Irreducible components of two-row Springer fibers for all classical types

We give an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. Given a cup diagram, we explicitly write down all flags contained in the component associated to the cup diagram. This generalizes results by Stroppel--Webster and Fung to all classical types.

Springer fibers via quiver varieties using Maffei-Nakajima isomorphism

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.

Irreducible components of two-row Springer fibers and Nakajima quiver varieties

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.

On calibrated representations of the degenerate affine periplectic Brauer algebra

We initiate the representation theory of the degenerate affine periplectic Brauer algebra on $n$ strands by constructing its finite-dimensional calibrated representations when $n=2$. We show that any such representation that is indecomposable and does not factor through a representation of the degenerate affine Hecke algebra occurs as an extension of two semisimple representations with one-dimensional composition factors; and furthermore, we classify such representations with regular eigenvalues up to isomorphism.

On the module structure of the center of hyperelliptic Krichever-Novikov algebras II

Let $R := R_{2}(p)=mathbb{C}[t^{pm 1}, u : u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $mathfrak{g}otimes R$ be the corresponding current Lie algebra. color{black} Here $mathfrak g$ is a finite dimensional simple Lie algebra defined over $mathbb C$ and begin{equation*} p(t)= t(t-alpha_1)cdots (t-alpha_{2n})=sum_{k=1}^{2n+1}a_kt^k. end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $mathfrak{g}otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $Omega_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(alpha_1,dots,alpha_{2n})$, which are the automorphism groups $mathbb G_n=text{Dic}_{n}$, $mathbb U_ncong D_n$ ($n$ odd), or $mathbb U_n$ ($n$ even) of the hyperelliptic curves begin{equation} S=mathbb{C}[t, u: u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $mathbb U_n=D_n$, where $n$ is odd.