On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ 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 $n$-tensor nilpotent degree of a finite group $G$ with respect to a subgroup $H$ of $G$. 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 $n$ 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 $n$, 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 $l_1$ and $l_2$ satisfy a linear upper bound in $n$, then the maximal level of composition factors of the tensor product of representations of levels $l_1$ and $l_2$ is $l_1+l_2$. 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

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