# nLab hafnian

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

$\mathrm{Hf}{F}^{i}:=\frac{1}{{2}^{k}k!}\sum _{\sigma \in {\Sigma }_{n}}\left(-1{\right)}^{\sum _{l=0}^{k-1}{i}_{\sigma \left(2l+1\right)}}{F}_{{i}_{\sigma \left(1\right)}{i}_{-\sigma \left(2\right)}}\cdots {F}_{{i}_{\sigma \left(2k-1\right)}{i}_{-\sigma \left(2k\right)}}$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

Created on October 9, 2012 19:11:41 by Zoran Škoda (161.53.130.104)