홀-리틀우드(Hall-Littlewood) 대칭함수

수학노트
Pythagoras0 (토론 | 기여)님의 2015년 8월 2일 (일) 17:31 판 (→‎관련논문)
둘러보기로 가기 검색하러 가기

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

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


관련논문

  • Feigin, Boris, and Igor Makhlin. “A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem.” arXiv:1505.04269 [math], May 16, 2015. http://arxiv.org/abs/1505.04269.
  • 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.
원본 주소 "https://wiki.mathnt.net/index.php?title=홀-리틀우드(Hall-Littlewood)_대칭함수&oldid=31838"