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Register a new userRepresentation stability of the cohomology of Springer varieties and some combinatorial consequences
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A sequence of $S_n$-representations ${V_n}$ is said to be uniformly representation stable if the decomposition of $V_n = bigoplus_{mu} c_{mu,n} V(mu)_n$ into irreducible representations is independent of $n$ for each $mu$---that is, the multiplicities $c_{mu,n}$ are eventually independent of $n$ for each $mu$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.
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In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.
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 give a proof of a conjecture of Lehrer and Shoji regarding the occurrences of the exterior powers of the reflection representation in the cohomology of Springer fibers. The actual theorem proved is a slight extension of the original conjecture to all nilpotent orbits and also takes into account the action of the component group. The method is to use Shojis approach to the orthogonality formulas for Green functions to relate the symmetric algebra to a sum over Green functions. In the second part of the paper we give an explanation of the appearance of the Orlik-Solomon exponents using a result from rational Cherednik algebras.
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.
Let $mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $mathfrak{g}$ over $mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $overline{ mathcal{O}}$ in $S mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broers result for the subregular nilpotent orbit. Along the way we give another proof of Broers result that $overline{ mathcal{O}}$ is normal. We also prove a result connecting a property of invariants related to flat bases to the question of when one copy of the adjoint representation is in the ideal in $S mathfrak{g}^*$ generated by another copy of the adjoint representation and the fundamental invariants.
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