Representation theory of groups

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:

WORLD SCIENTIFIC eBooks, 1989

Chapter 14. Characters and Algebraic Integers Chapter 15. Burnside's pq Theorem Chapter 16. Further Reading 16.1. Representation Theory of Symmetric Group 16.2. Representation Theory of GL 2 (F q) and SL 2 (F q) 16.3. Wedderburn Structure Theorem 16.4. Modular Representation Theory Bibliography CHAPTER 1

Journal of Geometry and Symmetry in Physics, 2009

Springer eBooks, 1980

American Mathematical Monthly, 1985

Mathematical Statistician and Engineering Applications

Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking algebraic structures with linear transformations.The representation theory has significant effects on algebra. Through the examination of the representations that go along with a group, it allows us to examine its composition and behaviour. One can learn more about the internal structures and underlying symmetries of groups by breaking representations down into irreducible parts. This has implicati...

The aim of the current dissertation is to address certain problems in the representation theory of simple Lie algebras and associated quantum algebras. In Part I, we study the simple Lie group of type G 2 from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer [25], relating the geometry of the partial flag varieties to cluster algebras in the case of G 2. In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra U (Lg) and the Yangian Y (g), associated with a simple Lie algebra g. The main result of this part gives a construction of an explicit isomorphism between completions of these algebras, thus strengthening the well known Drinfeld's degeneration homomorphism [10, 27]. In Part III, we give a proof of the monodromy conjecture of Toledano Laredo [50] for the case of g = sl 2. The monodromy conjecture relates two classes of representations of the affine braid group, the first arising from the quantum Weyl group operators of the quantum loop algebra and the second coming from the monodromy of the trigonometric Casimir connection, a flat connection introduced by Toledano Laredo [50]. To Prof. Valerio Toledano Laredo, for teaching me the monodromy side of quantum groups, for your constant availability and for having great patience with me during all our joint projects.

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.

This appendix consists of two sections. In the first we give a self-contained and fairly complete introduction to the representation and character theory of finite groups, including Frobenius's formula and a higher genus generalization. In the second we give several applications related to topics treated in this book.

2012