홀-리틀우드(Hall-Littlewood) 대칭함수

introduction

양의 정수 $n$에 대하여, $x=(x_1,\dots,x_n)$로 두자
Given a partition $\lambda$ such that $l(\lambda)\leq n$, write $x^{\lambda}$ for the monomial $x_1^{\lambda_1}\dots x_n^{\lambda_n}$
$m_i(\lambda)$ 는 $\lambda=(\lambda_n,\cdots,\lambda_1)$에서 $i$의 개수
define

\begin{equation} v_{\lambda}(q)=\prod_{i=0}^n \frac{(q)_{m_i}}{(1-q)^{m_i}}, \end{equation}

The Hall-Littlewood polynomial $P_{\lambda}(x;q)$ is defined as the symmetric function \cite{Macdonald95}

\begin{equation} P_{\lambda}(x;q)=\frac{1}{v_{\lambda}(q)} \sum_{w\in\mathfrak{S}_n} w\bigg(x^{\lambda}\prod_{i<j}\frac{x_i-qx_j}{x_i-x_j}\bigg), \end{equation} where the symmetric group $\mathfrak{S}_n$ acts on $x$ by permuting the $x_i$.

It follows from the definition that $P_{\lambda}(x;q)$ is a homogeneous polynomial of degree $\lvert\lambda \lvert$,

memo

spherical Macdonald functions

related items

Macdonald polynomials

expositions

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

articles

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.
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.