Hibi环是平坦环的充分条件与Schubert圈的平坦性
A sufficient condition for a Hibi ring to be level and levelness of Schubert cycles
摘要：
设$K$是域，$D$是有限分配格，$P$是$D$的所有并不可约元的集合.证明了如果对任何$x\in P$，$\{y\in P\mid y\geq x\}$是纯的，那么Hibi环$\RRRRR_K(D)$是水平的.利用这个结果和sagbi基理论的论证，我们证明了Grassmannian的Schubert子簇的齐次坐标环是水平的。
英文摘要：
Let $K$ be a field, $D$ a finite distributive lattice and $P$ the set of all join-irreducible elements of $D$. We show that if $\{y\in P\mid y\geq x\}$ is pure for any $x\in P$, then the Hibi ring $\RRRRR_K(D)$ is level. Using this result and the argument of sagbi basis theory, we show that the homogeneous coordinate rings of Schubert subvarieties of Grassmannians are level.
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