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⩽k of 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).