Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.2324

Special isogenies and tensor product multiplicities

Shrawan Kumar·John R. Stembridge
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that any bijection between two root systems that preserves angles (but not necessarily lengths) gives rise to inequalities relating tensor product multiplicities for the corresponding complex semisimple Lie groups (or Lie algebras). We explain the inequalities in two ways: combinatorially, using Littelmann's Path Model, and geometrically, using isogenies between algebaric groups defined over an algebraically closed field of positive characteristic.

Explore related subjects

math.RTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Shrawan Kumar, John R. Stembridge. 2007-06-15. Special isogenies and tensor product multiplicities. https://arxiv.org/abs/0706.2324

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

KEEP EXPLORING

Related papers

arXiv2026

$q$-Oper Structures on a Formal Punctured Disc

Let $G$ be a connected reductive complex algebraic group and let $q\in\mathbb{C}^{\times}$ be not a root of unity. We prove that every $(G,q)$-connection on a formal punctured disc admits a $(G,q)$-oper structure.

math.RT↗
arXiv2026

Normalization graphs and Dold-Kan equivalences for generalized Reedy categories

We introduce a combinatorial criterion for the projectivity of standard modules over generalized Reedy categories, formulated in terms of a finite bipartite graph whose normalized weightings determine normalization idempotents and hence projective splittings of standard modules. We apply this framework to recover several classical normalization constructions and Dold--Kan--type equivalences, and to obtain new Dold--Kan--type equivalences for categories of finite-dimensional affine, semilinear, and semiaffine spaces over a finite field. These equivalences also yield explicit descriptions of uniformly continuous representations of the corresponding infinite transformation monoids. Together with our previous work on Hom-spaces between standard modules, the results provide a combinatorial machinery, based mainly on graph theory and linear algebra, for establishing Dold--Kan--type equivalences for generalized Reedy categories.

math.RT↗
arXiv2026

Higher Cotorsion Pairs on Image-Nilpotent \(η\)-Extensions

Let \(\cB\) be an abelian category with enough projective and injective objects, and let $ \cA=\cB\ltimes_η\sF $ be an \(η\)-extension induced by a right exact endofunctor \(\sF\). We introduce uniform image-nilpotence for \(η\)-extensions and characterize it in terms of the nilpotence of the associative natural transformation \(η:\sF^2\to\sF\). Using the associated \(η\)-Loewy filtration, we prove a lifting theorem for right \(n\)-cotorsion pairs over image-nilpotent \(η\)-extensions and establish a corresponding completeness theorem. As applications, our results recover and unify related constructions for split nilpotent ring extensions, comma categories, formal triangular matrix rings, Morita context rings and classical trivial extensions.

math.RT↗