S-Space College of Natural Sciences (자연과학대학) Dept. of Mathematical Sciences (수리과학부) Theses (Master's Degree_수리과학부)
Trace polynomials of words in the free group of rank two
계수 2 자유군에서의 대각합 다항식
- Issue Date
- 서울대학교 대학원
- Trace polynomial; Free group of rank two; Special linear group,Cyclically reduced words; 대각합 다항식; 계수2 자유군; 특수 선형군; 순환 기약 워드
- 학위논문 (석사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2020. 8. 임선희.
- 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.