파피안(Pfaffian)

개요

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 \(\left(t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\right){}^2\)

\(A=(t_{i,j})\) 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\[t_{1,6} t_{2,5} t_{3,4}-t_{1,5} t_{2,6} t_{3,4}-t_{1,6} t_{2,4} t_{3,5}+t_{1,4} t_{2,6} t_{3,5}+t_{1,5} t_{2,4} t_{3,6}-t_{1,4} t_{2,5} t_{3,6}+t_{1,6} t_{2,3} t_{4,5}-t_{1,3} t_{2,6} t_{4,5}+t_{1,2} t_{3,6} t_{4,5}-t_{1,5} t_{2,3} t_{4,6}+t_{1,3} t_{2,5} t_{4,6}-t_{1,2} t_{3,5} t_{4,6}+t_{1,4} t_{2,3} t_{5,6}-t_{1,3} t_{2,4} t_{5,6}+t_{1,2} t_{3,4} t_{5,6}\]

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

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