Youngovi tabloi i kristali

master's thesis

• Title (english): Young tableaux and crystals
• Author: Sara Leljak
• Mentor: Slaven Kožić (mentor)
• Committee member: Slaven Kožić (predsjednik povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Committee member: Vjekoslav Kovač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-11-30, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Glavni cilj ovog rada bio je proučiti vezu između teorije kristalnih baza konačnodimenzionalnih \(U_q(\mathfrak{gl}_n)\)-modula i kombinatorike Youngovih dijagrama i Youngovih tablica. U prvom poglavlju proučili smo osnovne pojmove i rezultate iz područja Liejevih algebri te reprezentacija pripadnih kvantnih grupa. U drugom poglavlju smo uveli Kashiwarine operatore i pojam kristalne baze pomoću kojih smo zatim iskazali teorem o tenzorskom produktu kristala te ilustrirali njegove primjene na konkretnim primjerima. U trećem poglavlju smo se bavili Youngovim tablicama i kristalima pri čemu nam je glavni cilj bio proučiti vezu između \(U_q(\mathfrak{gl}_n)\)-kristala, koji su pridruženi \(U_q(\mathfrak{gl}_n)\)-modulima iz kategorije \(\mathcal{O}_{int}^{\geq 0}\), i Youngovih dijagrama. Preciznije, demonstrirali smo kako se spomenuta veza realizira primjenom dopustivih očitavanja na semistandardne Youngove tablice. Napokon, koristeći dobivenu realizaciju \(U_q(\mathfrak{gl}_n)\)-kristala pomoću Youngovih tablica, dokazali smo teorem o dekompoziciji tenzorskog produkta pripadnih modula. Na kraju poglavlja smo proučavali \(U_q(B_3)\)-kristale, dali primjere nekih kristala tog tipa i prikazali razlike do kojih dolazi prilikom realizacije kristala Liejevih algebri drugih tipova.

Abstract (english)

The main goal of this thesis was to study the connection between the crystal basis theory of finite dimensional \(U_q(\mathfrak{gl}_n)\)-modules and combinatorics of Young diagrams and Young tableaux. In the first chapter we studied basic concepts and results from the field of Lie algebras and representations of related quantum groups. In the second chapter we introduced Kashiwara’s operators and the notion of crystal basis, which we used to state the tensor product theorem for crystals and illustrated its applications with concrete examples. In the third chapter we considered Young tableaux and crystals and our main goal was to study the connection between \(U_q(\mathfrak{gl}_n)\)-crystals, which are associated with \(U_q(\mathfrak{gl}_n)\)-modules from category \(\mathcal{O}_{int}^{\geq 0}\) and Young diagrams. More precisely, we demonstrated how the mentioned connection is realized by applying admissible readings to semistandard Young tableaux. Finally, using the obtained realization of the \(U_q(\mathfrak{gl}_n)\)-crystals via Young tableaux, we proved the tensor product decomposition theorem for the corresponding modules. At the end of the chapter, we studied \(U_q(B_3)\)-crystals, gave examples of some crystals of that type and showed some differences that occur in realizations of crystals for Lie algebras of other types.

• Keywords: kristalne baze; Youngovi dijagrami; Youngove tablice
• Keywords (english): crystal basis; Young diagrams; Young tableaux
• Language: croatian
• URN:NBN: urn:nbn:hr:217:432902
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-03-26 14:33:36