Kerov functions for composite representations and Macdonald ideal

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.

26 pages

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• Kerov functions revisited
• On complete solution of the quantum Dell system
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• Can tangle calculus be applicable to hyperpolynomials?
• KNTZ trick from arborescent calculus and the structure of differential expansion
• On Hamiltonians for Kerov functions
• Pentad and triangular structures behind the Racah matrices
• On generalized Macdonald polynomials
• Algorithms for representations of quiver Yangian algebras
• Hopf superpolynomial from topological vertices
• Super-Hamiltonians for super-Macdonald polynomials