A simple proof of an identity concerning Pfaffians of skew symmetric matrices
Let \(A\) be a skew-symmetric \((n\times n)\)-matrix with coefficients in a commutative ring \(R\). For a subset \(I\subset \{1,\dots, n\}\) of cardinality \(m\), denote by \(A(I)\) the skew-symmetric \((m\times m)\)-matrix obtained from \(A\) by eliminating the rows and the columns not indexed by indices from \(I\). Denote by \(p(I)= Pf(A(I))\) the Pfaffian of \(A(I)\), where \(Pf(A(\emptyset))= 1\). The main result is Theorem 1. For any two subsets \(I_ 1, I_ 2\subset \{1,\dots, n\}\) and elements \(i_ 1, i_ 2,\dots, i_ t\in \{1,\dots, n\}\), with \(i_ 1< i_ 2<\cdots< i_ t\) and \(\{i_ 1,\dots, i_ t\}= I_ 1\Delta I_ 2= (I_ 1\backslash I_ 2)\cup (I_ 2\backslash I_ 1)\) one has \(\sum^ t_{\tau= 1} p(I_ 1 \Delta\{i_ \tau\}) p(I_ 2 \Delta\{i_ \tau\})= 0\). If \(A\) is a skew-symmetric \(((3m)\times (3m))\)-matrix, \(I_ 1= \{1,\dots, 2m+ 1\}\), \(I_ 2= \{1,\dots, m, 2m+ 2,\dots, 3m\}\), the obtained identity yields the Grassmann-Plücker identities.
scientific article; zbMATH DE number 755388