Subspaces where an immanant is convertible into its conjugate*

Abstract

Let n≥5 and let be an irreducible nonlinear character of Sn such that whenever σ is a transposition or a cycle of length three; furthermore let Tn be the (0, 1)-matrix of order n that has ones exactly on and below the upper neighbours of the main diagonal and denote by Eij the matrix of order n with 1 in position (i, j) and 0 elsewhere. Given i,jϵ{1,…,n}, with i+1<j, we prove that if j−i≠3, then in the subspace Mn (Tn +Eij there exist matrices for which the immanant is not convertible into the immanant by sign-affixing. Abusing language, we say that the space is -inconvertible, and show that spaces Mn (Tn +E25 ) and Mn (Tn +En−3,n ). We also state some sufficient fonditions for the subspace Mn (Tn ) to be external convertible. With some exceptions our theorems say that the coordinate subspaces found for the conversion of the permanent into the determinant by Gibson around 1970 are also best possible for other immanants.