UPODOBITVE GRUP

Adanič, Maja

| | SLO | ENG | Cookies and privacy

Bigger font | Smaller font

First page > Show document

Show document

Title:	UPODOBITVE GRUP
Authors:	ID Adanič, Maja (Author) ID Benkovič, Dominik (Mentor) More about this mentor...
Files:	UNI_Adanic_Maja_2010.pdf (330,45 KB) MD5: 78C7496014F8A3C0414CE61774CEF172 PID: 20.500.12556/dkum/0a364067-b13d-4fb3-9e28-962ded0d131b
Language:	Slovenian
Work type:	Undergraduate thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
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 χ, 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:	končne grupe, kompaktne grupe, upodobitve grup, unitarne upodobitve, značaji upodobitev.
Place of publishing:	Maribor
Publisher:	[M. Adanič]
Year of publishing:	2010
PID:	20.500.12556/DKUM-13604
UDC:	51(043.2)
COBISS.SI-ID:	17575944
NUK URN:	URN:SI:UM:DK:OENIILEG
Publication date in DKUM:	13.05.2010
Views:	3393
Downloads:	292
Metadata:
Categories:	FNM
:	Copy citation

Average score:	(0 votes)
Your score:	Voting is allowed only for logged in users.
Share:

Hover the mouse pointer over a document title to show the abstract or click on the title to get all document metadata.

Secondary language

Language:	English
Title:	GROUP REPRESENTATIONS
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(ℂ). Next, we prove that every finite subgroup of GLn(ℂ) 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 χ, that maps from group G to ℂ 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.
Keywords:	finite groups, compact groups, group representations, unitary representations, characters of representations.

Comments

Leave comment

You must log in to leave a comment.

Comments (0)

0 - 0 / 0

There are no comments!

Back