파피안(Pfaffian)

개요

• 교대행렬(alternating matrix, 또는 skew-symmetric matrix)의 행렬식은 어떤 다항식의 제곱이 되는 성질을 가진다
• 교대행렬에 대해, 이 행렬식의 제곱근의 하나를 파피안으로 정의한다.
• \( \operatorname{pf(A)}^2=\operatorname{det(A)}\)
• \(\operatorname{pf}(BAB^T)= \det(B)\operatorname{pf}(A)\)

교대행렬과 행렬식

• 2×2 교대행렬\[\left( \begin{array}{cc} 0 & t_{1,2} \\ -t_{1,2} & 0 \end{array} \right)\] 의 행렬식 \(t_{1,2}^2\)
  
• 4×4 교대행렬\[\left( \begin{array}{cccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 \end{array} \right)\], 행렬식 \(\left(t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\right){}^2\)
  
• 6×6 교대행렬\[\left( \begin{array}{cccccc} 0 & t_{1,2} & t_{1,3} & t_{1,4} & t_{1,5} & t_{1,6} \\ -t_{1,2} & 0 & t_{2,3} & t_{2,4} & t_{2,5} & t_{2,6} \\ -t_{1,3} & -t_{2,3} & 0 & t_{3,4} & t_{3,5} & t_{3,6} \\ -t_{1,4} & -t_{2,4} & -t_{3,4} & 0 & t_{4,5} & t_{4,6} \\ -t_{1,5} & -t_{2,5} & -t_{3,5} & -t_{4,5} & 0 & t_{5,6} \\ -t_{1,6} & -t_{2,6} & -t_{3,6} & -t_{4,6} & -t_{5,6} & 0 \end{array} \right)\],
  행렬식 \(\left(t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\right){}^2\)
  

 

 

파피안

• \(A=(t_{i,j})\) 로 주어진 교대행렬에 대하여 파피안을 다음과 같이 정의함\[\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}t_{\sigma(2i-1),\sigma(2i)}\]
  
• n=1인 경우\[t_{1,2}\]
  
• n=2인 경우\[t_{1,4} t_{2,3}-t_{1,3} t_{2,4}+t_{1,2} t_{3,4}\]
  
• n=3 인 경우\[t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\]
  

 

 

역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

 

 

메모

• number of perfect matchings on a planar rectangular lattice
• every non-zero term in the Pfaffian of the adjacency matrix of a graph G corresponds to a perfect matching.
• 통계물리에서 중요한 역할
• 도미노 타일링
• 다이머 모델
• http://www.science.uva.nl/onderwijs/thesis/centraal/files/f887198315.pdf

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxN2RmNDAyYzItYjlmYy00MzM5LWJkZmQtYjdjOWZhNjM3MTI0&sort=name&layout=list&num=50
• http://en.wikipedia.org/wiki/Talk%3APfaffian#Mathematica_code

 

 

관련된 항목들

• 사각격자의 도미노 타일링 (dimer problem)

 

사전 형태의 자료

• http://en.wikipedia.org/wiki/Pfaffian
• http://en.wikipedia.org/wiki/FKT_algorithm

 

리뷰, 에세이, 강의노트

• Hirota, Ryogo. "Determinants and Pfaffians." 数理解析研究所講究録 1302 (2003): 220-242. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1302-14.pdf

 

관련논문

• Wu, F. Y. 2006. “Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.” Physical Review E 74 (2): 020104. doi:10.1103/PhysRevE.74.020104.

 

관련도서

• Hirota, Ryogo, Atsushi Nagai, Jon Nimmo, and Claire Gilson. 2004. “Determinants and Pfaffians.” In The Direct Method in Soliton Theory. Cambridge Tracts in Mathematics. http://dx.doi.org/10.1017/CBO9780511543043.004.
• Barry M McCoy, Advanced Statistical Mechanics
  • The Pfaffian solution of the Ising model DOI:10.1093/acprof:oso/9780199556632.003.0011