On the number of points of nilpotent quiver varieties over finite fields

Abstract

We give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac Apolynomial. Finally we compute 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. T. BOZEC, O. SCHIFFMANN AND E. VASSEROT 2.3.1. The integrality 18 2.3.2. The value at 1 18 3. Nakajima's quiver varieties and their attracting subvarieties 20 3.1. Quiver varieties 20 3.2. Reminder on tori actions on varieties 22 3.2.1. The Bialynicki-Birula decomposition 22 3.2.2. Roots 23 3.3. The Bialynicki-Birula decompositions of M(v, w) 23 3.3.1. The cases of L 0 (v, w), L(v, w) and M 0 (v, w). 23 3.3.2. The cases of L 1 (v, w) and M 1 (v, w). 25 3.4. More on the Bialynicki-Birula decompositions of M(v, w) 26 3.4.1. The cases of L 0 (v, w), L(v, w). 26 3.4.2. The cases of L 1 (v, w). 29 3.4.3. Irreducible components of L 0 (v, w) and L 1 (v, w). 29 3.5. Counting polynomials of quiver varieties 30 4. Proof of the theorem 34 4.1. The stratification of L(v, w) 34 4.2. Proof of Theorem 1.4 35 5. Factorization of λ Q (q, z) and the strata in Λ v . 37 6. Appendix 38 6.1. The generalized quantum group 38 6.2. Character formulas 39 6.3. The generalized Kac-Moody algebra 42 References 44

Commuting matrices and volumes of linear stacks

“…The above statement in the case of quiver representations and Z = A 1 \ {0} or Z = {0} goes back to Hua [17] (see [26] for the plethystic formulation of these results). Other cases of the above result can be found in [27,28,33,29,5].…”

Classical limit of quantum Borcherds-Bozec algebras

“…By Proposition 4.1 (b), the character of V q (λ) is the same as the character of V (λ), which is given by (see, [7,3]) Theorem 4.6. The classical limit U 1 of U q (g) is isomorphic to the universal enveloping algebra U (g) as Q-algebras.…”

“…The quantum Borcherds-Bozec algebras were introduced by T. Bozec in his research of perverse sheaves theory for quivers with loops [1,2,3]. They can be treated as a further generalization of quantum generalized Kac-Moody algebras.…”

Translation quiver varieties

“…It was proved in [41] that the same formula is satisfied by the motivic classes of M(v, w). The virtual Poincaré polynomials of L(v, w) were computed in [5] by applying Bia lynicki-Birula decomposition to a torus action on M(v, w) (cf. Corollary 6.3).…”

“…Let us write down those formulas for completeness (cf. [25,5]). Define r(w, q −1 , z) = τ q −w•τ 1 k≥1 q χ(τ k ,τ k ) z τ k (q) τ k −τ k+1 where (1) τ = (τ i ) i∈Q 0 is a collection of partitions, (2) τ k = (τ i k ) i∈Q 0 ∈ N Q 0 for k ≥ 1, (3) z v = i∈Q 0 z v i i for v ∈ N Q 0 , (4) (q) v = i∈Q 0 (q) v i , (q) n = (q; q) n = n k=1 (1 − q k ) for v ∈ N Q 0 and n ∈ N, (5) χ is the Euler-Ringel form of the quiver Q. Theorem 6.11.…”