Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpringer fibers via quiver varieties using Maffei-Nakajima isomorphism
137
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
It is a remarkable theorem by Maffei--Nakajima that the Slodowy variety, which consists of certain complete flags, can be realized as certain Nakajima quiver variety of type A. However, the isomorphism is rather implicit as it takes to solve a system of equations in which variables are linear maps. In this paper, we construct solutions to this system and thus establish an explicit and efficient way to realize these quiver varieties in terms of complete flags in the corresponding Slodowy varieties. As Slodowy varieties contain Springer fibers naturally, we further provide an explicit description of irreducible components of two-row Springer fibers in terms of a family of kernel relations via quiver representations, which allows us to formulate a characterization of irreducible components of Springer fibers of classical type.
rate research
Read More
We give an explicit description of the irreducible components of two-row Springer fibers in type A as closed subvarieties in certain Nakajima quiver varieties in terms of quiver representations. By taking invariants under a variety automorphism, we obtain an explicit algebraic description of the irreducible components of two-row Springer fibers of classical type. As a consequence, we discover relations on isotropic flags that describe the irreducible components.
We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982.
We study the exotic t-structure on the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m+n,n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic t-structure, and enumerate them by crossingless (m,m+2n) matchings. We compute the Exts between the irreducible objects, and show that the resulting algebras are an annular variant of Khovanovs arc algebras. In subsequent work we will make a link with annular Khovanov homology, and use these results to give a positive characteristic analogue of some categorification results using two-block parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).
An affine Lie algebra acts on cohomology groups of quiver varieties of affine type. A Heisenberg algebra acts on cohomology groups of Hilbert schemes of points on a minimal resolution of a Kleinian singularity. We show that in the case of type $A$ the former is obtained by Frenkel-Kac construction from the latter.
We present a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a Riemann surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes at k punctures. Using the character table of GL_n(F_q) we calculate the E-polynomial of these character varieties and confirm that it is as predicted by our main conjecture. Then, using the character table of gl_n(F_q), we calculate the E-polynomial of certain associated comet-shaped quiver varieties, the additive analogues of our character variety, and find that it is the pure part of our conjectured mixed Hodge polynomial. Finally, we observe that the pure part of our conjectured mixed Hodge polynomial also equals certain multiplicities in the tensor product of irreducible representations of GL_n(F_q). This implies a curious connection between the representation theory of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically wild, quivers.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
