ScalingStacks

We first recall some basic definitions and facts about quiver flag varietes. We refer to [SW14] and [Prz19] for more details. Consider a quiver $Q$ with finite sets of vertices $Q_{0}$ and arrows $Q_{1}$ and source and target maps

Denote by $\Gamma=\mathbb{Z}_{\geq 0}Q_{0}$ and $\Gamma^{+}=\Gamma\backslash\{0\}$ the sets of (non-trivial) dimension vectors. The dimension vector of a $Q_{0}$-graded $k$-vector space $V=(V_{i})_{i\in Q_{0}}$ is $\dim(V)=(\dim(V_{i}))_{i\in Q_{0}}\in\Gamma.$ We denote by

the vector space of quiver representations with underlying $Q_{0}$-graded vector space $V.$ We fix a dimension vector $\mathbf{d}\in\Gamma$ and let $V$ be the standard vector space with $\dim V=\mathbf{d}.$ We often abbreviate $\operatorname{Rep}(\mathbf{d})=\operatorname{Rep}(V).$ There is a a natural conjugation action on $\operatorname{Rep}(\mathbf{d})$ by the group

A composition of a dimension vector $\mathbf{d}$ is a tuple $\underline{\mathbf{d}}=(\underline{\mathbf{d}}^{j})\in(\Gamma^{+})^{\ell_{\underline{\mathbf{d}}}}$ which sums to $\mathbf{d}.$ A composition $\underline{\mathbf{d}}$ is called complete if each $\underline{\mathbf{d}}^{j}$ is a unit vector. We write $\operatorname{Comp}(\mathbf{d})$ and $\operatorname{Compf}(\mathbf{d})$ for the set of (complete) compositions of $\mathbf{d}.$ A partial flag $\underline{V}$ of $V$ of type $\underline{\mathbf{d}}$ is a sequence of $Q_{0}$-graded $k$-vector spaces

such that $\dim V^{j}/V^{j-1}=\underline{\mathbf{d}}^{i}.$ We denote the smooth projective variety of such flags by $\operatorname{Fl}(V,\underline{\mathbf{d}})=\operatorname{Fl}(\underline{\mathbf{d}}).$ The partial flag $\underline{V}$ is called strictly $\rho$-stable for a quiver representation $\rho\in\operatorname{Rep}(\mathbf{d})$ if

We can hence consider the variety

The variety $\mathfrak{Q}(\underline{\mathbf{d}})$ is smooth and has a diagonal action by $\operatorname{GL}(\mathbf{d}).$ Projection yields a $\operatorname{GL}(\mathbf{d})$-equivariant proper map

For a representation $M=(V,\rho)$, the fiber $\mu_{\underline{\mathbf{d}}}^{-1}(\rho)$ is called a (partial) quiver flag variety and denoted by

We want to apply the general setup of Springer motives from Section 4.1 to the collection of maps $\mu_{\underline{\mathbf{d}}}:\mathfrak{Q}(\underline{\mathbf{d}})\to\operatorname{Rep}(\mathbf{d}).$ In particular, we are interested in settings where the condition (PT) and (FO) from Section 4.1 are fulfilled. We restrict our attention to the following two cases.

1. ($A$)

   the quiver $Q$ is a Dynkin quiver where the underlying graph is a Dynkin diagram of type $A.$

2. ($\widetilde{A}$)

   the quiver $Q$ is the cyclic quiver with cyclic orientation, so the underlying graph is a Dynkin diagram of type $\widetilde{A}$.

Let $Q$ be the cyclic quiver on $n$ vertices. We label vertices and arrows $Q_{0},Q_{1}=\mathbb{Z}/n$ such that $s(i)=i$ and $t(i)=i+1.$

Let $M=(V,\rho)$ be a representation of $M$ with dimension vector $\mathbf{d}=\dim V\in\Gamma$ and let $\underline{\mathbf{d}}\in\operatorname{Comp}(\mathbf{d})$ be a composition. he goal of this section is to prove the following theorem.

For $\underline{\mathbf{d}}\in\operatorname{Compf}(\mathbf{d})$ this was shown by Sauter [Sau16]. We generalize her approach to work for arbitrary $\underline{\mathbf{d}}\in\operatorname{Comp}(\mathbf{d}).$

By [Sch12, Proposition 3.24] every nilpotent representations $M=(V,\rho)$ of $Q$ is isomorphic to a direct sums of representations $E(i,l)$ defined as follows. For $i\in\mathbb{Z}/n$ and $l\in\mathbb{Z}_{\geq}{0}$ we denote by $E(i,l)=(V,\rho)$ the representation with basis $e_{i-(l-1)},\dots,e_{i-1},e_{i},$ such that $e_{j}\in V_{j}$ lives on the vertex $j\in\mathbb{Z}/n$ and $\rho_{j}(e_{j})=e_{j+1}$ and $\rho_{i}(e_{i})=0.$ The representation $E(i,l)$ is indecomposable and nilpotent with socle $\operatorname{soc}(E(i,l))=E(i,1)$ and radical filtration $\operatorname{rad}^{n}(E(i,l))=E(i,l-n).$

We discuss how to lift automorphims of the socle $\operatorname{soc}(M)$ to $M.$ Since $\rho$ restricted to $\operatorname{soc}(M)$ vanishes, we simply treat $\operatorname{soc}(M)$ as a $Q_{0}$-graded vector space. Choose $m>0$ such that $\operatorname{rad}^{m}(M)=0.$ Consider the flag obtained by intersecting the radical filtration of $M$ with the socle

Refine $I^{\prime}$ to a complete flag $I$ of $\operatorname{soc}(M)$ and denote by $B\subset P\subset\operatorname{GL}(\operatorname{soc}(M))$ the stabilizers of $I$ and $I^{\prime},$ respectively. Denote the automorphism group of $M$ by $\operatorname{Aut}(M)\subset\operatorname{GL}(V).$ Restriction yields a natural morphism $\operatorname{Res}:\operatorname{Aut}(M)\to P.$

Since $M=(V,\rho)$ is nilpotent, we have $\operatorname{soc}(M)=\ker(\rho).$ Hence, for every flag $\underline{V}=(0\subset V^{1}\subset\dots)$ that is strictly $\rho$-stable we have $V_{1}\subset\operatorname{soc}(M),$ see (6.1). We hence get a natural map

from the partial quiver flag variety to the Grassmannian of subspaces of $\operatorname{soc}(M)$ with dimension vector $\underline{\mathbf{d}}^{1}=\dim V^{1}.$ We construct an affine paving of $\operatorname{Fl}(M,\underline{\mathbf{d}})$ using $p$ inductively. We will show that the preimage under $p$ of a $B$-orbit in the Grassmannian has an affine paving and then deduce the claim.

Assume that we are in the cases ($A$) or ($\widetilde{A}$) of Section 6.2. For a fixed dimension vector $\mathbf{d}\in\Gamma$ we consider the collection of $\operatorname{GL}(\mathbf{d})$-equivariant maps

where $\underline{\mathbf{d}}\in\operatorname{Compf}(\mathbf{d})$ ranges over all complete compositions. Denote by $R_{\mathbf{d}}$ the quiver Hecke (KLR) algebra associated to $Q$ and $\mathbf{d}$ as defined by Khovanov–Lauda [KL09] and Rouquier [Rou08].

If we let $\underline{\mathbf{d}}\in\operatorname{Comp}(\mathbf{d})$ range over all compositions, the motivic extension algebra $E$ can be identified with the quiver Schur algebra $A_{\mathbf{d}}$ defined by Stroppel–Webster [SW14] and we get: