Characterization Of Characters Research Articles

Representations and Characters of Finite Groups.

Preface 1. General representation theory 2. Complex characters 3. Suzuki's theory of exceptional characters 4. Coherence and exceptional characters 5. The characterisation of characters 6. Isometries Appendix: Glauberman's Z*-theorem References Index of notation Index.

On an Identity Relating to Partitions and Repetitions of Parts

This note is concerned with a simple but rather surprising identity which emerged unexpectedly from the work of one of the authors on the characterisation of characters. Consider, for example, the seven partitions of 5. These are(1) 5, 4 1, 3 2, 3 12, 22 1, 2 13, 15and with each of these we can associate a product of factorials of the numbers of repetitions, respectively(2) 1!, (1!)(1!), (1!)(1!), (1!)(2!), (2!)(1!), (1!)(3!), 5!It is then seen that the product of all the numbers occurring in (1) coincides with that of all the numbers in (2).Generally, for any particular natural number n, the partitions can be written in the form1a1 2a2 3a3…,in which ak is the frequency of repetition of the part k, and are enumerated by the distinct sets {α} = {a1, a2, …,} with ak ≧ 0 and Σkak = n.

A characterization of generalized m-monomial characters

This paper continues a study of the character ring of a finite group begun in [S]. In [S] we characterized generalized permutation characters-that is, integral linear combinations of permutation characters of a group G. Our characterization involved, for each prime p dividing 1 G ] and each cyclic p’subgroup (x) of G, p-Sylow subgroups P, of C,(x) and pX of N&x) chosen so that P, E P,. To determine if a generalized character x of G is a generalized permutation character, one checks whether for each p and (x) as above a certain generalized character xX of P, extends to a suitable p-integral combination of characters of i),. In this paper we will characterize integral linear combinations of characters AC where A is a linear character of a subgroup of G and the order of A divides a fixed integer m. We will call such characters IZG m-monomial characters. Integral linear combinations of m-monomial characters are called generalized m-monomial characters. We may assume, of course, that m divides ( G I. When m= 1, an m-monomial character is precisely a transitive permutation character, integral linear combinations of which were characterized in ]5] as described above. When m = ] G ], the m-monomial characters include all monomial characters of G, and by Brauer’ s induction theorem, any generalized character of G is generalized m-monomial. Thus, the two extreme cases m = 1 and m = ] G 1 have been dealt with, so the purpose of this paper is to establish intermediate theorems between the main theorem of [S] and Braucr’s characterization of characters. Since we work mostly with p-integral rather than integral linear combinations of characters the main theorem of this paper in the case m = ] G 1 will yield not exactly Brauer’s characterization of characters but rather a p-integral version, due to J. Thompson, which says that a class function x on G is a 123

The Green Correspondence and Brauer's Characterization of Characters

The Green Correspondence and Brauer's Characterization of Characters

A characterization of characters which arise from -normalizers.

A characterization of characters which arise from -normalizers.