"홀-리틀우드(Hall-Littlewood) 대칭함수"의 두 판 사이의 차이

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잔글 (Pythagoras0 사용자가 Hall-Littlewood functions 문서를 홀-리틀우드(Hall-Littlewood) 대칭함수 문서로 옮겼습니다.)
1번째 줄: 1번째 줄:
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==introduction==
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==개요==
 
* 양의 정수 $n$에 대하여, $x=(x_1,\dots,x_n)$로 두자
 
* 양의 정수 $n$에 대하여, $x=(x_1,\dots,x_n)$로 두자
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* Given a partition $\lambda$ such that $l(\lambda)\leq n$, write $x^{\lambda}$ for the monomial $x_1^{\lambda_1}\dots x_n^{\lambda_n}$
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* 분할 $\lambda=(\lambda_n\geq \cdots\geq \lambda_1\geq 0)$에 대하여, $x^{\lambda}$는 단항식 $x_1^{\lambda_1}\dots x_n^{\lambda_n}$
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* $m_i(\lambda)$ 는 $\lambda=(\lambda_n,\cdots,\lambda_1)$에서 $i$의 개수
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* $m_i(\lambda)$ 는 $\lambda$에서 $i$의 개수
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* define
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* 다음과 같이 $v$를 정의
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\begin{equation}
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$$
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v_{\lambda}(q)=\prod_{i=0}^n \frac{(q)_{m_i}}{(1-q)^{m_i}},
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v_{\lambda}(q)=\prod_{i=0}^n \frac{(q)_{m_i}}{(1-q)^{m_i}}
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\end{equation}
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$$
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* The Hall-Littlewood polynomial $P_{\lambda}(x;q)$ is defined as the symmetric function \cite{Macdonald95}
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* 홀-리틀우드 다항식 $P_{\lambda}(x;q)$은 다음과 같이 정의
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\begin{equation}
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$$
 
P_{\lambda}(x;q)=\frac{1}{v_{\lambda}(q)}
 
P_{\lambda}(x;q)=\frac{1}{v_{\lambda}(q)}
 
\sum_{w\in\mathfrak{S}_n}  
 
\sum_{w\in\mathfrak{S}_n}  
 
w\bigg(x^{\lambda}\prod_{i<j}\frac{x_i-qx_j}{x_i-x_j}\bigg),
 
w\bigg(x^{\lambda}\prod_{i<j}\frac{x_i-qx_j}{x_i-x_j}\bigg),
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\end{equation}
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$$
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where the symmetric group $\mathfrak{S}_n$ acts on $x$ by permuting the $x_i$.
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여기서 대칭군 $\mathfrak{S}_n$는 $x$에 $x_i$의 치환으로 작용
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* It follows from the definition that $P_{\lambda}(x;q)$ is a homogeneous polynomial of degree $\lvert\lambda \lvert$,
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* $P_{\lambda}(x;q)$는 차수가 $\lvert\lambda \lvert$인 동차다항식이다
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* $q=0$일 때, [[슈르 다항식(Schur polynomial)]]을 얻는다
  
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==memo==
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==예==
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===변수의 개수가 2이고, 4의 분할인 경우===
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* 슈르다항식 $s_{\lambda}$과 홀-리틀우드 다항식 $P_{\lambda}$를 같이 나타냄
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$$
 +
\begin{array}{c|c|c}
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\lambda & s_{\lambda }(x) & P_{\lambda }(x;t) \\
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\hline
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\{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 & -t x_2 x_1^3-t x_2^2 x_1^2-t x_2^3 x_1+x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\
 +
\{3,1\} & x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 & x_1 x_2 \left(-t x_2 x_1+x_1^2+x_2 x_1+x_2^2\right) \\
 +
\{2,2\} & x_1^2 x_2^2 & \frac{(t-1)^2 (t+1) x_1^2 x_2^2}{(t;t)_2} \\
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\{2,1,1\} & 0 & 0 \\
 +
\{1,1,1,1\} & 0 & 0
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\end{array}
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$$
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==메모==
 
* spherical Macdonald functions
 
* spherical Macdonald functions
  
  
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==related items==
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==관련된 항목들==
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* [[Macdonald polynomials]]
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* [[대칭다항식]]
  
  
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==expositions==
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==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
 
* 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
  
  
  
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==articles==
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==관련논문==
 
* 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.
 
* 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.
 
* 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.
40번째 줄: 58번째 줄:
 
* 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.
 
* 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.
 
* 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.
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[[분류:개인노트]]
 
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[[분류:math and physics]]
 
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[[분류:math]]
 

2014년 9월 22일 (월) 23:51 판

개요

  • 양의 정수 $n$에 대하여, $x=(x_1,\dots,x_n)$로 두자
  • 분할 $\lambda=(\lambda_n\geq \cdots\geq \lambda_1\geq 0)$에 대하여, $x^{\lambda}$는 단항식 $x_1^{\lambda_1}\dots x_n^{\lambda_n}$
  • $m_i(\lambda)$ 는 $\lambda$에서 $i$의 개수
  • 다음과 같이 $v$를 정의

$$ v_{\lambda}(q)=\prod_{i=0}^n \frac{(q)_{m_i}}{(1-q)^{m_i}} $$

  • 홀-리틀우드 다항식 $P_{\lambda}(x;q)$은 다음과 같이 정의

$$ P_{\lambda}(x;q)=\frac{1}{v_{\lambda}(q)} \sum_{w\in\mathfrak{S}_n} w\bigg(x^{\lambda}\prod_{i<j}\frac{x_i-qx_j}{x_i-x_j}\bigg), $$ 여기서 대칭군 $\mathfrak{S}_n$는 $x$에 $x_i$의 치환으로 작용


예

변수의 개수가 2이고, 4의 분할인 경우

  • 슈르다항식 $s_{\lambda}$과 홀-리틀우드 다항식 $P_{\lambda}$를 같이 나타냄

$$ \begin{array}{c|c|c} \lambda & s_{\lambda }(x) & P_{\lambda }(x;t) \\ \hline \{4\} & x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 & -t x_2 x_1^3-t x_2^2 x_1^2-t x_2^3 x_1+x_1^4+x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1+x_2^4 \\ \{3,1\} & x_2 x_1^3+x_2^2 x_1^2+x_2^3 x_1 & x_1 x_2 \left(-t x_2 x_1+x_1^2+x_2 x_1+x_2^2\right) \\ \{2,2\} & x_1^2 x_2^2 & \frac{(t-1)^2 (t+1) x_1^2 x_2^2}{(t;t)_2} \\ \{2,1,1\} & 0 & 0 \\ \{1,1,1,1\} & 0 & 0 \end{array} $$


메모

  • spherical Macdonald functions


관련된 항목들


리뷰, 에세이, 강의노트


관련논문

  • 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.
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