In previous work we showed that two apparently unrelated formulas for the Hall–Littlewood
polynomials of type
are, in fact, closely related. The first is the tableau formula obtained by specializing
in the
Haglund–Haiman–Loehr formula for Macdonald polynomials. The second is the type
instance of Schwer’s formula (rephrased and rederived by Ram) for Hall–Littlewood
polynomials of arbitrary finite type; Schwer’s formula is in terms of so-called alcove
walks, which originate in the work of Gaussent and Littelmann and of the author
with Postnikov on discrete counterparts to the Littelmann path model. We showed
that the tableau formula follows by “compressing” Ram’s version of Schwer’s formula.
In this paper, we derive new tableau formulas for the Hall–Littlewood polynomials of
type
and
by compressing the corresponding instances of Schwer’s formula.
Keywords
Hall–Littlewood polynomials, Macdonald polynomials, alcove
walks, Schwer's formula, the Haglund–Haiman–Loehr formula
