Trace Formulas on Finite Groups

The European Physical Journal Conferences

A few elements of the formalism of finite group representations are recalled. As to avoid a too mathematically oriented approach the discussed items are limited to the most essential aspects of the linear and matrix representations of standard use in chemistry and physics.

Чебышевский сборник, 2019

We study realization fields and integrality of characters of discrete and finite subgroups of 2 (C) and related lattices with a focus on on the integrality of characters of finite groups. Theory of characters of finite and infinite groups plays the central role in the group theory and the theory of representations of finite groups and associative algebras. The classical results are related to some arithmetic problems: the description of integral representations are essential for finite groups over rings of integers in number fields, local fields, or, more generally, for Dedekind rings. A substantial part of this paper is devoted to the following question, coming back to W. Burnside: whether every representation over a number field can be made integral. Given a linear representation : → () of finite group over a number field /Q, is it conjugate in () to a representation : → () over the ring of integers ? To study this question, it is possible to translate integrality into the setting of lattices. This question is closely related to globally irreducible representations; the concept introduced by J. G. Thompson and B. Gross, was developed by Pham Huu Tiep and generalized by F. Van Oystaeyen and A.E. Zalesskii, and there are still many open questions. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, splitting fields, and, in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras.

1998

In this paper we investigate the representations of reductive groups over a finite field, introduced in 1987 by D.Kazhdan and G.Laumon. We show that generically these representations are irreducible and that their character is equal to the function obtained by traces of Frobenius morphism on the stalks of the corresponding Lusztig's character sheaf. This, together with the corresponding result by

1997

Eichler's trace formula for traces of the Brandt-Eichler matrices is proved for arbitrary totally definite orders in central simple algebras of prime index over global fields. A formula for type numbers of such orders is proved as an application.

2002

Sei V ein endlich-dimensionaler, komplexer Vektorraum. Eine Teilmenge X in V hat die Trennungseigenschaft, falls das Folgende gilt: Für je zwei linear unabhängige lineare Funktionen l, m auf V existiert ein Punkt x in X mit l(x) = 0 und m(x) = 0. Wir interessieren uns für den Fall V = C[x, y] n , d.h. V ist eine irreduzible Darstellung von SL 2 . Die Teilmengen, die wir untersuchen, sind Bahnabschlüsse von Elementen aus C[x, y] n . Wir beschreiben die Bahnen, die die Trennungseigenschaft erfüllen:

Transactions of the American Mathematical Society, 1982

We prove an alternating sum formula for multiplicities in L2(T \ G), where G is a semisimple Lie group of split rank one with finite center and T is a discrete cocompact torsion free subgroup. Introduction. Let G be a connected semisimple Lie group with finite center. Let G = K ■ A ■ N be an Iwasawa decomposition of G and g = k © a © n the corresponding decomposition at the algebra level. Let M denote the centralizer of A in K. We will assume throughout this paper that G has rank one, that is, dim(a) = 1. We will not assume that G is linear. Let &(G) denote the set of equivalence classes of irreducible unitary representations of G. Given a homomorphism p: %-> C, where % is the center of ^l(g) (the universal enveloping algebra of G), let SM(C7) denote the (finite) subset of &(G) of those classes with infinitesimal character p. If S is any subset of &(G) we write S^ = S n êy(C7). Let T be a discrete, torsion free subgroup of G such that r \ G is compact. The right regular representation decomposes into a direct sum ttt = 2ues(G) "A03) ' w and Hr(co) < co, for any co E &(G). The purpose of this paper is to prove an alternating sum formula connecting the multiplicities nr(co) of representations having the same infinitesimal character. We now describe our result. Let K and M be as above. Let /*: R(K)-> R(M) denote the restriction map from the representation ring of K to that of M. If t E S (A) and co E &(G), then [t : co] stands for the multiplicity of t in the restriction of to to A. If rj E R(K) we abuse notation by writing [n : to] = Sm, • [t, : co], where i\ = Sm, • t,. Let &C(G) denote those classes in &(G) corresponding to the irreducible unitary principal series or the complementary series and let &2(G) denote the discrete series of G. Let a: &(G)->Z be defined as follows: a(co) = 0, if co £ &C(G); a(co) = nr(co)-d(u) • \ol(T \ G), if co E &2(G); and a(co) = «r(co) if co E ë(G)-(&C(G) u &2(G)). Here ¿/(to) denotes the formal degree of co. (We observe that d(u) and vol(r \ G) depend on the choice of the Haar measure on G, but their product does not). Finally let A be a fixed infinitesimal character of G. We prove (Theorem 1.2) that if r/ £ ker(/'*), then (i) 2 «(«)•[>:«]-a "6ESA(G)

2013

Let G be a finite group and let k be a field of char(k) = 2. We explicitly describe the set of trace forms of G-Galois algebras over k when the virtual 2-cohomological dimension vcd2(k) of k is at most 1. For fields with vcd2(k) ≤ 2 we give a cohomological criterion for the orthogonal sum of a trace form of a G-Galois algebra with itself to be isomorphic to another such form.

Representation Theory of Group, 2021

Representation theory of groups: the first part concern the representation theory of finite groups (with also character theory) and the second part the discussion is focused on the Lie Groups/Algebras. The topics are: Introduction Basic notions One dimensional representations (abelian groups) Character of a representation Reduction and induction of representations Non finite groups ( Algebraic sets , Lie algebra) Solvability and (semi-)simplicity Root systems and Dynkin diagrams Universal enveloping Lie algebra At the end many exercises are solved

1999

We introduce the concept of a modulator, which leads to a family of character formulae, each generalizing the Kirillov formula. For a suitable choice of modulator, this enables one to understand the Plancherel measure of a compact Lie group as arising from a partition of the identity on the dual of its Lie algebra.

Representation Thoery of Finite Groups, 2012

In this article, the character tables of some finite groups are constructed. The Mathematical knowledge needed to do such constructions is developed within the first three chapters. The modular theory approach to representation theory was adopted in this work; hence strong emphasis was placed on modular representation. Use was also made of the computer system GAP (see ) in this work.