Noncommutative Catalan numbers and orthogonality

Noncommutative Catalan numbers C_n were introduced in a joint work with Vladimir Retakh 8 years ago as certain elements of a free Laurent polynomial algebra in n+1 variables x_0,...,x_n. Even though C_n are multi-parameter deformations of their commutative counterparts c_n, they have many properties in common. In our forthcoming joint paper with Misha Gekhtman and Vladimir Retakh we discovered that C_n, similarly to c_n, are moments for an orthogonal basis of polynomials in one variable with noncommutative coefficients. This observation allowed us to further generalize both commutative and noncommutative Catalan numbers as well as commutative and noncommutative orthogonality.

If time permits, I will explain a rather surprising connection of this new orthogonality theory with (commutative and noncommutative) cluster structures of type A.