Subsection 6.4. (0N0C) · ScalingStacks

6.4. Quiver Hecke and quiver Schur algebras

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

\mu_{\underline{\mathbf{d}}}:\mathfrak{Q}(\underline{\mathbf{d}})\to\operatorname{Rep}(\mathbf{d})

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].

0MZE

Theorem 6.5. There is an equivalence of categories

\operatorname{DM}^{Spr}_{\operatorname{GL}(\mathbf{d})}(\operatorname{Rep}(\mathbf{d}),\mathbb{Q})\cong\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(R_{\mathbf{d}})

between Springer motives with respect to $\mu_{\underline{\mathbf{d}}}:\mathfrak{Q}(\underline{\mathbf{d}})\to\operatorname{Rep}(\mathbf{d})$ for complete compositions $\underline{\mathbf{d}}\in\operatorname{Compf}(\mathbf{d})$ and the perfect derived category of graded modules of the quiver Hecke algebra.

0MZF

Proof. The motivic extension algebra $E$ can be identified with $R_{\mathbf{d}}.$ To see this, one can adapt the proof of Varagnolo–Vasserot [VV11, Theorem 3.6] from the context of equivariant Borel–Moore homology to Chow groups. Their arguments apply unchanged making use of the fact that the partial quiver flag varieties admit affine pavings and hence their equivariant Borel–Moore homology and Chow groups coindice. By Proposition 6.1 conditions (PT) and (PO) hold and the statement follows by Theorem 4.9. ∎

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:

0MZG

Theorem 6.6. There is an equivalence of categories

\operatorname{DM}^{Spr}_{\operatorname{GL}(\mathbf{d})}(\operatorname{Rep}(\mathbf{d}),\mathbb{Q})\cong\operatorname{D}^{\mathbb{Z}}_{\operatorname{perf}}(A_{\mathbf{d}})

between Springer motives with respect to $\mu_{\underline{\mathbf{d}}}:\mathfrak{Q}(\underline{\mathbf{d}})\to\operatorname{Rep}(\mathbf{d})$ for compositions $\underline{\mathbf{d}}\in\operatorname{Comp}(\mathbf{d})$ and the perfect derived category of graded modules of the quiver Schur algebra.

Jens Niklas Eberhardt, Catharina Stroppel

Original source: arXiv:2109.00305v2