We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions. 1. Schur positivity conjectures The ring of...

By the Choi matrix criteria it is easy to determine if a specific linear map completely positive, but establish whether positive much less straightforward. In this paper we consider classes of maps, determined by structural conditions on an associated matrix, for which positivity and complete coincide. The basis our proofs lies in representation ⁎-linear maps going back work R.D. Hill enables u...

We study a heat kernel e defined by a self-adjoint Hamiltonian H acting on a Hilbert space H, and a unitary representation U(g) of a symmetry group G of H, normalized so that the ground vector of H is invariant under U(g). The triple [H, U(g), H] defines a twisted partition function Zg and a twisted Gibbs expectation ( } ) g , Zg=TrH (U(g) e&;H) and ( } ) g=TrH (U(g) } e) TrH (U(g) e). We say t...