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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/09/2008

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In this paper, we find a condition that characterizes when two Camina
$p$-groups of nilpotence class 2 form a Brauer pair.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/12/2008

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/12/2008

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/12/2008

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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

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## Extreme lattices and vexillar designs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/12/2008

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Number Theory#Mathematics - Algebraic Geometry#Mathematics - Group Theory#Primary 11G18, 141130, 14H25, 14M41, 20B15, 20C15, 30F10, Secondary
11R58, 12D05, 12E30, 12F10, 20E22

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...

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## Products of conjugacy classes and fixed point spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Group Theory#Mathematics - Representation Theory#20D05 (Primary), 20C20, 20C15 (Secondary)

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/06/2015

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/04/2007

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2013

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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

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## Reflection factorizations of Singer cycles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## Faithfulness of the Lawrence representation of braid groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/09/2005

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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

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## Arithmetical Properties of Finite Groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/09/2005

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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

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## Decompositions of stochastic processes based on irreductible group representations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/09/2005

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2005

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We first review some invariant theoretic results about the finite subgroups
of SU(2) in a quick algebraic way by using the McKay correspondence and quantum
affine Cartan matrices. By the way it turns out that some parameters
(a,b,h;p,q,r) that one usually associates with such a group and hence with a
simply-laced Coxeter-Dynkin diagram have a meaningful definition for the
non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula
for the determinant of the Cartan matrix to all cases. Returning to invariant
theory we show that for each irreducible representation i of a binary
tetrahedral, octahedral, or icosahedral group one can find a homomorphism into
a finite complex reflection group whose defining reflection representation
restricts to i.; Comment: 19 pages

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## Character tables of association schemes based on attenuated spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/01/2011

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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

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