arXiv 1409.01742014-09math.RT

不变子空间簇的不可约分支的边界

The boundary of the irreducible components for invariant subspace varieties

Justyna Kosakowska, Markus Schmidmeier

发表信息：: Mathematische Zeitschrift, 290 (2018), 953-972
DOI：: 10.1007/s00209-018-2047-8

摘要：

给定划分$α$，$β$，$γ$，Jordan型$α$，$β$，$γ$的幂零线性算子的短正合序列$0\to N_α\to N_β\to N_γ\to 0$分别定义仿射簇的可构造子集$\mathbb V_{α,γ}^β$.几何上，品种$\mathbb V_{α,γ}^β$是特别感兴趣的，因为它们自然发生，因为它们通常由几个不可约的组成部分。实际上，$(α,β,γ)$形状的每个Littlewood-Richardson（LR-）表$Γ$贡献一个不可约分量$\overline{\mathbb V}_Γ$。我们考虑LR-表上的偏序$Γ\leq_{\sf bound}^*\widetildeΓ$，它是$\mathbb V_{\widetildeΓ}\cap \overline{\mathbb V}_Γ\neq \emptyset$给出的关系的传递闭包.本文将边界关系与由代数、组合和几何条件给出的偏序进行了比较。已知在$α$的部分至多为两个的情况下，所有那些偏序是等价的。我们证明了这些偏序在$β\setminusγ$是水平和垂直条带的情况下也是等价的。此外，我们还讨论了订单一般如何不同。

英文摘要：

Given partitions $α$, $β$, $γ$, the short exact sequences $0\to N_α\to N_β\to N_γ\to 0$ of nilpotent linear operators of Jordan types $α$, $β$, $γ$, respectively, define a constructible subset $\mathbb V_{α,γ}^β$ of an affine variety. Geometrically, the varieties $\mathbb V_{α,γ}^β$ are of particular interest as they occur naturally and since they typically consist of several irreducible components. In fact, each Littlewood-Richardson (LR-) tableau $Γ$ of shape $(α,β,γ)$ contributes one irreducible component $\overline{\mathbb V}_Γ$. We consider the partial order $Γ\leq_{\sf bound}^*\widetildeΓ$ on LR-tableaux which is the transitive closure of the relation given by $\mathbb V_{\widetildeΓ}\cap \overline{\mathbb V}_Γ\neq \emptyset$. In this paper we compare the boundary relation with partial orders given by algebraic, combinatorial and geometric conditions. It is known that in the case where the parts of $α$ are at most two, all those partial orders are equivalent. We prove that those partial orders are also equivalent in the case where $β\setminusγ$ is a horizontal and vertical strip. Moreover, we discuss how the orders differ in general.

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