Vol. 31, No. 1, 1969

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Symmetric positive definite multilinear functionals with a given automorphism

Marvin David Marcus and Stephen J. Pierce

Vol. 31 (1969), No. 1, 119–132

Let V be an n-dimensional vector space over the real numbers R and let φ be a multilinear functional,

φ : m× V → R

i.e., φ(x1,,xm) is linear in each xj separately, j = 1,,m. Let H be a subgroup of the symmetric group Sm. Then φ is said to be symmetric with respect to H if

φ (x σ(1),⋅⋅⋅ ,xσ(m)) = φ(x1,⋅⋅⋅ ,xm)

for all σ H and all xj V,j = 1,,m. (In general, the range of φ may be a subset of some vector space over R.) Let T : V V be a linear transformation. Then T is an automorphism with respect to φ if

φ(T x1,⋅⋅⋅ ,Txm ) = φ (x1,⋅⋅⋅ ,xm)

for all xi V,i = 1,,m. It is easy to verify that the set A(H,T) of all φ which are symmetric with respect to H and which satisfy (3) constitutes a subspace of the space of all multilinear functionals symmetric with respect to H. We denote this latter set by Mm(V,H,R).

We shall say that φ is positive definite if

φ(x,⋅⋅⋅ ,x) > 0

for all nonzero x in V , and we shall denote the set of all positive definite φ in A(H,T) by P(H,T). It can be readily verified that P(H,T) is a convex cone in U(H,T).

Our main results follow.

Mathematical Subject Classification
Primary: 15.60
Received: 7 September 1967
Revised: 21 May 1969
Published: 1 October 1969
Marvin David Marcus
Stephen J. Pierce