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The Poisson degeneracy locus of a flag variety

Élie Casbi, Aria Masoomi, Milen Yakimov

Abstract

We present a comprehensive study of the degeneracy loci of the full flag varieties of all complex semisimple Lie groups equipped with the standard Poisson structures. The reduced Poisson degeneracy loci are shown to stratify under the action of the canonical maximal torus into open Richardson varieties $\mathcal{R}_v^w$ for pairs of Weyl group elements $v \leq w$ that extend the covering relation of the Bruhat order. Four different combinatorial descriptions of those pairs are given, and it is shown that their Bruhat intervals are power sets. The corresponding closed Richardson varieties $\overline{\mathcal{R}_v^m}$ are shown to be isomorphic to $(\mathbb{C}\mathbb{P}^1)^d$ for $d \geq 0$ in a compatible way with the stratification. As a consequence, we obtain that the reduced Poisson degeneracy loci of all full flag varieties are connected, and all of their irreducible components are isomorphic to $(\mathbb{C}\mathbb{P}^1)^n$ for some $n \geq 0$; they are not equidimensional in general. Using the framework of projected Richardson varieties, these results are extended to all partial flag varieties. The top dimension of irreducible components of the reduced Poisson degeneracy locus in the full flag case is proved to be equal to the cardinality of Kostant's cascade of roots and the reflective length of the longest Weyl group element. It is shown that the Poisson degeneracy loci of flag varieties are not reduced in general.

The Poisson degeneracy locus of a flag variety

Abstract

We present a comprehensive study of the degeneracy loci of the full flag varieties of all complex semisimple Lie groups equipped with the standard Poisson structures. The reduced Poisson degeneracy loci are shown to stratify under the action of the canonical maximal torus into open Richardson varieties for pairs of Weyl group elements that extend the covering relation of the Bruhat order. Four different combinatorial descriptions of those pairs are given, and it is shown that their Bruhat intervals are power sets. The corresponding closed Richardson varieties are shown to be isomorphic to for in a compatible way with the stratification. As a consequence, we obtain that the reduced Poisson degeneracy loci of all full flag varieties are connected, and all of their irreducible components are isomorphic to for some ; they are not equidimensional in general. Using the framework of projected Richardson varieties, these results are extended to all partial flag varieties. The top dimension of irreducible components of the reduced Poisson degeneracy locus in the full flag case is proved to be equal to the cardinality of Kostant's cascade of roots and the reflective length of the longest Weyl group element. It is shown that the Poisson degeneracy loci of flag varieties are not reduced in general.
Paper Structure (32 sections, 41 theorems, 244 equations, 2 figures)

This paper contains 32 sections, 41 theorems, 244 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The degeneracy loci of Poisson schemes
  3. Poisson homogeneous spaces, symplectic leaves and Poisson degeneracy loci
  4. Results
  5. background material on semisimple Lie groups
  6. Weyl groups and root systems
  7. Length, reflection length and reduced expressions
  8. The full flag variety $G/B_{+}$ and Richardson varieties
  9. Partial flag varieties and projected Richardson varieties
  10. A stratification of the Poisson degeneracy loci of flag varieties
  11. The standard Poisson structure on $G/B_+$
  12. Product decompositions and the reflective length
  13. The reduced Poisson degeneracy locus $D_0(G/B_+,\pi)_{\mathop{\mathrm{ {\mathrm{red}}}}\nolimits}$ and open Richardson varieties
  14. The index set $\mathop{\mathrm{ {\mathcal{ECR}} }}\nolimits(W) \subset W \times W$
  15. $T$-orbit stratifications of reduced Poisson degeneracy loci
  16. ...and 17 more sections

Key Result

Lemma 2.1

For each $w \in W$, one has Moreover, $w$ is an involution if and only if $w$ can be written as a product of reflections associated to pairwise orthogonal roots, and in that case we have $r=l_{\Delta}(w)$.

Figures (2)

  • Figure 1: Each edge represents an irreducible Richardson variety which is isomorphic to a ${{\mathbb C} {\mathbb P}^1}$
  • Figure 2: The 11 irreducible components $({{\mathbb C} {\mathbb P}^1})^{2}$ of the reduced Poisson degeneracy locus in the case of $SL_{4}({\mathbb C})$ in the background of the Bruhat graph. The blue faces correspond to the irreducible components for the roots $\alpha_{1},\alpha_{3}$, the red ones correspond to $\alpha_{2},\alpha_{1}+\alpha_{2}+\alpha_{3}$, and yellow ones to $\alpha_{1}+\alpha_{2},\alpha_{2}+\alpha_{3}$.

Theorems & Definitions (81)

  • Lemma 2.1: Carter, Carter
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 71 more