Hankel Determinants and Laurent Biorthogonal of Narayana with q-Narayana Polynomials and Operator Space Theory

Abstract

We consider a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arise from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. We study several different types of completely bounded polynomials on the operator space which form the operator space duals endowed with related operator structures. Of special interest are what we call Haagerup, Kronecker, and Schur polynomials – polynomials associated with different types of matrix products.