Relative interval tilting, higher Auslander staircase corners and rational Dyck posets

Abstract

We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras. Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn--Pham singularities.