In its most schematic form, the Weyl character formula can be expressed by the ratio P Q of two multinomials P ≡ a(1)+a(2) . . . a(D) and Q ≡ b(1)+b(2) . . . b(D) where D is the order of Weyl groupW (Gr) for a Lie algebra Gr of rank r. Each and every one of a(k)’s and b(k)’s is obtained by the action of a Weyl reflection, i.e. an element of W (Gr). We, instead, show that there is a way to obtai...

We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X . Further consequences f...

A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S(n), the symmetric group on any number, n, of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2p i...

Let G be a finite group and V be a finite G–module. We present upper bounds for the cardinalities of certain subsets of Irr(GV ), such as the set of those χ ∈ Irr(GV ) such that, for a fixed v ∈ V , the restriction of χ to 〈v〉 is not a multiple of the regular character of 〈v〉. These results might be useful in attacking the non–coprime k(GV )–problem.

Computational methods for finding the character table of a finite group require a stock of reducible characters and techniques for decomposing these characters into irreducible characters. A method for generating reducible characters from a representation of a group is proposed. Several new techniques for computing the irreducible decompositions of a set of characters are given; these technique...