[Paper Review] Schubert slices in the combinatorial geometry of flag domains

Ana-Maria Brecan|arXiv (Cornell University)|Nov 2, 2014

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper provides a combinatorial characterization of Iwasawa-Schubert varieties that intersect flag domain orbits transversally in finite points, using generalized strictly pairing conditions on Weyl group elements. It computes the exact number of such Schubert varieties for all real forms of $SL(n,\mathbb{C})$, including explicit constructions for $SL(n,\mathbb{R})$, $SL(m,\mathbb{H})$, and $SU(p,q)$, and gives a precise algorithmic description of the intersection points via flag data and canonical rearrangements of Weyl group elements.

ABSTRACT

Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. Certain group theoretically defined compact complex submanifolds, which are regarded as cycles, are of basic importance for their complex geometric and representation theoretic properties. It is known that there are optimal Schubert varieties which intersect the cycles transversally in finitely many points and in particular determine them in homology. Here we give a precise description of these Schubert varieties in terms of certain subsets of the Weyl group and compute their total number for all the real forms of SL(n,C). Furthermore, we give an explicit description of the points of intersection in terms of flags and their number.

Motivation & Objective

• To provide a complete combinatorial description of Schubert varieties that intersect flag domain orbits transversally in finitely many points.
• To compute the total number of such Schubert varieties for all real forms of $SL(n,\mathbb{C})$, including $SL(n,\mathbb{R})$, $SL(m,\mathbb{H})$, and $SU(p,q)$.
• To give an explicit algorithmic description of the points of intersection between these Schubert varieties and the open orbits in terms of flags and Weyl group data.
• To define and study the canonical lifting of the base cycle $C_0$ to a higher-dimensional flag manifold and relate it to the generalized strictly pairing condition.
• To establish a homology class formula expressing the base cycle class as a sum of Schubert classes indexed by these Iwasawa-Schubert varieties.

Proposed method

• Introduces the concept of Iwasawa-Borel subgroups $B_I$ as Borel subgroups of $G$ containing an Iwasawa component $A_0N_0$, leading to the definition of Iwasawa-Schubert varieties as closures of $B_I$-orbits in $Z=G/P$.
• Uses the Iwasawa decomposition $G_0 = K_0A_0N_0$ to define a canonical parametrization $\alpha$ of open orbits in flag domains, which encodes the structure of the base cycle $C_0$.
• Defines the generalized strictly pairing condition on Weyl group elements $w$ via block-wise rearrangement of sequences involving $h_j < q$, $g_j > n-q$, and $k_j \in \{q+1,\dots,p\}$, ensuring that the canonical rearrangement $\hat{w}$ satisfies the standard strictly pairing condition.
• Applies a canonical rearrangement procedure to $w$ that reorders elements in each block to satisfy the strictly pairing condition, with the length difference $\sum a_i b_i$ accounting for the dimension drop between $S_{\hat{w}}$ and $S_w$.
• Uses the projection map $\pi$ from the complexified flag manifold to $Z=G/P$ to relate $S_{\hat{w}}$-invariant cycles to $S_w$ in the original flag domain, ensuring transversal intersection with the open orbit.
• Employs the Barlet space $C^q(D)$ and the smoothness of the cycle space at $C_0$ to justify the use of transversal slices and the homology class formula $[C_0^\alpha] = \sum_{S \in \mathcal{S}_{C_0}^\alpha} [S]$.

Experimental results

Research questions

• RQ1 Which Schubert varieties in $G/P$ intersect a given flag domain orbit $D_{a,b}$ transversally in finitely many points?
• RQ2 What is the total number of such Schubert varieties for each real form of $SL(n,\mathbb{C})$?
• RQ3 How can the points of intersection between these Schubert varieties and the open orbit be explicitly described in terms of flags and Weyl group data?
• RQ4 What is the precise combinatorial condition on Weyl group elements $w$ that ensures $S_w$ is an Iwasawa-Schubert variety for a given flag domain?
• RQ5 How does the canonical lifting of the base cycle $C_0$ to a higher-dimensional flag manifold relate to the generalized strictly pairing condition?

Key findings

• A Schubert variety $S_w$ is an Iwasawa-Schubert variety for the open orbit $D_{a,b}$ if and only if $w$ satisfies the generalized strictly pairing condition, which ensures that its canonical rearrangement $\hat{w}$ satisfies the standard strictly pairing condition.
• The number of such Schubert varieties of dimension $pq$ intersecting the open orbit $D_{a,b}$ is given by the binomial coefficient $\binom{f_1 + f_2}{f}$, where $f_1 = \min(a_1, b_1)$, $f_2 = \min(a_2, b_2)$, and $f = \max(f_1, f_2)$.
• For the case $Z = Gr(5,11)$ with $G_0 = SU(7,4)$ and parameters $a_1=3, a_2=1, b_1=2, b_2=5$, the number of such Schubert varieties is $\binom{3}{2} = 3$, explicitly realized by the sequences $1281011345679$, $1389112456710$, and $2389101456711$.
• The intersection point of $S_w$ with the open orbit $D_{a,b}$ is uniquely determined by the generalized strictly pairing condition algorithm applied to $w$ and the canonical parametrization $\alpha$.
• The homology class of the base cycle $C_0^\alpha$ in $Z=G/P$ is equal to the sum of the Schubert classes of all Iwasawa-Schubert varieties in $\mathcal{S}_{C_0}^\alpha$, i.e., $[C_0^\alpha] = \sum_{S \in \mathcal{S}_{C_0}^\alpha} [S]$.
• The canonical lifting $\hat{C}_0$ of the base cycle $C_0$ to the flag manifold $G/B$ is associated with the canonical rearrangement $\hat{w}$ of $w$, and the dimension drop $\dim S_{\hat{w}} - \dim S_w = \sum a_i b_i$ precisely characterizes the generalized pairing condition.