[Paper Review] 2-block Springer fibers: Convolution algebras, coherent sheaves, and embedded TQFT

Catharina Stroppel, Ben Webster|arXiv (Cornell University)|Feb 14, 2008

Algebraic structures and combinatorial models9 citations

TL;DR

This paper studies the geometry of 2-block Springer fibers via torus actions and Bialynicki-Birula theory, proving Fung's conjecture on component intersections. It constructs a convolution algebra on cohomologies of component intersections and Bialynicki-Birula cells, showing isomorphism to a generalized arc algebra, and connects this to coherent sheaf approaches in link homology.

ABSTRACT

For a fixed 2-block Springer fiber, we describe the structure of its irreducible components and their relation to the Bialynicki-Birula paving, following work of Fung. That is, we consider the space of complete flags in C^n preserved by a fixed nilpotent matrix with 2 Jordan blocks, and study the action of diagonal matrices commuting with our fixed nilpotent. In particular, we describe the structure of each component, its set of torus fixed points, and prove a conjecture of Fung describing the intersection of any pair. Then we define a convolution algebra structure on the direct sum of the cohomologies of pairwise intersections of irreducible components and closures of C^*-attracting sets (that is, Bialynicki-Birula cells), and show this is isomorphic to a generalization of the arc algebra of Khovanov defined by the first author. We investigate the connection of this algebra to Cautis & Kamnitzer's recent work on link homology via coherent sheaves and suggest directions for future research.

Motivation & Objective

• To understand the structure of irreducible components in 2-block Springer fibers and their relation to torus actions.
• To verify Fung's conjecture on the intersection of any two irreducible components in the 2-block case.
• To define a convolution algebra on cohomologies of component intersections and C*-attracting sets (Bialynicki-Birula cells).
• To establish an isomorphism between this convolution algebra and a generalized arc algebra introduced by the first author.
• To explore connections between this algebraic structure and Cautis & Kamnitzer's coherent sheaf framework for link homology.

Proposed method

• Utilizes the action of diagonal matrices commuting with a fixed nilpotent matrix of two Jordan blocks.
• Applies Bialynicki-Birula theory to decompose the Springer fiber into cells (attracting sets) and analyze their closures.
• Analyzes the set of torus fixed points on each irreducible component and their combinatorial structure.
• Constructs a convolution algebra on the direct sum of cohomologies of pairwise intersections of components and Bialynicki-Birula cells.
• Employs techniques from algebraic geometry and representation theory to prove the isomorphism between the convolution algebra and the generalized arc algebra.
• Draws connections to Cautis & Kamnitzer's work on coherent sheaves and link homology, suggesting new research directions in categorified quantum groups and link invariants.

Experimental results

Research questions

• RQ1 What is the precise structure of the irreducible components of a 2-block Springer fiber, and how do they relate to the Bialynicki-Birula paving?
• RQ2 Does Fung's conjecture on the intersection of any two irreducible components in the 2-block case hold true?
• RQ3 What is the algebraic structure of the convolution algebra defined on the cohomologies of intersections of components and Bialynicki-Birula cells?
• RQ4 Is this convolution algebra isomorphic to a generalized arc algebra as conjectured?
• RQ5 How does this algebraic construction relate to Cautis & Kamnitzer's coherent sheaf approach to link homology?

Key findings

• The paper proves Fung's conjecture on the intersection of any two irreducible components in the 2-block Springer fiber, confirming their transverse intersection pattern.
• Each irreducible component is shown to have a well-defined structure with a specific set of torus fixed points, determined by the Bialynicki-Birula decomposition.
• The convolution algebra on the direct sum of cohomologies of component intersections and Bialynicki-Birula cells is isomorphic to a generalized arc algebra defined by the first author.
• The construction provides a geometric realization of the generalized arc algebra via cohomology of algebraic varieties in the Springer fiber.
• The results suggest a deep connection between this convolution algebra and Cautis & Kamnitzer's framework of coherent sheaves in link homology, opening new pathways for categorified representation theory.
• The work establishes a bridge between geometric representation theory (Springer fibers, torus actions) and topological quantum field theory via algebraic structures on cohomology.