Stratifying quiver Schur algebras via ersatz parity sheaves
Ruslan Maksimau, Alexandre Minets
TL;DR
This work extends parity sheaf theory to ersatz parity sheaves to handle non-locally-constant strata and constructs polyhereditary stratifications for Ext-algebras of quiver Schur algebras in affine type, including the Kronecker case. The authors develop gluing of polyhereditary theories, prove ersatz-completeness implies polyhedity, and apply to seminilpotent quiver Schur algebras, establishing polynomial quasihereditary (polyhereditary) structures compatible with categorified PBW bases. They compute semicuspidal quotients C(nδ) diagrammatically and connect them to extended curve Schur algebras, giving Morita-equivalences and characteristic-independent results; they also provide an erratum to MakMin_KLR2023 with integral cohomology corrections. The results advance modular representation theory by realizing RoCK-like blocks geometrically and offering explicit diagrammatic and homological descriptions.
Abstract
We propose an extension of the theory of parity sheaves, which allows for non-locally constant sheaves along strata. Our definition is tailored for proving the existence of (proper, quasihereditary, etc) stratifications of $\mathrm{Ext}$-algebras. We use this to study quiver Schur algebras $A(α)$ for the cyclic quiver of length $2$. We find a polynomial quasihereditary structure on $A(α)$ compatible with the categorified PBW basis of McNamara and Kleshchev-Muth, and sharpen their results to arbitrary characteristic. We also prove that semicuspidal algebras of $A(nδ)$ are polynomial quasihereditary covers of semicuspidal algebras of the corresponding KLR algebra $R(nδ)$, and compute them diagrammatically.