Kerov functions revisited

Abstract

The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots and of topological vertices in DIM-based network theories, which could merge into a unified theory still to be developed. The Macdonald functions do the same for hyperpolynomials and refined vertices, but merging appears to be more problematic. For a detailed study of this problem, more knowledge is needed about the Macdonald polynomials than is usually available. As a preparation for the discussion of the knot/vertices relation, we summarize the relevant facts and open problems about the Macdonald and, more generally, Kerov functions. Like Macdonald polynomials, they are triangular combinations of Schur functions, but orthogonal in a more general scalar product. We explain that the parameters of the measure can be considered as a set of new time variables, and the Kerov functions are actually expressed through the Schur functions of these variables as well. Despite they provide an infinite-parametric extension of the Schur and Macdonald polynomials, the Kerov functions, and even the skew Kerov functions continue to satisfy the most important relations, like Cauchy summation formula, the transposition identity for reflection of the Young diagram and expression of the skew functions through the generalized Littlewood-Richardson structure constants. Since {\it these} are the properties important in most applications, one can expect that the Kerov extension exists for most of them, from the superintegrable matrix and tensor models to knot theory.