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

HfF i:=12 kk! σΣ n(1) l=0 k1i σ(2l+1)F i σ(1)i σ(2)F i σ(2k1)i σ(2k) 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

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