The stabilizer of immanants

Immanants are homogeneous polynomials of degree n in n² variables associated to the irreducible representations of the symmetric group S_n of n elements. We describe immanants as trivial S_n modules and show that any homogeneous polynomial of degree n on the space of n×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 Purificação 3], we prove that for n?6 the identity component of the stabilizer of any immanant (except determinant, permanent, and π=(4,1,1,1)) is Δ(S_n)?T(GL_n×GL_n)?Z₂, where T(GL_n×GL_n) is the group consisting of pairs of n×n diagonal matrices with the product of determinants 1, acting by left and right matrix multiplication, Δ(S_n) is the diagonal of S_n×S_n, acting by conjugation (S_n is the group of symmetric group) and Z₂ acts by sending a matrix to its transpose. Based on the work of Purificação 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 Δ(S_n)?T(GL_n×GL_n)?Z₂.