Projections of Richardson Varieties
Projections of Richardson Varieties
复制标题
理查森品种的预测
DOI：
10.1515/crelle-2012-0045
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
David E. Speyer
中科院分区：
文献类型：
--
作者：
A. Knutson;T. Lam;David E. Speyer
当Schubert簇在完全广义旗流形G/B到部分旗流形G/P$的投影也是Schubert簇时，Richardson簇（Schubert簇与相反的Schubert簇的交叉）的投影不总是Richardson簇。由Richardson簇投影的G/P分层出现在全正性理论中，也来自Poisson和非对易几何。
在本文中，我们表明，许多几何性质的理查森品种持有更普遍的投影理查森品种，他们是正常的，科恩-麦考利，有合理的奇点，并兼容Frobenius分裂相对于标准的分裂。事实上，我们表明，预计理查森品种是唯一的兼容分裂的子品种，提供了一个例子，最近的定理[施韦德，Kumar-梅塔]，一个弗罗贝纽斯分裂计划只有1000多兼容分裂的子品种。(The G/B情形由[Hague]处理，我们对他的证明作了一些简化。）
Richardson簇的一个组合类似物是W中相应的Bruhat区间的序复形;这个复形被称为EL-可壳球[Bjorner-Wachs '82]。证明了这样的复形在W/W_P上的Bruhat序复形的投影也是一个可壳球。这需要广泛的分析“P-布鲁哈特秩序”，推广的k-布鲁哈特秩序[Bergeron Sottile '98]。在这种情况下，G/P是极小的（例如格拉斯曼），我们表明，其Grobner退化需要每个投影理查森品种的斯坦利-Reisner计划的相应的球。
While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry.
In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, Cohen-Macaulay, have rational singularities, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem [Schwede, Kumar-Mehta] that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by [Hague], whose proof we simplify somewhat.)
One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an EL-shellable ball [Bjorner-Wachs '82]. We prove that the projection of such a complex into the order complex of the Bruhat order on W/W_P is again a shellable ball. This requires extensive analysis of "P-Bruhat order", a generalization of the k-Bruhat order of [Bergeron-Sottile '98]. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Grobner degeneration takes each projected Richardson variety to the Stanley-Reisner scheme of its corresponding ball.