbf{G}$ be a connected reductive complex algebraic group with split real form $(G,\sigma)$. Consider a strict wonderful $\mathbf{G}$-variety $\bf{X}$ equipped with its $\sigma$-equivariant real structure, and let $X$ be the corresponding real locus. Further, let $E$ be a real differentiable $G$-vector bundle over $X$. In this paper, we introduce a distribution character for the regular representation of $G$ on the space of smooth sections of $E$, and show that on a certain open subset of $G$ of transversal elements it is locally integrable and given by a sum over fixed points.; Comment: Revised version, 22 pages

## Fitting heights of solvable groups with no nontrivial prime power character degrees

Lewis, Mark L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
We construct solvable groups where the only degree of an irreducible character that is a prime power is $1$ and that have arbitrarily large Fitting heights. We will show that we can construct such groups that also have a Sylow tower. We also will show that we can construct such groups using only three primes.; Comment: 6 pages

## Irreducible representations and Artin L-functions of quasi-cyclotomic fields

Bae, Sunghan; Hu, Yong; Yin, Linsheng
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields.; Comment: 17 pages

## Controlling composition factors of a finite group by its character degree ratio

Cossey, James P.; Nguyen, Hung Ngoc
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
For a finite nonabelian group $G$ let $\rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by $\rat(G)$ in some sense. Specifically, if $S$ different from the simple linear groups $\PSL_2(q)$ is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of $\rat(G)$. Furthermore, when the groups $\PSL_2(q)$ are not composition factors of $G$, we prove that $|G:\Oinfty(G)|\leq \rat(G)^{21}$ where $\Oinfty(G)$ denotes the solvable radical of $G$.; Comment: 16 pages, 1 table

## Reflection factorizations of Singer cycles

Lewis, Joel Brewster; Reiner, Victor; Stanton, Dennis
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^(n - 1). Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and transvections. The results suggest several open problems and questions, which are discussed at the end.; Comment: Historical references added; final version to appear in J. Algebraic Combinatorics

## Faithfulness of the Lawrence representation of braid groups

Zheng, Hao
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
The Lawrence representation $L_{n,m}$ is a family of homological representation of the braid group $B_n$, which specializes to the reduced Burau and the Lawrence-Krammer representation when $m$ is 1 and 2. In this article we show that the Lawrence representation is faithful for $m \geq 2$.; Comment: 9 pages, 6 figures

## Arithmetical Properties of Finite Groups

Shi, Wujie
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
Let $G$ be a finite group and $Ch_i(G)$ some quantitative sets. In this paper we study the influence of $Ch_i(G)$ to the structure of $G$. We present a survey of author and his colleagues' recent works.; Comment: 7 pages

## A Characterization of $L_2(2^f)$ in Terms of Character Zeros

Qian, Guohua; Shi, Wujie
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.; Comment: 11 pages

## Decompositions of stochastic processes based on irreductible group representations

Peccati, Giovanni; Pycke, Jean-Renaud
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
Let G be a topological compact group acting on some space Y. We study a decomposition of Y-indexed stochastic processes, based on the orthogonality relations between the characters of the irreducible representations of G. In the particular case of a Gaussian process with a G-invariant law, such a decomposition gives a very general explanation of a classic identity in law - between quadratic functionals of a Brownian bridge - due to Watson (1961). Several relations with Karhunen-Lo\`{e}ve expansions are discussed, and some applications and extensions are given - in particular related to Gaussian processes indexed by a torus.; Comment: 27 pages

## Kazhdan--Lusztig cells and the Frobenius--Schur indicator

Geck, Meinolf
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.45%
Let $W$ be a finite Coxeter group. It is well-known that the number of involutions in $W$ is equal to the sum of the degrees of the irreducible characters of $W$. Following a suggestion of Lusztig, we show that this equality is compatible with the decomposition of $W$ into Kazhdan--Lusztig cells. The proof uses a generalisation of the Frobenius--Schur indicator to symmetric algebras, which may be of independent interest.; Comment: 12 pages; the new version contains some additions in Section 3

## Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations

Suter, Ruedi
Tipo: Artigo de Revista Científica
The set of subspaces of a given dimension in an attenuated space has a structure of a symmetric association scheme and this association scheme is called an association scheme based on an attenuated space. Association schemes based on attenuated spaces are generalizations of Grassmann schemes and bilinear forms schemes, and also $q$-analogues of non-binary Johnson schemes. Wang, Guo and Li computed the intersection numbers of association schemes based on attenuated spaces. The aim of this paper is to compute character tables of association schemes based on attenuated spaces using the method of Tarnanen, Aaltonen and Goethals. Moreover, we also prove that association schemes based on attenuated spaces include as a special case the $m$-flat association scheme, which is defined on the set of cosets of subspaces of a constant dimension in a vector space over a finite field.; Comment: 15 pages