Let G be a finite group and Ω a set of n elements. Assume that G acts faithfully on Ω and let V be a vector space over the complex field C, with dimV = m ≥ 2. It is shown that for each irreducible constituent χ of permutation character of G, the symmetry class of tensors associated with G and χ is non-trivial. This extends a result of Merris and Rashid (see [6, Theorem 2]).

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

Character polynomials are used to study the restriction of a polynomial representation general linear group its subgroup permutation matrices. A simple formula is obtained for computing inner products class functions given by character polynomials. symmetric and alternating tensors computed using generating with Eulerian factorizations. These compute Weyl modules, which exhibit duality. By taki...

We prove that a finite group G $G$ has normal Sylow p $p$ -subgroup P $P$ if, and only every irreducible character of appearing in the permutation ( 1 ) $({\bf 1}_P)^G$ with multiplicity coprime to degree . This confirms prediction by Malle Navarro from 2012. Our proof above result depends on reduction simple groups ultimately combinatorial analysis properties branching coefficients for symmetr...

The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an O(1/r) complexity (r is the number of passes of all blocks). However, such bounds are at least linearly depend on K (the number of variable blocks), and are a...

1.1 Let A = kG, the group algebra of some finite group where the characteristic of the field k divides |G|. In contrast to working over the complex field, the kG-modules are not usually semisimple. If a Sylow p-subgroup of G is not cyclic then there are infinitely many indecomposable kG-modules, and we usually enjoy little control over the category of such modules. It is therefore instructive t...

In this paper, we define a new type of product of association schemes (and of the related objects, permutation groups and orthogonal block structures), which generalizes the direct and wreath products (which are referred to as “crossing” and “nesting” in the statistical literature.) Given two association schemes Qr for r = 1, 2, each having an inherent partition Fr (that is, a partition whose e...

Let G be a group. A G-set is a finite set provided with a left action of g 4 G. For a G-set X and g g G we write X s x g X : gx s x . The permutation character of a G-set X is the function p from G to the ring X Z of integers that sends g to the cardinality aX g of X . We say that two G-sets X and Y are linearly equï alent if p s p . By representation X Y theory of finite groups, two G-sets X a...

Problem 6.6 in the Kourovka Notebook [9], posed by H. Wielandt, reads as follows. ' Let P, Q be permutation representations of a finite group G with the same character. Suppose P(G) is a primitive permutation group. Is Q{G) necessarily primitive? Equivalently: Let A, B be subgroups of a finite group G such that for each class C of conjugate elements of G their intersection with C has the same c...

Let Sn denote the full symmetric permutation group of degree n. For each a E e, let (J (a) be the corresponding permutation matrix, i.e., Q(a) = (0 iUU»)' If e is any subgroup of S n, then Q is a faithful representation of e whose c haracter, e, counts the number of fixed points . In trus note, we in vestigate a red uction of fr, the character of the rth Kronecker power of (J . The reduction of...