Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.00421

Kleshchev multipartitions, affine Mirkovi\'c-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$

Samantha Allen·Jack Isaac·Corinne Moscariello·Robert Muth·Bella Deborah Uwase·Lucas Walton
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct explicit isomorphisms between three models for the $B(\infty)$ crystal in type ${\tt A}_1^{(1)}$: affine Mirkovi\'c--Vilonen polytopes, Kleshchev multipartitions, and a new model we call upper ledge diagrams. We also present some clarifying results on these crystals, giving a direct method for completing an affine MV polytope from the data of one of its boundary root partitions, and a non-iterative recognition theorem which characterizes Kleshchev multipartitions in type ${\tt A}_1^{(1)}$. We apply these results to the representation theory of KLR algebras, where they yield a combinatorial dictionary between cuspidal- and cellular-theoretic frameworks, along with some augmented branching rules for real root functors of induction and restriction.

Explore related subjects

math.RTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Samantha Allen, Jack Isaac, Corinne Moscariello, Robert Muth, Bella Deborah Uwase, Lucas Walton. 2026-05-29. Kleshchev multipartitions, affine Mirkovi\'c-Vilonen polytopes, and representations of KLR algebras in type ${\tt A}^{(1)}_1$. https://arxiv.org/abs/2606.00421

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↗