Ernst Snapper

Numerical polynomials for arbitrary characters

Abstract Let G be a finite group. To each permutation representation (G, X) of G and each class function χ of G we associate a “generalized cycle index” P(G, X; χ) ϵ k[x1,…, xm], where k is the field of the complex numbers and m is the number of elements in the finite set X. If the representation (G, X) is faithful and χ is the trivial character 1G of G, P(G, X; 1G) is the usual cycle index of combinatorial mathematics. We prove that, if χ is a generalized character of G, the coefficients of P(G, X; χ) are rational numbers and the substitution x i → Σ j i jx j transforms the polynomial P(G, X; χ) into a numerical polynomial. This generalizes Theorem 1.1 of [8] from the trivial character 1G to all generalized characters. We apply the theory to de Bruijns theory of counting [3] and show that the class function which dominates [3] is actually a generalized character of G. We finish by showing how the theory of the generalized cycle index P(G, X; χ) gives the powerful theorem of Frobenius, which states that every simple character of the symmetric group is an integral linear combination of transitive permutation characters.

Characteristic polynomials of a permutation representation

Abstract Let G be a finite group. To each permutation representation ( G , X ) of G and class function χ of G we associate the χ- characteristic polynomial g χ ( x ) of ( G , X ) which is a polynomial in one variable with complex numbers as coefficients. The coefficients of g χ ( x ) are investigated in terms of the “exterior powers” of ( G , X ). If χ is the principal character 1 G of G , the coefficients of g χ ( x ) are non-negative integers; and if furthermore G has odd order, the k th coefficient is the number of orbits of G acting on the subsets of X with k elements. Quadratic and linear relations among the exterior powers of ( G , X ) have been derived and it is shown how the polynomial g χ ( x ) can be computed from the cycle index of ( G , X ).

The polynomial of a permutation representation

Abstract Let (G, D) be a permutation representation of a finite group G acting on a finite set D. The cycle index of this representation is a polynomial P(G,D;x1,…,xm) in several variables x1,…,xm with rational numbers as coefficients (see [1]). The restriction, made in [1], that the representation (G, D) is faithful, is unnecessary and we put no restriction on (G, D) whatsoever. We replace each variable xi of the cycle index P(G, D; x1,…,xm) by the polynomial Σjxj, where j runs through the divisors of i. For instance, x1→x1; x2→x1+2x2; x12→x1+2x2+3x3+4x4+6x6+12x12; etc. The resulting polynomial q(G, D; x1,…,xm) still has rational numbers as coefficients and has the additional property: o Theorem 1. If all the variables x1,…xm of the polynomial q(G, D; x1,…,xm) are replaced by integers (=whole numbers), the value of q is also an integer. The proof of Theorem 1 is based on [2]. We then investigate the polynomial q(G, D; x1,…,xm) further in the case that (G, D) is the regular representation, i.e., when D=G and G acts on G by left multiplication. We prove: Theorem 2. If (G, D) is the regular representation, Theorem 1 is equivalent to the following theorem of Frobenius: The number of solutions in G of the equation xi=1 is divisible by the greatest common divisor of i and the order of G. Because of Theorem 2, we consider Theorem 1 as the extension of the above Frobenius theorem to all permutation representations. The polynomial q(G, D; x1,…,xm) for arbitrary permutation representations has been further investigated in [4].

Quadratic spaces over finite fields and codes

Abstract Let V be a finite-dimensional quadratic space over a finite field GF ( ϱ ) of characteristic different from 2. It is shown that, even if V is singular, the geometry of V is completely determined by the number of points on the unit sphere, the “sphere of the nonsquares,” and the “0-sphere.” For ϱ = 3, this implies that two codes over GF (3) with the same weight enumerator are isometric.

metric vector spaces

This chapter focuses on metric vector spaces. Affine geometry is the study of incidence and parallelism. A vector space, provided with an inner product, is called a metric vector space, a vector space with metric or even a geometry. It is very important to adopt the geometric attitude toward metric vector spaces. This is done by taking the pictures and language from Euclidean geometry. Two vectors A and B of a metric vector space are orthogonal—perpendicular—if AB = 0. The definition does not exclude the possibility that one or both of the vectors A, B is equal to the origin 0. In Euclidean geometry, the only vector orthogonal to all vectors is the origin. A metric vector space is called nonsingular if the origin is the only vector which is orthogonal to all vectors. In general, two mathematical structures are called “equivalent” or isomorphic if there exists a one-to-one mapping from one of the structures onto the other one which preserves all intrinsic structure. Such a mapping is called an isomorphism and hence, two structures are isomorphic if there exists an isomorphism between them.

metric affine spaces

This chapter focuses on metric affine spaces. A metric affine space is an affine space ( X , V , k ) where V is a metric vector space. Two affine subspaces of X are called orthogonal—perpendicular—if their direction spaces are orthogonal. The chapter discusses a theorem that considers the rigid motions which leave a point of X fixed. It states that these motions form a group and describe the group fully. It follows immediately from the theorem that the structure of the group of motions which leave a point fixed does not depend on the choice of that point. In Euclidean geometry, it is customary to define a Euclidean transformation as a one-to-one function of X onto itself that preserves distance. In other words, one never mentions that a Euclidean transformation has to be an affine transformation. Each theorem for metric vector spaces has an analogue for metric affine spaces. Usually, all that one has to do to find the analogue is to investigate the behavior of the translations of X . The Cartan-Dieudonne theorem for affine spaces is also proven in the chapter.

chapter 1 – affine geometry

Publisher Summary This chapter focuses on the study of affine geometry. Affine geometry is what remains after practically all ability to measure length, area, and angles, has been removed from Euclidean geometry. The whole theory of homothetic figures lies within affine geometry. The notions of translation and magnification—these are the dilations—are in the domain of affine geometry and, more generally, affine transformations—one-to-one, onto functions which preserve parallelism—constitute an affine notion. The chapter discusses axioms for affine geometry. A division ring for which multiplication is commutative is called a field. A model for n-dimensional affine space is reviewed. The affine sub-spaces of dimension n -1 are called hyperplanes. As the intersection of two affine subspaces can be empty and an affine subspace is never empty, the intersection of two affine subspaces is not always an affine subspace. With the exception of the empty intersection, the intersection of affine subspaces is an affine subspace.