  Immanants are polynomial functions of n by n matrices attached to irreducible
characters of the symmetric group S_n, or equivalently to Young diagrams of
size n. Immanants include determinants and permanents as extreme cases. Valiant
proved that computation of permanents is a complete problem in his algebraic
model of NP theory, i.e., it is VNP-complete. We prove that computation of
immanants is VNP-complete if the immanants are attached to a family of diagrams
whose separation is $\Omega(n^\delta)$ for some $\delta>0$. We define the
separation of a diagram to be the largest number of overhanging boxes contained
in a single row. Our theorem proves a conjecture of Buergisser for a large
variety of families, and in particular we recover with new proofs his
VNP-completeness results for hooks and rectangles.
