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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.
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 o
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
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
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 ge