Tableau Formulas for One-Row Macdonald Polynomials of Types $C_n$ and $D_n$

TL;DR

This paper derives explicit tableau formulas for one-row Macdonald polynomials of types C_n and D_n using hypergeometric series transformations, linking algebraic structures to combinatorial representations.

Contribution

It introduces new tableau formulas for these polynomials by applying hypergeometric transformation techniques, expanding the combinatorial understanding of Macdonald polynomials.

Findings

Explicit tableau formulas for types C_n and D_n Macdonald polynomials.

Connection between deformed W-algebra correlation functions and tableau formulas.

Abstract

We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some transformation formulas for the basic hypergeometric series involving very well-poised balanced ${}_{12}W_{11}$ series. We remark that the correlation functions of the deformed $\mathcal{W}$ algebra generators automatically give rise to the tableau formulas when we principally specialize the coordinate variables.