Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.06890

BGG Resolutions, Koszulity, and Stratifications, Part I: the nilBrauer Algebra

Fan Zhou
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we homologically construct a (functorial) BGG resolution of the finite-dimensional simple module of the nilBrauer algebra by using infinity-categorical methods following the reconstruction-from-stratification philosophy, e.g. appearing in Ayala-Mazel-Gee-Rozenblyum. To do so, we prove a fact of independent interest, that half of the nilBrauer algebra is Koszul. This BGG resolution categorifies a character formula of Brundan-Wang-Webster. More generally, we have a (functorial) ``BGG spectral sequence'' which converges to any desired module; this spectral sequence is secretly a resolution when the desired module is finite-dimensional. This spectral sequence also categorifies the character formulae of Brundan-Wang-Webster for any (possibly infinite-dimensional) simple module. We expect the methods used here for producing BGG resolutions to be applicable to other (graded) triangular-based algebras also, especially diagrammatic ones.

Explore related subjects

math.RTmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Fan Zhou. 2024-02-10. BGG Resolutions, Koszulity, and Stratifications, Part I: the nilBrauer Algebra. https://arxiv.org/abs/2402.06890

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Whittaker supermodules over Lie superalgebras

In this paper, we develop a general theory of quasi-Whittaker supermodules over Lie superalgebras induced from an arbitrary ideal. We determine the quasi-Whittaker vectors in universal supermodules, establish an irreducibility criterion, and classify several families of irreducible supermodules. The odd part produces a new irreducibility phenomenon absent from the Lie algebra setting. As applications, we determine all irreducible quasi-Whittaker supermodules over the $N=1$ super Schr\"odinger algebra and the $N=1$ $\frac{3}{2}$-conformal Galilei superalgebra, and over the complete spectrum-generating superalgebra in a special case.

math.RT↗
arXiv2026

Rankin--Selberg integrals of opposite conductor--one newforms

Let $F$ be a nonarchimedean local field of characteristic zero and let $n\geq2$. For $r=n,n+1$, let $\Pi_r$ be an irreducible tempered representation of ${\rm GL}_r(F)$ of conductor one and with trivial central character. We evaluate the Rankin--Selberg integral of opposite newforms in $\Pi_{n+1}\times \Pi_n$ explicitly and show that its central value is nonzero. As an application, this implies a case of Disegni--Zhang's conjecture on the nonvanishing of local relative characters.

math.RT↗
arXiv2026

Obstructions to Jacobi-Finiteness of Quivers with Potentials

We show that Jacobi-finite potentials need not exist on finite $2$-acyclic quivers. Our main tool is a matrix-valued Golod--Shafarevich--Vinberg inequality for quotients of completed path algebras by finitely many, possibly nonhomogeneous, topological relations. Applied to cyclic derivatives, it yields a potential-dependent obstruction to the finite-dimensionality of completed Jacobian algebras. We then construct a purely quiver-level criterion excluding every Jacobi-finite potential on a given quiver, and exhibit a family of quivers for which every potential has an infinite-dimensional Jacobian algebra.

math.RT↗