The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics computation because connections with the Hong-Ou-Mandel effect Boson Sampling problem. In that context, it would be useful to know distribution or other immanants for random matrices, but seems a difficult problem.We advance this program by calculating average squared modulus generic immanan...

In the context of statistical physics, Chandrasekharan and Wiese recently introduced the fermionant Fermk, a determinant-like function of a matrix where each permutation π is weighted by −k raised to the number of cycles in π . We show that computing Fermk is #P-hard under polynomial-time Turing reductions for any constant k > 2, and is ⊕P-hard for k = 2, where both results hold even for the ad...

Abstract Immanants are functions on square matrices generalizing the determinant and permanent. Kazhdan–Lusztig immanants, which indexed by permutations, involve $q=1$ specializations of Type A polynomials, were defined Rhoades Skandera (2006, Journal Algebra 304, 793–811). Using results Haiman (1993, American Mathematical Society 6, 569–595) Stembridge (1991, Bulletin London 23, 422–428), show...

The immanant d () associated with the irreducible character of the symmetric group S n , indexed by the partition of n, acting on an nn matrix A = a ij ] is deened by d (A) = X 2Sn () n Y i=1 a ii(i) : For a tree T on n vertices, let L(T) denote its Laplacian matrix. Let x be an indeterminate variable and I be the n n identity matrix. The immanantal polynomial of T corresponding to d is deened ...

Given a Hermitian, non-negativede nite kernelK and a character of the symmetric group on n letters, de ne the corresponding immanant functionK [x1; : : : ; xn ] := P ( ) Qn i=1 K(xi; x (i)), where the sum is over all permutations of f1; : : : ; ng. When is the sign character (resp. the trivial character), then K is a determinant (resp. permanent). The function K is symmetric and non-negative, a...

We show that dual canonical basis elements of the quantum polynomial ring in n variables can be expressed as specializations of dual canonical basis elements of 0-weight spaces of other quantum polynomial rings. Our results rely upon the natural appearance in the quantum polynomial ring of Kazhdan-Lusztig polynomials, Rpolynomials, and certain single and double parabolic generalizations of thes...

In recent papers, formulas are obtained for directional derivatives, of all orders, of the determinant, the permanent, the m-th compound map and the m-th induced power map. This paper generalizes these results for immanants and for other symmetric powers of a matrix.