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arXiv · 2003.02916

Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties

Igor Makhlin
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Abstract

The main goal of this paper is to give explicit descriptions of two maximal cones in the Gröbner fan of the Plücker ideal. These cones correspond to the monomial ideals given by semistandard and PBW-semistandard Young tableaux. For the first cone, as an intermediate result we obtain the description of a maximal cone in the Gröbner fan of any Hibi ideal. For the second, we generalize the notion of Hibi ideals by associating an ideal with every interpolating polytope. This is a family of polytopes that generalizes the order and chain polytopes of a poset (à la Fang--Fourier--Litza--Pegel). We then describe a maximal cone in the Gröbner fan of each of these ideals. We also establish some useful facts concerning PBW-semistandardness, in particular, we prove that it provides a new Hodge algebra structure on the Plücker algebra.

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Igor Makhlin. 2021-09-22. Gröbner fans of Hibi ideals, generalized Hibi ideals and flag varieties. https://arxiv.org/abs/2003.02916

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