Using Du’s characterization of the dual canonical basis of the coordinate ring O(GL(n,C)), we express all elements of this basis in terms of immanants. We then give a new factorization of permutations w avoiding the patterns 3412 and 4231, which in turn yields a factorization of the corresponding Kazhdan-Lusztig basis elements C w(q) of the Hecke algebra Hn(q). Using the immanant and factorizat...

The fermionant Fermn(x̄) = ∑ σ∈Sn (−k)c(π) ∏n i=1 xi,j can be seen as a generalization of both the permanent (for k = −1) and the determinant (for k = 1). We demonstrate that it is VNP-complete for any rational k , 1. Furthermore it is #P -complete for the same values of k. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (whe...

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatori-ally.

Let Z be a character of the symmetric group 5P~. The immanant of an n x n matrix A = [-a•] with respect to X is Zw~snZ(w)al,,~(t)...an, w~,). Goulden and Jackson conjectured, and Greene recently proved, that immanants of Jacobi-Trudi matrices are polynomials with nonnegative integer coefficients. This led one of us (Stembridge) to formulate a series of conjectures involving immanants , some of ...

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatorially. We also present a β-extension of the theorem and a simple determinantal expression for the irreducible characters of the symmetric group.

In recent papers, the authors obtained formulas for directional derivatives of all orders, of the immanant and of the m-th ξ-symmetric tensor power of an operator and a matrix, when ξ is a character of the full symmetric group. The operator norm of these derivatives was also calculated. In this paper, similar results are established for generalized matrix functions and for every symmetric tenso...