[Paper Review] Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology

Ruslan Maksimau|arXiv (Cornell University)|Sep 11, 2019

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper proves that flag versions of quiver Grassmannians for Dynkin quivers (types A, D, E) have no odd integral cohomology, establishing a stronger result than prior work by working over ℤ. For types A and D, it further shows these varieties admit α-partitions into affine spaces, and for rigid representations, it proves a diagonal decomposition, implying vanishing odd cohomology. The result confirms a conjecture on fibers of geometric KLR algebra constructions and provides a direct geometric proof independent of representation theory.

ABSTRACT

We prove the conjecture that flag versions of quiver Grassmannians (also known as Lusztig's fibers) for Dynkin quivers (types $A$, $D$, $E$) have no odd cohomology groups over an arbitrary ring. Moreover, for types A and D we prove that these varieties have affine pavings. We also show that to prove the same statement for type E, it is enough to check this for indecomposable representations. We also give a flag version of the result of Cerulli Irelli-Esposito-Franzen-Reineke on rigid representations: we prove that flag versions of quiver Grassmannians for rigid representations have a diagonal decomposition. In particular, they have no odd cohomology groups.

Motivation & Objective

• To establish that flag versions of quiver Grassmannians for Dynkin quivers have no odd integral cohomology over ℤ.
• To show that for types A and D, these varieties admit α-partitions into affine spaces.
• To prove that flag quiver Grassmannians for rigid representations have a diagonal decomposition, implying vanishing odd cohomology.
• To provide a direct geometric proof of a conjecture on fibers of KLR algebra constructions, independent of KLR algebra representation theory.

Proposed method

• Uses geometric methods to analyze the fibers of the map π, which are flag versions of quiver Grassmannians.
• Applies the concept of α-partitions to decompose varieties into affine spaces in types A and D.
• Employs vector bundles with zero-locus sections to construct diagonal classes in the Chow ring.
• Leverages the vanishing of Ext¹ for rigid representations to construct a bundle with a section vanishing exactly on the diagonal.
• Uses Chern class theory and exact sequences of vector bundles to show that the diagonal class lies in the subring generated by pullbacks.
• Reduces the general case to indecomposable representations via reduction techniques in the quiver representation theory.

Experimental results

Research questions

• RQ1 Do flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology over ℤ?
• RQ2 Do these varieties admit α-partitions into affine spaces in types A and D?
• RQ3 Do flag quiver Grassmannians for rigid representations have a diagonal decomposition in the Chow ring?
• RQ4 Is it sufficient to verify the odd cohomology vanishing for indecomposable representations in type E?
• RQ5 Can the conjecture on fibers of KLR algebra constructions be proven geometrically without relying on KLR algebra theory?

Key findings

• Flag quiver Grassmannians for Dynkin quivers (A, D, E) have no odd integral cohomology, i.e., H^odd(𝒢, ℤ) = 0.
• For types A and D, these varieties admit α-partitions into affine spaces, implying a cellular structure.
• Flag quiver Grassmannians for rigid representations have a diagonal decomposition in the Chow ring, meaning [Δ] = c_top(ℰ) for a suitable vector bundle ℰ.
• The diagonal decomposition implies that such varieties have no odd cohomology over ℤ.
• For type E, the odd cohomology vanishing reduces to checking the property for indecomposable representations due to reduction techniques.
• The result confirms Conjecture 1.1 on fibers of KLR algebra constructions over ℤ, providing a geometric proof independent of KLR algebra representation theory.