论文标题
在两个偏旗品种的总积极性的概念上
On two notions of total positivity for partial flag varieties
论文作者
论文摘要
给定的整数$ 1 \ le k_1 <\ cdots <k_l \ le n-1 $，令$ \ text {fl} _ {k_1，\ dots，k_l; n} $表示类型$ a $ a $ partial标志品种，由子链的所有链条$（v_ {k_1} \ subset $ subset \ cdots）组成$ \ mathbb {r}^n $，其中每个$ v_k $都有尺寸$ k $。 Lusztig（1994，1998）引入了完全积极的部分$ \ text {fl} _ {k_1，\ dots，k_l; n}^{> 0} $作为部分标志的子集，可以由完全积极的$ n \ times n $ n $矩阵表示，并定义为完全不合格的部分$ \ text {fl} _ {k_1，\ dots，k_l; n}^{\ ge 0} $作为$ \ text {flext {flext {flext {f text {f text {k_1，\ dots，k_l; n}^{> 0} $的关闭。另一方面，按照后尼科夫（2007），我们定义$ \ text {fl} _ {k_1，\ dots，k_l; n}^{Δ> 0} $和$ \ text {flext {fl} _ {k_1，k_1，\ dots，\ dot $ \ text {fl} _ {k_1，\ dots，k_l; n} $，其中所有plücker坐标分别为正和非负坐标。从以下定义来看，Lusztig的总阳性意味着Plücker的积极性，并且自然要问这两个阳性概念何时同意。 Rietsch（2009）证明了他们在Grassmannian $ \ text {fl} _ {k; n} $的情况下同意，而Chevalier（2011）表明，这两个概念对于$ \ text {fl} _ {1,3; 4} $而言是不同的。我们表明，通常，这两个概念同意以及仅当$ k_1，\ dots，k_l $是连续的整数。我们根据线性代数和总阳性理论的经典结果给出了基本证明（包括格拉曼尼亚人）。 We also show that the cell decomposition of $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ coincides with its matroid decomposition if and only if $k_1,\dots,k_l$ are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and $ \ text {fl} _ {1,3; 4} $。最后，我们确定哪些阳性概念与旋转索引集的订单$ n $的自然作用兼容。
Given integers $1 \le k_1 < \cdots < k_l \le n-1$, let $\text{Fl}_{k_1,\dots,k_l;n}$ denote the type $A$ partial flag variety consisting of all chains of subspaces $(V_{k_1}\subset\cdots\subset V_{k_l})$ inside $\mathbb{R}^n$, where each $V_k$ has dimension $k$. Lusztig (1994, 1998) introduced the totally positive part $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$ as the subset of partial flags which can be represented by a totally positive $n\times n$ matrix, and defined the totally nonnegative part $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ as the closure of $\text{Fl}_{k_1,\dots,k_l;n}^{>0}$. On the other hand, following Postnikov (2007), we define $\text{Fl}_{k_1,\dots,k_l;n}^{Δ>0}$ and $\text{Fl}_{k_1,\dots,k_l;n}^{Δ\ge 0}$ as the subsets of $\text{Fl}_{k_1,\dots,k_l;n}$ where all Plücker coordinates are positive and nonnegative, respectively. It follows from the definitions that Lusztig's total positivity implies Plücker positivity, and it is natural to ask when these two notions of positivity agree. Rietsch (2009) proved that they agree in the case of the Grassmannian $\text{Fl}_{k;n}$, and Chevalier (2011) showed that the two notions are distinct for $\text{Fl}_{1,3;4}$. We show that in general, the two notions agree if and only if $k_1, \dots, k_l$ are consecutive integers. We give an elementary proof of this result (including for the case of Grassmannians) based on classical results in linear algebra and the theory of total positivity. We also show that the cell decomposition of $\text{Fl}_{k_1,\dots,k_l;n}^{\ge 0}$ coincides with its matroid decomposition if and only if $k_1,\dots,k_l$ are consecutive integers, which was previously only known for complete flag varieties, Grassmannians, and $\text{Fl}_{1,3;4}$. Finally, we determine which notions of positivity are compatible with a natural action of the cyclic group of order $n$ that rotates the index set.
