Youngovi dijagrami i reprezentacije simetrične grupe

master's thesis

• Title (english): Young diagrams and representations of a symmetric group
• Author: Luka Fran
• Mentor: Slaven Kožić (mentor)
• Committee member: Slaven Kožić (predsjednik povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Luka Grubišić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-09-21, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavni cilj ovog rada bio je proučiti reprezentacije simetrične grupe i njihovo djelovanje na Youngove tablice. U prva dva poglavlja definirali smo pojam tenzorskog produkta i naveli nekoliko svojstava koja tenzorski produkt zadovoljava. Prezentirali smo što je reprezentacija grupe te koja su njezina obilježja. Nakratko smo približili Schurovu lemu i objasnili njezin značaj za reprezentacije grupa. Definirali smo karakter reprezentacije i što je centralna funkcija te ih detaljno proučili za reprezentacije konačne grupe. Definicije i rezultati iz ova dva poglavlja bili su nam potrebni da bismo mogli proučiti reprezentacije simetrične grupe. Nakon toga, u trećem poglavlju smo preusmjerili fokus na reprezentacije simetrične grupe, definirali Youngove dijagrame te uveli i analizirali djelovanje simetrične grupe na Youngove tablice. Predstavili smo Spechtove module i kroz nekoliko propozicija objasnili njihovu vezu s teorijom reprezentacija. U ovom poglavlju smo predstavili glavne pojmove, konstrukcije i rezultate koji su nam bili potrebni da definiramo i proučimo djelovanje simetrične grupe na Youngove tablice, što smo željeli prezentirati ovim radom.

Abstract (english)

The main goal of this thesis was to study representations of the symmetric groups and their action on Young tableaux. In the first two chapters of the thesis, we define tensor products and recall some of their properties. We introduced the notion of representation of a group and we presented its properties. We took a closer look at Schur’s lemma and explained its connection with group representations. Finally, we defined the character of representation and class functions on a group, and we investigated closely these notions in the case of finite groups. The definitions and results from these two chapters present important tools for studying representations of symmetric groups. In the third chapter we turned our focus on representations of symmetric groups, we defined Young diagrams and introduced and analyzed the action of symmetric groups on Young tableaux. We presented the notion of the Specht module and through several propositions explained its connection with representation theory. In this chapter, we presented main concepts, constructions, and results that were necessary to define and investigate the action of a symmetric group on Young tableaux, which was the main goal of this thesis.

• Keywords: Youngove tablice; tenzorski produkt; Schurova lema; Spechtovi moduli
• Keywords (english): Young tableaux; tensor products; Schur’s lemma; Specht module
• Language: croatian
• URN:NBN: urn:nbn:hr:217:988332
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-10-26 12:37:06