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Preprint
A basis for a quotient of symmetric polynomials
arXiv (Cornell University)
01 Oct 2019
Abstract
Consider the ring $\mathcal{S}$ of symmetric polynomials in $k$ variables
over an arbitrary base ring $\mathbf{k}$. Fix $k$ scalars
$a_{1},a_{2},\ldots,a_{k}\in\mathbf{k}$. Let $I$ be the ideal of $\mathcal{S}$
generated by $h_{n-k+1}-a_{1},h_{n-k+2}-a_{2},\ldots,h_{n}-a_{k}$, where
$h_{i}$ is the $i$-th complete homogeneous symmetric polynomial. The quotient
ring $\mathcal{S}/I$ generalizes both the usual and the quantum cohomology of
the Grassmannian. We show that $\mathcal{S}/I$ has a $\mathbf{k}$-module basis
consisting of (residue classes of) Schur polynomials fitting into an $\left(
n-k\right) \times k$-rectangle; and that its multiplicative structure constants
satisfy the same $S_3$-symmetry as those of the Grassmannian cohomology. We
prove a Pieri rule and a "rim hook algorithm", and conjecture a positivity
property generalizing that of Gromov-Witten invariants. We construct two
further bases of $\mathcal{S}/I$ as well. We also study the quotient of the
whole polynomial ring (not just the symmetric polynomials) by the ideal
generated by the same $k$ polynomials as $I$.
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Details
- Title
- A basis for a quotient of symmetric polynomials
- Creators
- Darij Grinberg
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862240104721