Jennifer Morse - A new basis for Macdonald polynomials

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Jennifer Morse - A new basis for Macdonald polynomials

JENNIFER MORSE, University of Pennsylvania, Philadelphia, Pennsylvania 19104 USA

A new basis for Macdonald polynomials

We will present a set of multivariate symmetric polynomials,

$\begin{displaymath}A_{\lambda}^{(k)}=\sum_{\mu ; \ell(\mu) \leq k}v_{\mu \lambda}^{(k)}(q) S_{\mu} (x;t), \end{displaymath}$

with $v_{\mu \lambda}^{(k)}(q)$ a polynomial in q with positive integer coefficients and $\ell(\lambda) \leq k$ . We conjecture that for any partition $\mu$ with $\ell(\mu) \leq k$ , the Hall-Littlewood polynomials can be expanded in this basis as

$\begin{displaymath}H_{\mu}(x;q,t) = \sum_{\lambda ; \ell(\lambda) \leq k} c_{\lambda \mu}^{(k)}(q) A_{\lambda}^{(k)}, \end{displaymath}$

where $c_{\lambda \mu}^{(k)}(q)$ is also $\in \bbd N[q]$ . The $A_{\lambda}^{(k)}$ basis provides a natural mechanism to divide the set of standard tableaux into families and is loosely related to the atomic decomposition of Lascoux and Schützenberger. We will discuss properties of this basis that are associated to tableaux combinatorics, creation operators, Pieri formulas and the Macdonald polynomials.