A generalization of Fulton's conjecture for arbitrary groups
type: Item
instance of: scholarly article
title: A generalization of Fulton's conjecture for arbitrary groups
zbMATH Open document ID: 1258.14008
author: Prakash Belkale
author: Shrawan Kumar
author: Nicolas Ressayre
published in: Mathematische Annalen
publication date: 2012-09-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.4379
review text: Let \(L\) be a connected reductive complex algebraic group. Its irreducible representation are parametrized, up to isomorphism, by their highest weights. The Littlewood-Richardson coefficients \(c_{\lambda,\mu}^\nu\) is defined as the multiplicity of \(V(\nu)\) in \(V(\lambda)\otimes V(\mu)\). For \(L=\mathrm{GL}(r)\), Fulton conjectured that, if \( c_{\lambda,\mu}^\nu=1\) then \(c_{n\lambda,n\mu}^{n\nu}=1\) for all integers \(n\). This result was proved by \textit{A. Knutson, T. Tao} and \textit{C. Woodward} [J. Am. Math. Soc. 17, No. 1, 19--48 (2004; Zbl 1043.05111)]; the converse follows by the saturation theorem of Knutson-Tao. The direct generalization of the above theorem to arbitrary reductive groups is false. It also known that the saturation theorem fails. The authors prove a geometric generalization of this theorem. First of all, the previous result can be restated in the following way: let \(\lambda,\mu,\nu\) be three dominant weight of \(\mathrm{GL}(r)\). If \(\dim([V(\lambda)\otimes V(\mu)\otimes V(\nu)]^{\mathrm{SL}(r)})=1\), then \(\dim([V(n\lambda)\otimes V(n\mu)\otimes V(n\nu)]^{\mathrm{SL}(r)})=1\) for all \(n\). Moreover one can reduce itself to the case of polynomial representations. One can consider \(\mathrm{SL}(r)\) as the semisimple part of a Levi subgroup of \(\mathrm{GL}(k+r)\). The Schubert varieties in the Grassmannian \(Gr(r,r+k)\) are parametrized by dominant weights of \(\mathrm{SL}(r)\), i.e. sequences \(\lambda=(\lambda_1\geq \lambda_2\geq\dots\geq \lambda_r\geq0)\), with the condition \(\lambda_1\leq k\). Let \(\sigma_\lambda\in H^*(\mathrm{Gr}(r,r+k),\mathbb{Z})\) be the class of the Schubert variety associated to \(\lambda\). The structure constants for the intersection product in \(H^*(\mathrm{Gr}(r,r+k),\mathbb{Z})\) are the Littlewood-Richardson coefficients. Let \(\lambda_0=(k\geq \dots \geq k)\). The above result can be restated as follows : if \(L=\mathrm{SL}(r)\) and \(\sigma_\lambda\cdot \sigma_\mu\cdot\sigma_\nu=\sigma_{\…
reviewed by: Alessandro Ruzzi
Mathematics Subject Classification ID: 14C17
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 6087539
zbMATH Keywords: representation theory
zbMATH Keywords: intersection theory
MaRDI profile type: Publication
OpenAlex ID: W2139493149
arXiv ID: 1004.4379
Wikidata QID: Q123022316
cites work: Q4658945
cites work: Geometric proof of a conjecture of Fulton
cites work: Eigenvalue problem and a new product in cohomology of flag varieties
cites work: Q3941610
cites work: Generic singularities of certain Schubert varieties
cites work: Higher-dimensional algebraic geometry
cites work: Q4342000
cites work: Q3129746
cites work: Schubert varieties and degeneracy loci
cites work: Q4143433
cites work: Q4440466
cites work: Q4775956
cites work: The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone
cites work: Kac-Moody groups, their flag varieties and representation theory
cites work: Q4442702
cites work: The red book of varieties and schemes
cites work: Geometric invariant theory and generalized eigenvalue problem. II
cites work: A short geometric proof of a conjecture of Fulton
DOI: 10.1007/S00208-011-0728-2
description: scientific article; zbMATH DE number 6087539