"홀-리틀우드(Hall-Littlewood) 대칭함수"의 두 판 사이의 차이
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| 19번째 줄: | 19번째 줄: | ||
** $t=1$일 때, [[단항 대칭 다항식 (monomial symmetric polynomial)]] $m_{\lambda}$을 얻는다 | ** $t=1$일 때, [[단항 대칭 다항식 (monomial symmetric polynomial)]] $m_{\lambda}$을 얻는다 | ||
| − | == | + | ==테이블== |
| − | + | * 슈르다항식 $s_{\lambda}$, 단항 대칭 다항식 $m_{\lambda}$, 홀-리틀우드 다항식 $P_{\lambda}$ | |
| − | * 슈르다항식 $s_{\lambda}$ | + | ===$n=2,d=3$=== |
$$ | $$ | ||
\begin{array}{c|c|c} | \begin{array}{c|c|c} | ||
| − | \lambda & s_{\lambda }(x) & P_{\lambda }(x;t) \\ | + | \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ |
\hline | \hline | ||
| − | \{ | + | \{3\} & x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 & x_1^3+x_2^3 & -t x_2 x_1^2-t x_2^2 x_1+x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 \\ |
| − | \{ | + | \{2,1\} & x_2 x_1^2+x_2^2 x_1 & x_1 x_2 \left(x_1+x_2\right) & x_2 x_1^2+x_2^2 x_1 \\ |
| − | + | \{1,1,1\} & 0 & 0 & 0 | |
| − | \{ | ||
| − | |||
\end{array} | \end{array} | ||
$$ | $$ | ||
| + | ===$n=2,d=4$=== | ||
| + | $$ | ||
| + | \begin{array}{c|c|c} | ||
| + | \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ | ||
| + | \hline | ||
| + | \{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 & x_1^4+x_2^4 & -t x_2 x_1^3-t x_2^2 x_1^2-t x_2^3 x_1+x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\ | ||
| + | \{3,1\} & x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 & x_1 x_2 \left(x_1^2+x_2^2\right) & -t x_2^2 x_1^2+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 \\ | ||
| + | \{2,2\} & x_1^2 x_2^2 & x_1^2 x_2^2 & x_1^2 x_2^2 \\ | ||
| + | \{2,1,1\} & 0 & 0 & 0 \\ | ||
| + | \{1,1,1,1\} & 0 & 0 & 0 | ||
| + | \end{array} | ||
| + | $$ | ||
==메모== | ==메모== | ||
2015년 4월 1일 (수) 04:10 판
개요
- 양의 정수 $n$에 대하여, $x=(x_1,\dots,x_n)$로 두자
- 분할 $\lambda=(\lambda_n, \cdots, \lambda_1),\, \lambda_n\geq \cdots\geq \lambda_1\geq 0$에 대하여, $x^{\lambda}$는 단항식 $x_1^{\lambda_1}\dots x_n^{\lambda_n}$를 나타냄
- $m_i(\lambda)$ 는 $\lambda$에서 $i$의 개수
- 다음과 같이 $v_{\lambda}$를 정의
$$ v_{\lambda}(t)=\prod_{i=0}^n \frac{(t)_{m_i}}{(1-t)^{m_i}} $$
- 홀-리틀우드 다항식 $P_{\lambda}(x;t)$은 다음과 같이 정의
$$ P_{\lambda}(x;t)=\frac{1}{v_{\lambda}(t)} \sum_{w\in\mathfrak{S}_n} w\bigg(x^{\lambda}\prod_{i<j}\frac{x_i-tx_j}{x_i-x_j}\bigg), $$ 여기서 대칭군 $\mathfrak{S}_n$는 $x$에 $x_i$의 치환으로 작용
- $P_{\lambda}(x;t)$는 차수가 $\lvert\lambda \lvert$인 동차대칭다항식이다
- 특수화
- $t=0$일 때, 슈르 다항식(Schur polynomial) $s_{\lambda}$을 얻는다
- $t=1$일 때, 단항 대칭 다항식 (monomial symmetric polynomial) $m_{\lambda}$을 얻는다
테이블
- 슈르다항식 $s_{\lambda}$, 단항 대칭 다항식 $m_{\lambda}$, 홀-리틀우드 다항식 $P_{\lambda}$
$n=2,d=3$
$$ \begin{array}{c|c|c} \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ \hline \{3\} & x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 & x_1^3+x_2^3 & -t x_2 x_1^2-t x_2^2 x_1+x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 \\ \{2,1\} & x_2 x_1^2+x_2^2 x_1 & x_1 x_2 \left(x_1+x_2\right) & x_2 x_1^2+x_2^2 x_1 \\ \{1,1,1\} & 0 & 0 & 0 \end{array} $$
$n=2,d=4$
