Upodobitve grup

diplomsko delo

Maja Adanič (Author), Dominik Benkovič (Mentor)

Abstract

V diplomskem delu najprej predstavimo osnovne definicije teorije grup, ki jih potrebujemo skozi celotno diplomsko delo. Sledi definicija upodobitve grupe, ki pravi, da je upodobitev grupe G nad vektorskim prostorom V homomorfizem iz grupe G v linearno grupo GL(V). Nato povemo nekaj o G-invariantnih upodobitvah in unitarnih prostorih ter zapišemo, da je unitarna upodobitev homomorfizem iz grupe G v unitarno grupo Un(â%%). Nadalje sledi izrek, da je vsaka končna podgrupa grupe GLn(â%%) konjugirana k podgrupi unitarne grupe in da je vsaka matrična upodobitev končne grupe G konjugirana k unitarni upodobitvi. V nadaljevanju vpeljemo kompaktne grupe in dokažemo izrek, da sta unitarna in ortogonalna grupa kompaktni. V poglavju Nerazcepne upodobitve pokažemo, da je vsaka upodobitev končne grupe G direktna vsota nerazcepnih upodobitev. Prav tako je v diplomi dokazan izrek, da je vsaka nerazcepna upodobitev grupe G enodimenzionalna, če je G Abelova grupa. V nadaljevanju obravnavamo značaj upodobitve grupe. Značaj je funkcija %, ki slika iz grupe G v â%% in je sled matrične upodobitve. S pomočjo nekaterih primerov predstavimo tabelo značajev. Na koncu predstavimo upodobitev grupe SU2 in s pomočjo te grupe dokažemo dejstva, ki so veljala za končne grupe, tudi za kompaktne grupe.

Keywords

matematika;končne grupe;kompaktne grupe;upodobitve grup;unitarne upodobitve;značaji upodobitev;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2010
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Adanič]
UDC:	51(043.2)
COBISS:	17575944
Views:	2416
Downloads:	235
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	GROUP REPRESENTATIONS
Secondary abstract:	Basic definitions of group theory are presented at the beginning of the graduation thesis. Next, we define a representation of a group G on a vector space V as a homomorphism from group G to the general linear group GL(V). Then we study G-invariant representations and unitary spaces. We also introduce unitary representation as a homomorphism from group G to unitary group Un(%). Next, we prove that every finite subgroup of GLn(%) is conjugate to a subgroup of a unitary group and every matrix representation of a finite group G is conjugate to a unitary representation. Further, we introduce compact groups and prove that unitary and orthogonal groups are compact. In chapter 5 we show that every representation of a finite group G is a direct sum of irreducible representations. We conclude that each irreducible representation of a finite abelian group G is one-dimensional. We also consider characters of a group representation. A character is a function %, that maps from group G to % and it is a trace of matrix representation. Through some examples we present the character table. At the end we present a representation of group SU2 and we obtain some results on finite groups and compact groups.
Secondary keywords:	finite groups;compact groups;group representations;unitary representations;characters of representations.;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	55 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	18458