Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0909.2935

Rigidity of tilting modules

Henning Haahr Andersen·Masaharu Kaneda
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $U_q$ denote the quantum group associated with a finite dimensional semisimple Lie algebra. Assume that $q$ is a complex root of unity of odd order and that $U_q$ is %the quantum group version obtained via Lusztig's $q$-divided powers construction. We prove that all regular projective (tilting) modules for $U_q$ are rigid, i.e., have identical radical and socle filtrations. Moreover, we obtain the same for a large class of Weyl modules for $U_q$. On the other hand, we give examples of non-rigid indecomposable tilting modules as well as non-rigid Weyl modules. These examples are for type $B_2$ and in this case as well as for type $A_2$ we calculate explicitly the Loewy structure for all regular Weyl modules. We also demonstrate that these results carry over to the modular case when the highest weights in question are in the so-called Jantzen region. At the same time we show by examples that as soon as we leave this region non-rigid tilting modules do occur.

Explore related subjects

math.RTmath.QA

Keep this discovery

BibTeXRIS

Henning Haahr Andersen, Masaharu Kaneda. 2009-09-16. Rigidity of tilting modules. https://arxiv.org/abs/0909.2935

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↗