The stabilizer of immanants

Immanants are homogeneous polynomials of degree n in n(2) variables associated to the irreducible representations of the symmetric group G(n) of n elements. We describe immanants as trivial G(n) modules and show that any homogeneous polynomial of degree n on the space of n x n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by permutation matrices is a linear combination of immanants. Building on works of Duffner [5] and Purificacao [3], we prove that for n >= 6 the identity component of the stabilizer of any immanant (except determinant, permanent, and pi = (4, 1, 1, 1)) is Delta(G(n)) (sic) T(GL(n) x GL(n)) (sic) Z(2). where T(GL(n) x GL(n)) is the group consisting of pairs of n x n diagonal matrices with the product of determinants 1, acting by left and right matrix multiplication, Delta(G(n)) is the diagonal of G(n) x G(n), acting by conjugation (G(n) is the group of symmetric group) and Z(2) acts by sending a matrix to its transpose. Based on the work of Purificacao and Duffner [4], we also prove that for n >= 5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is Delta(G(n)) (sic) T(GL(n) x GL(n)) (sic) Z(2). (C) 2011 Elsevier Inc. All rights reserved.