Abstract

We work here with the linear span ?t(k) of Hall-Littlewood polynomials indexed by partitions whose first part is no larger than k. The sequence of spaces ?t(k) yields a filtration of the space ? of symmetric functions in an infinite alphabet X. In joint work with Lascoux [4] we gave a combinatorial construction of a family of symmetric polynomials {A?(k) [X; t]} ?1?k, with ?[t]-integral Schur function expansions, which we conjectured to yield a basis for ?t(k). Our primary motivation for this construction is to provide a positive integral factorization of the Macdonald q, t-Kostka matrix. More precisely, we conjecture that the connection coefficients expressing the Hall-Littlewood or Macdonald polynomials belonging to ?t(k) in terms of the basis {A?(k) [X; t]} ?1?k are polynomials in ?[q, t]. We give here a purely algebraic construction of a new family {s?(k) [X; t]}?1?k of polynomials in ?t(k) which we conjecture is identical to {A?(k) [X; t]} ?1?k. We prove that {s?(k) [X; t]} ?1?k is in fact a basis of ?t(k) and derive several further properties including that s?(k) [X; t] reduces to the Schur function s?[X] for sufficiently large k. We also state a number of conjectures which reveal that the polynomials {s?(k) [X; t]} ?1 ?k are in fact the natural analogues of Schur functions for the space ?t(k). © 2003 Elsevier Science (USA). All rights reserved.