Youngova rešetka

master's thesis

• Title (english): Young's lattice
• Author: Filip Kustura
• Mentor: Vedran Krčadinac (mentor)
• Committee member: Vedran Krčadinac (predsjednik povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Tomislav Pejković (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-03-01, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj rad bavi se područjem algebarske kombinatorike. Točnije, fokus je usmjeren na Youngovu rešetku, odnosno skup particija prirodnih brojeva zajedno s odgovarajućim parcijalnim uređajem. U radu su predstavljeni Youngovi dijagrami kojima se prikazuju particije prirodnih brojeva te se proučavaju posebni podskupovi skupa particija i njihova svojstva. Korištenjem koncepta \(q\)-binomnih koeficijenata istražuju se neka svojstva Youngove rešetke. Razmatraju se brojna svojstva i rekurzivne relacije koje zadovoljavaju \(q\)-binomni koeficijenti, uključujući simetriju i unimodalnost, a dane su i mnoge kombinatorne interpretacije \(q\)-binomnih koeficijenata. Također, istražuju se Hasseove šetnje u Youngovoj rešetki čija važnost leži u tome što je skup svih permutacija konačnog skupa ekvipotentan sa specifičnim podskupom skupa Hasseovih šetnji. Koncept koji se nadovezuje na Youngovu rešetku su Youngove tablice, važan alat u teoriji reprezentacija simetrične grupe. U radu je također opisan algoritam RSK koji uspostavlja bijekciju između permutacija konačnog skupa i uređenih parova standardnih Youngovih tablica istog oblika.

Abstract (english)

The area of interest in this thesis is algebraic combinatorics. The focus is directed towards the analysis of Young's lattice, that is, the set of partitions of natural numbers together with the corresponding partial order. In the thesis, Young's diagrams which represent partitions of natural numbers are presented and special subsets of the set of partitions and their properties are studied. Using the concept of \(q\)-binomial coefficients, some properties of Young's lattice are investigated. Numerous properties and recursive relations for \(q\)-binomial coefficients are discussed, including symmetry and unimodality, and many combinatorial interpretations of \(q\)-binomial coefficients are given. Also, Hasse walks in Young's lattice are discussed. The importance of Hasse walks lies in the fact that the set of all permutations of a finite set is equipotent with a specific subset of the set of Hasse walks. A concept related are Young tableaus, an important tool in the representation theory of the symmetric group. The thesis also gives a description of the RSK algorithm that establishes a bijection between permutations of a finite set and pairs of standard Young tableau of the same form.

• Keywords: algebarska kombinatorika; Youngovi dijagrami; Hasseove šetnje
• Keywords (english): algebraic combinatorics; Young's diagrams; Hasse walks.
• Language: croatian
• URN:NBN: urn:nbn:hr:217:972994
• Study programme: Title: Computer Science and Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar računarstva i matematike (sveučilišni magistar računarstva i matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-03-13 12:01:32