Schur function analogs for a filtration of the symmetric function space

We work here with the linear span Λt(k) of Hall-Littlewood polynomials indexed by partitions whose first part is no larger than k. The sequence of spaces Λt(k) yields a filtration of the space Λ 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 {Aλ(k)[X; t]}λ1 ≤ k, with N[t]-integral Schur function expansions, which we conjectured to yield a basis for Λt(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 Λt(k) in terms of the basis {Aλ(k)[X; t]}λ1 ≤ k are polynomials in N [q, t]. We give here a purely algebraic construction of a new family {sλ(k)[X; t]}λ1 ≤ kof polynomials in Λt(k) which we conjecture is identical to {Aλ(k)[X; t]}λ1 ≤ k. We prove that {sλ(k) [X; t]}λ1 ≤ k is in fact a basis of Λt(k) and derive several further properties including that sλ(k)[X; t] reduces to the Schur function sλ[X] for sufficiently large k. We also state a number of conjectures which reveal that the polynomials {sλ(k)[X; t]}λ1 ≤ k are in fact the natural analogues of Schur functions for the space Λt(k).