O teoriji upodobitev končnih grup

delo diplomskega seminarja

Tjaž Silovšek (Author), Pavle Saksida (Mentor)

Abstract

V diplomskem delu obravnavamo upodobitve končnih grup. Povemo, da je upodobitev homomorfizem iz grupe G v linearno grupo GL(V), stopnja upodobitve pa je enaka dimenziji prostora V. Nato povemo, kaj so G-invariantni podprostori in nerazcepne upodobitve. Definiramo ekvivalenčno relacijo med upodobitvami in dokažemo, da se nerazcepnost in razgradljivost ohranjata na ekvivalenčnih razredih. Prvi večji cilj je rezultat, da je poljubna upodobitev končne grupe ekvivalentna direktni vsoti nerazcepnih upodobitev. Nato s teorijo upodobitev dokažemo nekaj rezultatov iz linearne algebre. Dokažemo, da je poljubna nerazcepna upodobitev abelove grupe prve stopnje. Za konec si pogledamo karakterje upodobitev, to so preslikave, ki vsakemu elementu iz grupe priredijo sled njegove upodobitve. Pokažemo, da lahko s karakterji povemo, ali je upodobitev nerazcepna. Zadnji večji rezultat pa je, da je poljubna upodobitev ekvivalentna natanko eni direktni vsoti nerazcepnih upodobitev, pri čemer so posamezni elementi v vsoti določeni do ekvivalenčnega razreda natančno.

Keywords

matematika;končne grupe;upodobitve grup;nerazcepne upodobitve;karakterji upodobitev;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[T. Silovšek]
UDC:	512
COBISS:	77512195
Views:	681
Downloads:	72
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Representation theory of finite groups
Secondary abstract:	In this work we deal with representations of finite groups. First, we say that a representation is a homomorphism from a group G to a linear group GL(V) and the degree of the representation is equal to the dimension of the space V. We then tell what G-invariant subspaces and irreducible representations are. We further define the equivalence relation between the representations and prove that the irreducibility and decomposability are preserved on the equivalence classes. The first major goal is to prove that any representation of a finite group is equivalent to the direct sum of irreducible representations. Then, with the theory of representations, we prove some results from linear algebra. We prove that any irreducible representation of the abelian group is of the first degree. Finally, we look at the characters of the representations. The character is a mapping that returns the trace of the representation evaluated at element for each element in the group. We show that we can tell with characters whether representation is irreducible. The last major result, however, is that any representation is equivalent to exactly one direct sum of irreducible representations, with the individual elements in the sum determined to the equivalence class exactly.
Secondary keywords:	mathematics;finite groups;group representations;irreducible representations;characters of representations;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	24 str.
ID:	13495795