Schur function analogs for a filtration of the symmetric function space

L. Lapointe; J. Morse

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arXiv · math/0111192

Schur function analogs for a filtration of the symmetric function space

L. Lapointe·J. Morse

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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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BibTeX RIS

L. Lapointe, J. Morse. 2001-11-17. Schur function analogs for a filtration of the symmetric function space. https://arxiv.org/abs/math/0111192

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