Reprezentacije konačnih grupa

master's thesis

• Title (english): Representations of finite groups
• Author: Fran Perica
• Mentor: Pavle Pandžić (mentor)
• Committee member: Pavle Pandžić (predsjednik povjerenstva)
• Committee member: Petar Žugec (član povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Marko Tomislav Cvitaš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Prvo poglavlje je namijenjeno kao dobar uvod u teoriju reprezentacija jer se kreće samo s bazičnim znanjem linearne algebre. Uvode se osnovne definicije i brzo se prelazi na primjere odnosno na reprezentacije Abelove grupe \(\mathfrak{G}_{3}\). Zatim se detaljnije uvode karakteri reprezentacija koji su od velike važnosti jer karakter neke reprezentacije govori dosta o samoj reprezentaciji. Uz uvod u projekcijske formule, također je i detaljno pokazan postupak pronalaska tablice karaktera za određene reprezentacije i to je napravljeno za \(\mathfrak{G}_{4}\). Zatim imamo još dobrih primjera kao što su tablice karaktera od \(\mathfrak{U}_{5}\) i posebito \(\mathfrak{G}_{5}\) koji nam dobro pokazuje kako evoluira tablica karaktera kada mijenjamo broj elemenata u nekoj permutacijskoj grupi. Završava se s kratkim uvodima u grupnu algebru, realnim i induciranim reprezentacijama.

Abstract (english)

The first chapter is intended as a gentle introduction to the theory of representations because it only assumes basic knowledge of linear algebra. Basic definitions are introduced followed by an example where we explore representations of the Abelian group \(\mathfrak{G}_{3}\). Next concept is the characters of representations which are of great importance in the theory as they give a lot of information about the representation itself. Along with introducing projection formulas here we show how to construct character tables for different representations, in this case on the group \(\mathfrak{G}_{4}\). What follows is a pair of good examples as we construct more character tables, for \(\mathfrak{U}_{5}\) and more importantly for \(\mathfrak{G}_{5}\) where we can see how the tables change when we add elements to a permutation group. In the end there is a short introduction to group algebra, real and induced representations.

• Keywords: linearna algebra; Abelovae grupa; karakteri reprezentacija; grupna algebra; realne reprezentacije; inducirane reprezentacije
• Keywords (english): linear algebra; Abelian group; characters of representations; group algebra; real representations; induced representations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:378845
• Study programme: Title: Mathematics and Physics Education; specializations in: Mathematics and Physics Education; Course: Mathematics and Physics Education; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: magistar/magistra edukacije matematike i fizike (magistar/magistra edukacije matematike i fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-11-04 12:46:37