On the number of points of the Lusztig nilpotent variety over a finite field

2012/12/16 by Olivier Schiffmann, Schiffmann, Olivier
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1212.3772

openalex publication_date 2012/12/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this note we give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver without edge loops, in terms of Kac's A-polynomial. We conjecture a similar result for quivers in which edge loops are allowed. Finally, we give a formula for the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.

Citations

• Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima)
• Motivic Donaldson-Thomas invariants and McKay correspondence
• Lectures on canonical and crystal bases of Hall algebras
• Positivity of Kac polynomials and DT-invariants for quivers

Related

• Number of points on abelian and Jacobian varieties over finite fields
• Curves in abelian varieties over finite fields
• Counting Quiver Representations over Finite Fields Via Graph Enumeration
• Number of points on a family of curves over a finite field
• Counting curves over finite fields