93,474 research outputs found
On the tensor degree of finite groups
We study the number of elements and of a finite group such that
in the nonabelian tensor square
of . This number, divided by , is called the tensor degree of and
has connection with the exterior degree, introduced few years ago in [P.
Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra
39 (2011), 335--343]. The analysis of upper and lower bounds of the tensor
degree allows us to find interesting structural restrictions for the whole
group.Comment: 10 pages, accepted in Ars Combinatoria with revision
Remarks on the tensor degree of finite groups
The present paper is a note on the tensor degree of finite groups, introduced
recently in literature. This numerical invariant generalizes the commutativity
degree through the notion of nonabelian tensor square. We show two
inequalities, which correlate the tensor and the commutativity degree of finite
groups, and, indirectly, structural properties will be discussed.Comment: 5 pages; to appear with revisions in Filoma
On The Relative N-Tensor Nilpotent Degree of Finite Groups
In this paper, we introduce the relative -tensor nilpotent degree of a
finite group with respect to a subgroup of . The aim of this paper
is to investigate this concept and give some results on this topic
Tensor envelopes of regular categories
We extend the calculus of relations to embed a regular category A into a
family of pseudo-abelian tensor categories T(A,d) depending on a degree
function d. Under the condition that all objects of A have only finitely many
subobjects, our main results are as follows:
1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N
is semisimple provided that A is exact and Mal'cev. Thereby, we produce many
new semisimple, hence abelian, tensor categories.
2. Using lattice theory, we give a simple numerical criterion for the
vanishing of N.
3. We determine all degree functions for which T(A,d) is Tannakian. As a
result, we are able to interpolate the representation categories of many series
of profinite groups such as the symmetric groups S_n, the hyperoctahedral
groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed
finite field.
This paper generalizes work of Deligne, who first constructed the
interpolating category for the symmetric groups S_n. It also extends (and
provides proofs for) a previous paper math.CT/0605126 on the special case of
abelian categories.Comment: v1: 52 pages; v2: 52 pages, proof of Lemma 7.2 fixed, otherwise minor
change
Level, rank, and tensor growth of representations of symmetric groups
We develop a theory of levels for irreducible representations of symmetric
groups of degree analogous to the theory of levels for finite classical
groups. A key property of level is that the level of a character, provided it
is not too big compared to , gives a good lower bound on its degree, and,
moreover, every character of low degree is either itself of low level or
becomes so after tensoring with the sign character. Furthermore, if and
satisfy a linear upper bound in , then the maximal level of
composition factors of the tensor product of representations of levels
and is . To prove all of this in positive characteristic, we
develop the notion of rank, which is an analogue of the notion of rank of
cross-characteristic representations of finite classical groups. We show, using
modular branching rules and degenerate affine Hecke algebras, that the level
and the rank agree, as long as the level is not too large. We exploit
Schur-Weyl duality, modular Littlewood-Richardson coefficients and tilting
modules to prove a modular analogue of the Murnaghan-Littlewood theorem on
Kronecker products for symmetric groups. As an application, we obtain
representation growth results for both ordinary and modular representations of
symmetric and alternating groups analogous to those for finite groups of Lie
type
Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley
Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois\u27 theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama\u27s lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn\u27s theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside\u27s paqb Theorem; Injective modules
Connected components of compact matrix quantum groups and finiteness conditions
We introduce the notion of identity component of a compact quantum group and
that of total disconnectedness. As a drawback of the generalized Burnside
problem, we note that totally disconnected compact matrix quantum groups may
fail to be profinite. We consider the problem of approximating the identity
component as well as the maximal normal (in the sense of Wang) connected
subgroup by introducing canonical, but possibly transfinite, sequences of
subgroups. These sequences have a trivial behaviour in the classical case. We
give examples, arising as free products, where the identity component is not
normal and the associated sequence has length 1.
We give necessary and sufficient conditions for normality of the identity
component and finiteness or profiniteness of the quantum component group. Among
them, we introduce an ascending chain condition on the representation ring,
called Lie property, which characterizes Lie groups in the commutative case and
reduces to group Noetherianity of the dual in the cocommutative case. It is
weaker than ring Noetherianity but ensures existence of a generating
representation. The Lie property and ring Noetherianity are inherited by
quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an
example arising from the compact real form of U_q(sl_2) for q<0.Comment: 43 pages. Changes in the introduction. The relation between our and
Wang's notions of central subgroup has been clarifie
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