홀-리틀우드(Hall-Littlewood) 대칭함수

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 \(P_{\lambda}\)

\(n=2,d=3\)

\[ \begin{array}{c|c|c} \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ \hline \{3\} & x_1^3+x_2 x_1^2+x_2^2 x_1+x_2^3 & x_1^3+x_2^3 & x_1^3+(1-t) x_1^2 x_2+(1-t) x_1 x_2^2 +x_2^3 \\ \{2,1\} & x_2 x_1^2+x_2^2 x_1 & x_1 x_2 \left(x_1+x_2\right) & x_2 x_1^2+x_2^2 x_1 \\ \{1,1,1\} & 0 & 0 & 0 \end{array} \]

\(n=2,d=4\)

\[ \begin{array}{c|c|c} \lambda & s_{\lambda }(x) & m_{\lambda }(x) & P_{\lambda }(x;t) \\ \hline \{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 & x_1^4+x_2^4 & x_1^4+(1-t) x_1 x_2^3+(1-t) x_1^2 x_2^2+(1-t) x_1^3 x_2+x_2^4 \\ \{3,1\} & x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 & x_1^3 x_2+x_1 x_2^3 & x_1^3 x_2+(1-t) x_1^2 x_2^2+x_1 x_2^3 \\ \{2,2\} & x_1^2 x_2^2 & x_1^2 x_2^2 & x_1^2 x_2^2 \\ \{2,1,1\} & 0 & 0 & 0 \\ \{1,1,1,1\} & 0 & 0 & 0 \end{array} \]

메모

• spherical Macdonald functions

관련된 항목들

• 대칭다항식

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxS25WNEVDRlRvSW8/edit

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

• Boris Feigin, Igor Makhlin, A Combinatorial Formula for Affine Hall-Littlewood Functions via a Weighted Brion Theorem, arXiv:1505.04269 [math.CO], May 16 2015, http://arxiv.org/abs/1505.04269
• Piotr Pragacz, A Gysin formula for Hall-Littlewood polynomials, arXiv:1403.0788[math.AG], March 04 2014, http://arxiv.org/abs/1403.0788v8, Proc. Amer. Math. Soc. 143 (2015) no.11, 4705-4711
• François Bergeron, A q-Analog of Foulke's conjecture, http://arxiv.org/abs/1602.08134v2
• Duval, Antoine, and Vincent Pasquier. “Pieri Rules, Vertex Operators and Baxter Q-Matrix.” arXiv:1510.08709 [math-Ph, Physics:nlin], October 29, 2015. http://arxiv.org/abs/1510.08709.
• Wheeler, Michael, and Paul Zinn-Justin. “Refined Cauchy/Littlewood Identities and Six-Vertex Model Partition Functions: III. Deformed Bosons.” arXiv:1508.02236 [math-Ph], August 10, 2015. http://arxiv.org/abs/1508.02236.
• 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.