Mešanje z naključnimi transpozicijami : delo diplomskega seminarja

Miščič, Matevž (author), Jezernik, Urban (mentor)

V diplomski nalogi dokažemo, da pri mešanju z naključnimi transpozicijami pride do odreza ob času $\frac{1}{2}n\log{n}$. Pri dokazu zgornje meje odreza uporabimo nekomutativno Fourierovo transformacijo, za njeno uporabo pa predstavimo teorijo upodobitev končnih grup s posebnim poudarkom na simetrične grupe. Klasificiramo Spechtove module in pokažemo, da standardni politabloidi tvorijo njihove baze. Spodnjo mejo odreza dokažemo z verjetnostnimi metodami. Predstavimo tudi nekaj nadaljnjih primerov odreza pri sprehodih po grupah.

• Language: Slovenian
• Keywords: upodobitve grup, simetrične grupe, naključni sprehodi, naključne transpozicije
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-139856
• UDC: 512
• COBISS.SI-ID: 120837379
• Publication date in RUL: 08.09.2022

Secondary language

• Language: English
• Title: Shuffling by random transpositions

Abstract

In this thesis we prove that in the case of random transposition shuffling cutoff occurs at time $\frac{1}{2}n\log{n}$. The upper bound is proved using noncommutative Fourier transform. To understand it representation theory of finite groups is presented with emphasis on symmetric groups. Specht modules are classified and it is shown that standard polytabloids form their bases. Lower bound is proved using methods from probability. We also discuss some further examples of cutoff for random walks on groups.

• Keywords: group representations, symmetric groups, random walks, random transpositions