nLab hafnian

Hafnian is a special determinant-like expression evaluated for the pair $(F,I)$ of a matrix corresponding to an element $F$ of symplectic Lie algebra $sp(2n)$ and a subsequence $\mathbf{i} = (i_1,\ldots,i_{2k})$ of the ascending sequence $(-n,\ldots,-1, 1,\ldots,n)$. Let $\Sigma_n$ be the symmetric group on $n$ letters. Then

$$Hf F^{\mathbf{i}} := \frac{1}{2^k k!}\sum_{\sigma\in\Sigma_n} (-1)^{\sum_{l = 0}^{k-1} i_{\sigma(2 l + 1)}} F_{i_{\sigma(1)}i_{-\sigma(2)}}\cdots F_{i_{\sigma(2k-1)}i_{-\sigma(2k)}}$$

Related entries include determinant, Pfaffian, Pfaffian line bundle

• J.-G. Luque, J.-Y. Thibon, Pfaffian and hafnian identities in shuffle algebras, math.CO/0204026

• sec 4.4. in: A. I. Molev, Yangians and their applications, in “Handbook of Algebra”, Vol. 3, (M. Hazewinkel, Ed.), Elsevier, 2003, pp. 907-959 http://arxiv.org/abs/math.QA/0211288