Publications
Detailed Information
Trace polynomials of words in the free group of rank two : 계수 2 자유군에서의 대각합 다항식
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 임선희 | - |
dc.contributor.author | 박현수 | - |
dc.date.accessioned | 2020-10-13T04:02:02Z | - |
dc.date.available | 2020-10-13T04:02:02Z | - |
dc.date.issued | 2020 | - |
dc.identifier.other | 000000161223 | - |
dc.identifier.uri | https://hdl.handle.net/10371/170699 | - |
dc.identifier.uri | http://dcollection.snu.ac.kr/common/orgView/000000161223 | ko_KR |
dc.description | 학위논문 (석사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2020. 8. 임선희. | - |
dc.description.abstract | Procesi's theorem guarantees that traces in a two generator subgroup of $\ssl$ are polynomials in traces of the generators. These polynomials are called trace polynomials and defined for words in the free group of rank two. Let $\cw$ denote the set of cyclically reduced words in $F_2$.
Improving Jorgensen's algorithm, we classify all words in $\cw$ with the word lengths less than nine via their trace polynomials. Then we check whether they are in $\sim$-equivalence defined from the operation Mirror, Left shift, and Inverse on $\cw$. We prove that two words of the same trace polynomials are $\sim$-equivalent when the word lengths are less than nine. We also show, by counterexamples, this result does not hold for the word lengths greater than eight. As a corollary, we verify Wang's conjecture for the word lengths less than nine. | - |
dc.description.tableofcontents | Introduction 1
1 Preliminaries 2 1.1 Traces in SL(2, C) 2 1.2 Free group of rank two 3 1.3 Trace polynomial of words in F2 5 1.3.1 Traces in two-generator subgroups of SL(2, C) 5 1.3.2 Definition of the trace polynomial 6 1.3.3 Basic properties of trace polynomials 8 2 Computation of trace polynomials 11 2.1 Existence of the trace polynomial 11 2.1.1 2r-vectors and multiplicative groups 11 2.1.2 Proof of the existence of trace polynomials 12 2.2 Algorithms computing trace polynomials 15 2.2.1 Algorithm 1 : recursive method 15 2.2.2 Algorithm 2 : alternating formula 18 3 Trace polynomials of cyclically reduced words in F2 25 3.1 Properties of trace polynomial 25 3.2 Trace polynomial and equivalence class on C 30 3.2.1 Injectivity of α as a map on equivalence classes 35 3.2.2 Conjecture of Wang 37 3.3 Proof of the main theorem 38 3.3.1 Classifying words via trace polynomials 38 3.3.2 The sizes of -equivalence classes 40 3.3.3 The case when n is one of 1, 2, 3, 5 and 7 41 3.3.4 The case when n is one of 4, 6 and 8 43 3.3.5 The case when n is greater than or equal to 9 49 Conclusion 51 List of Tables 52 Bibliography 88 Abstract (in Korean) 89 | - |
dc.language.iso | eng | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject | Trace polynomial | - |
dc.subject | Free group of rank two | - |
dc.subject | Special linear group,Cyclically reduced words | - |
dc.subject | 대각합 다항식 | - |
dc.subject | 계수2 자유군 | - |
dc.subject | 특수 선형군 | - |
dc.subject | 순환 기약 워드 | - |
dc.subject.ddc | 510 | - |
dc.title | Trace polynomials of words in the free group of rank two | - |
dc.title.alternative | 계수 2 자유군에서의 대각합 다항식 | - |
dc.type | Thesis | - |
dc.type | Dissertation | - |
dc.contributor.AlternativeAuthor | Hyeonsu Park | - |
dc.contributor.department | 자연과학대학 수리과학부 | - |
dc.description.degree | Master | - |
dc.date.awarded | 2020-08 | - |
dc.identifier.uci | I804:11032-000000161223 | - |
dc.identifier.holdings | 000000000043▲000000000048▲000000161223▲ | - |
- Appears in Collections:
- Files in This Item:
Item View & Download Count
Items in S-Space are protected by copyright, with all rights reserved, unless otherwise indicated.