The Stabilizer Of Immanants

Abstract: We describe immanants as trivial modules of the symmetric group and show that any homogeneous polynomial of degree n on the space of n by 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. we prove that the identity component of the stabilizer of any immanant (except determinant, permanent) is the expected one. We also prove that for n>5 the stabilizer of the immanant of any non-symmetric partition (except determinant and permanent) is again the expected one.