Tensor inequalities, ξ-functions and inequalities involving immanants Original Research Article

If f is a complex valued function with domain image, the symmetric group on {1,2,…,n}, then we define the matrix function df(·) in the usual way. If α is a partition of n and λα denotes the associated irreducible character of image, then dλα(·), which we abbreviate to dα(·), is known as an immanant. The permanent dominance conjecture of Lieb specialized to the symmetric groups asserts that if α is a partition of n, then dα(A)less-than-or-equals, slantdeg(α) per (A) for each image, the set of n×n positive semi-definite Hermitian matrices. Recently it was shown (T.H. Pate, Proc. London Math. Soc. 76 (2) (1998) 307–358.) that if α is a partition of n of the form (p,q,r,2s,1t), then dα(A)less-than-or-equals, slantdeg(α)per(A) for each image. We improve upon this result by showing that this same inequality holds for all image when α is a partition of the form (p,qw,r,2s,1t) where 0less-than-or-equals, slantwless-than-or-equals, slant2. Thus, permanent dominance holds for all immanants whose associated partitions are of the form (p,q2,r). We also show that permanent dominance holds for all immanants whose associated partitions are of the form (n+p,nk) as long as n is sufficiently large. Inequalities involving immanants have often been obtained using a special kind of class function known as a Ψ-function. Our results on immanants are obtained by analyzing a new set of class functions, called ξ-functions, that are more fundamental than the Ψ-functions in the sense that each Ψ-function is a non-negative linear combination of ξ-functions. The ξ-functions arise from a study of tensor contractions and a special collection of quadratic forms defined on spaces of bi-symmetric tensors.