Schur function analogs for a filtration of the symmetric function space

2001/11/17 by L. Lapointe, Lapointe, L., J. Morse +1
Mathematics · #05E05 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05E05

paper · pdf · doi:10.48550/arxiv.math/0111192

24 pages

arxiv created 2001/11/17 · arxiv updated 2009/11/30

Abstract

We consider a filtration of the symmetric function space given by Λ^(k)_t, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ^(k)_t. We prove several properties for the k-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when k is large. We also show that the connection coefficients for the k-Schur function basis with the Macdonald polynomials belonging to Λ^(k)_t are polynomials in q and t with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory.

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