# Kerov functions for composite representations and Macdonald ideal

## TL;DR

This paper explores Kerov functions, a broad generalization of Macdonald polynomials, focusing on their behavior for conjugate and composite Young diagram representations, especially at the Macdonald locus where formulas simplify.

## Contribution

It analyzes the structure of Kerov functions for composite representations and examines the special properties and simplifications at the Macdonald locus.

## Key findings

- Simplified formulas at the Macdonald locus for conjugate and composite representations
- Deviations from Schur functions highlight challenges in link hyperpolynomial theory
- Complex N-dependence in Kerov functions compared to the uniformization in Macdonald case

## Abstract

Kerov functions provide an infinite-parametric deformation of the set of Schur functions, which is a far-going generalization of the 2-parametric Macdonald deformation. In this paper, we concentrate on a particular subject: on Kerov functions labeled by the Young diagrams associated with the conjugate and, more generally, composite representations. Our description highlights peculiarities of the Macdonald locus (ideal) in the space of the Kerov parameters, where some formulas and relations get drastically simplified. However, even in this case, they substantially deviate from the Schur case, which illustrates the problems encountered in the theory of link hyperpolynomials. An important additional feature of the Macdonald case is uniformization, a possibility of capturing the dependence on $N$ for symmetric polynomials of $N$ variables into a single variable $A=t^N$, while in the generic Kerov case the $N$-dependence looks considerably more involved.