$$ \begin{array}{c|c|c} \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ \hline \{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 & x_1^4+x_2^4 & -t x_2 x_1^3-t x_2^2 x_1^2-t x_2^3 x_1+x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\ \{3,1\} & x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 & x_1 x_2 \left(x_1^2+x_2^2\right) & -t x_2^2 x_1^2+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 \\ \{2,2\} & x_1^2 x_2^2 & x_1^2 x_2^2 & x_1^2 x_2^2 \\ \{2,1,1\} & 0 & 0 & 0 \\ \{1,1,1,1\} & 0 & 0 & 0 \end{array} $$
메모
- spherical Macdonald functions
관련된 항목들
매스매티카 파일 및 계산 리소스
리뷰, 에세이, 강의노트
- Macdonald, I. G. 1992. “Schur Functions: Theme and Variations.” In Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), 498:5–39. Publ. Inst. Rech. Math. Av. Strasbourg: Univ. Louis Pasteur. http://www.ams.org/mathscinet-getitem?mr=1308728. http://emis.u-strasbg.fr/journals/SLC/opapers/s28macdonald.pdf
관련논문
- Cori, Robert, Pasquale Petrullo, and Domenico Senato. “Hall-Littlewood Symmetric Functions via Yamanouchi Toppling Game.” arXiv:1412.0444 [math], December 1, 2014. http://arxiv.org/abs/1412.0444.
- Borodin, Alexei. “On a Family of Symmetric Rational Functions.” arXiv:1410.0976 [math], October 3, 2014. http://arxiv.org/abs/1410.0976.
- Venkateswaran, Vidya. 2014. “A P-Adic Interpretation of Some Integral Identities for Hall-Littlewood Polynomials.” arXiv:1407.3755 [math], July. http://arxiv.org/abs/1407.3755.
- Frechette, Claire, and Madeline Locus. 2014. “Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials.” arXiv:1407.2880 [math], July. http://arxiv.org/abs/1407.2880.
- Griffin, Michael J., Ken Ono, and S. Ole Warnaar. 2014. “A Framework of Rogers-Ramanujan Identities and Their Arithmetic Properties.” arXiv:1401.7718 [math], January. http://arxiv.org/abs/1401.7718.
- Bartlett, Nick, and S. Ole Warnaar. “Hall-Littlewood Polynomials and Characters of Affine Lie Algebras.” arXiv:1304.1602 [math], April 4, 2013. http://arxiv.org/abs/1304.1602.
- Lenart, Cristian. “Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams.” Discrete Mathematics 311, no. 4 (2011): 258–75. doi:10.1016/j.disc.2010.11.010.
- Warnaar, S. Ole. 2007. “Rogers-Szego Polynomials and Hall-Littlewood Symmetric Functions.” arXiv:0708.3110 [math], August. http://arxiv.org/abs/0708.3110.
- Warnaar, S. Ole. “Hall-Littlewood Functions and the $A_2$ Rogers-Ramanujan Identities.” Advances in Mathematics 200, no. 2 (2006): 403–34. doi:10.1016/j.aim.2004.12.001.
- Jouhet, Frédéric, and Jiang Zeng. “New Identities for Hall-Littlewood Polynomials and Applications.” The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan 10, no. 1 (2005): 89–112. doi:10.1007/s11139-005-3508-3.
- Andrews, George E., Anne Schilling, and S. Ole Warnaar. “An $A_2$ Bailey Lemma and Rogers-Ramanujan-Type Identities.” Journal of the American Mathematical Society 12, no. 3 (1999): 677–702. doi:10.1090/S0894-0347-99-00297-0.
- Kirillov, Anatol N. ‘New Combinatorial Formula for Modified Hall-Littlewood Polynomials’. arXiv:math/9803006, 2 March 1998. http://arxiv.org/abs/math/9803006.
- Jing, Naihuan. ‘Vertex Operators and Hall-Littlewood Symmetric Functions’. Advances in Mathematics 87, no. 2 (June 1991): 226–48. doi:10.1016/0001-8708(91)90072-F.
- Stembridge, John R. “Hall-Littlewood Functions, Plane Partitions, and the Rogers-Ramanujan Identities.” Transactions of the American Mathematical Society 319, no. 2 (1990): 469–98. doi:10.2307/2001250.