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Full exceptional collections on the isotropic Grassmannians

Lyalya Guseva, Alexander Novikov

TL;DR

The paper proves that Kuznetsov–Polishchuk exceptional collections on isotropic Grassmannians \\mathrm{IGr}(k,V)\$ for a symplectic space $V$ are full and consist of vector bundles. It introduces three new combinatorial/homological tools—generalized staircase complexes, symplectic staircase complexes, and secondary staircase complexes—to control mutations and pushforwards via Borel–Bott–Weil theory, enabling an induction on the block structure \\mathcal{A}_t(l). The authors provide explicit descriptions of the KP objects and their duals, prove vector-bundle realizations, and demonstrate fullness by embedding twists into a growing ambient subcategory \\mathcal{D}_t$; parity considerations are essential, with even and odd cases treated by distinct staircase families. Collectively, these results extend the reach of KP-type full exceptional collections to all isotropic Grassmannians of type $C$, including the Lagrangian case, and supply a robust framework for related geometric representation-theoretic constructions.

Abstract

We prove that the Kuznetsov--Polishchuk exceptional collections on rational homogeneous spaces of the symplectic groups $\mathrm{Sp}(2n,\mathbb{C})$ are full and consist of vector bundles. To achieve this, we construct several classes of complexes, which we call generalized staircase complexes, symplectic staircase complexes and secondary staircase complexes -- each of which may be of independent interest.

Full exceptional collections on the isotropic Grassmannians

TL;DR

The paper proves that Kuznetsov–Polishchuk exceptional collections on isotropic Grassmannians \\mathrm{IGr}(k,V)\V; parity considerations are essential, with even and odd cases treated by distinct staircase families. Collectively, these results extend the reach of KP-type full exceptional collections to all isotropic Grassmannians of type , including the Lagrangian case, and supply a robust framework for related geometric representation-theoretic constructions.

Abstract

We prove that the Kuznetsov--Polishchuk exceptional collections on rational homogeneous spaces of the symplectic groups are full and consist of vector bundles. To achieve this, we construct several classes of complexes, which we call generalized staircase complexes, symplectic staircase complexes and secondary staircase complexes -- each of which may be of independent interest.
Paper Structure (34 sections, 57 theorems, 331 equations, 2 figures)

This paper contains 34 sections, 57 theorems, 331 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Kuznetsov--Polishchuk construction
  4. Sketch of the proof of Theorem \ref{['theo:intro']}
  5. Sketch of the proof of Theorem \ref{['lemma:fullness_induction_step']}
  6. Organization of the paper
  7. Notation
  8. Preliminaries
  9. Semiorthogonal decompositions and exceptional collections
  10. Schur functors and Young diagrams
  11. Symplectic Schur functors
  12. Notation
  13. Decomposition of tensor products of Schur functors
  14. Equivariant bundles on flag varieties
  15. Borel--Bott--Weil theorems
  16. ...and 19 more sections

Key Result

Theorem 1.2

Let $V$ be a $2n$-dimensional symplectic vector space over an algebraically closed field of characteristic zero and let $1\le k\le n$ be an integer. Then the Kuznetsov--Polishchuk exceptional collection in the bounded derived category $\mathrm{D}^{b}(\mathrm{IGr}(k, V))$ of coherent sheaves on $\mat

Figures (2)

  • Figure 1: Graphs of $\mathbb{d}$ for $\alpha=(3,2,2,1)$
  • Figure 2: Young diagram for the partition (7, 6, 6, 3, 2) with the shaded outer strip

Theorems & Definitions (128)

  • Conjecture 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • Theorem 1.6
  • proof
  • Theorem 1.7
  • proof
  • Remark 1.8
  • ...and 118 more