Permutation-operator inequalities on certain symmetry classes of tensors

AbstractIf A∈T(m, N), the real-valued N-linear functions on Em, and σ∈SN, the symmetric group on {…,N}, then we define the permutation operator Pσ: T(m, N) → T(m, N) such that Pσ(A)(x1,x2,…,xN = A(xσ(1),xσ(2),…, xσ(N)). Suppose Σqi=1ni = N, where the ni are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈Pσ(A1⊗A2⊗⋯⊗Aq),A1⊗A2⊗⋯ ⊗ Aq〉 ⩾ 0 for Ai∈S(m, ni), where S(m, ni) denotes the subspace of T(m, ni) consisting of all the fully symmetric members of T(m, ni). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of SN. We let TG(m, N) denote all A∈T(m, N) such that Pσ(A) = A for all σ∈G. We define the symmetrizer SG: T(m, N)→TG(m,N) such that SG(A) = 1/|G|Σσ∈G Pσ(A). Suppose H is a subgroup of G and A∈TH(m, N). Clearly ‖SG‖(A) ‖⩽ ‖A‖. We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of TH(m,N), then can we explicitly present a constant C>0 such that ‖ SG(A)‖⩾C‖A‖ for all A∈D