Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator

Abstract

We extend the classical Schur–Weyl duality between representations of the groups $${SL(n, \mathbb{C})}$$ and $${\mathfrak{S}_N}$$ to the case of $${SL(n, \mathbb{C})}$$ and the infinite symmetric group $${\mathfrak{S}_\mathbb{N}}$$ . Our construction is based on a “dynamic,” or inductive, scheme of Schur–Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which has not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand–Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter–Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.