A generalization of Fulton's conjecture for arbitrary groups
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 \(\sigma_\lambda\cdot \sigma_\mu\cdot\sigma_\nu=\sigma_{\lambda_0}\) then \(\dim([V(n\lambda)\otimes V(n\mu)\otimes V(n\nu)]^{\mathrm{SL}(r)})=1\) for all \(n\) (here \(\cdot\) is the intersection produit). The authors show an example where this statement fails for generic \(G/P\) and \(L\). However, they prove that results became true if the intersection product is replaced by the deformed product of \(G/P\) introduced by \textit{P. Belkale} and \textit{S. Kumar}, [Invent. Math. 166, No. 1, 185--228 (2006; Zbl 1106.14037)] (where \(P\) is a parabolic subgroup of the reductive group \(G\)). In the general statement \(\mathrm{SL}(r)\) is remplaced by the semisimple part of a Levi subgroup of \(P\). Finally, the authors show that the converse of this results is not true in general.
scientific article; zbMATH DE number 6087539
scientific article