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2 votes
0 answers
131 views

Pfaffian generalization

The identity $$\left| \begin{array}{cccc} x & y_1 & y_2 & y_3 \\ z_1 & 0 & a & b \\ z_2 & -a & 0 & c \\ z_3 & -b & -c & 0 \\ \end{array} \right|=\...
11 votes
1 answer
952 views

Detailed modern references for basic properties of Pfaffians over commutative rings

Pfaffians are important to algebraic combinatorics, at least. This is to propose the making of a 'wiki' list, more modern, precise and compressed than e.g. the relevant Wikipedia page (nothing against ...
5 votes
2 answers
389 views

Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...