M. Antónia Duffner

Linear transformations that preserve immanants

Abstract We characterize the linear transformations on matrices that preserve the immanants, i.e., d χ ( T ( X )) d χ ( X ) for all X , where χ is an irreducible character of degree greater than one on S n . We present a necessary and sufficient condition for T to be invertible if it preserves the generalized matrix function d c , and prove that the preservers of immanants are always nonsingular.

On the Relation Between the Determinant and the Permanent on Symmetric Matrices

Let H n ( F ) be the space of n -square symmetric matrices over the field F . We generalize the main result of [M.H. Lim (1979). A note on the relation between the determinant and the permanent. Linear and Multilinear Algebra , 7 , 145-147], proving that the determinant is not convertible into the permanent on H n ( F ), provided that n ⩽ 3, F has at least n elements and the characteristic of F is not 2. The case n = 2 is also studied.Let H n ( F ) be the space of n -square symmetric matrices over the field F . We generalize the main result of [M.H. Lim (1979). A note on the relation between the determinant and the permanent. Linear and Multilinear Algebra , 7 , 145-147], proving that the determinant is not convertible into the permanent on H n ( F ), provided that n ⩽ 3, F has at least n elements and the characteristic of F is not 2. The case n = 2 is also studied.

On the conversion of an immanant into another

Let χ and λ be distinct irreducible complex characters of Sn where n≥3 and {χ, λ} ≠ {[3,1], [2,12]}, if n=4. We prove that there is no linear transformation T of Mn (C) that converts the immanant d λ into the immanant d χ, i.e., such that d χ(T(X)) = d λ(X), for all X∊Mn (C).

Subspaces where an immanant is convertible into its conjugate

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.

Linear preservers of immanants on symmetric matrices

Abstract Let Q n ( C ) denote the space of the n -square skew-symmetric complex matrices and let χ be an irreducible nonlinear complex character of the symmetric group S n , with χ ≠ [ n - 1 , 1 ] , [ 2 , 1 n - 2 ] . We describe the linear operators of Q n ( C ) that preserve the immanant d χ .

A note on singular matrices satisfying certain polynomial identities

Let Sn be the symmectric group of degree n G a subgroup of Sn a field and λ and -valued character of G. We describe the singular matrices A that satisfy and if χ is an irreducible character of Sn , we describe the singular matrices A and B such that