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Geometric proof of a conjecture of Fulton. (English) Zbl 1129.14063
The irreducible polynomial representations of $$\text{GL}(r)$$ are indexed by the partitions $$\lambda=(\lambda_1\geq \ldots\geq\lambda_r\geq 0)$$ with at most $$r$$ nonzero parts. Let $$V_\lambda$$ denote the representation corresponding to $$\lambda$$. The Littlewood-Richardson coefficients $$\{c_{\lambda,\mu}^\nu\}$$ are defined via the tensor product decompositions: $$V_\lambda\otimes V_\mu=\sum c_{\lambda,\mu}^\nu V_\nu$$. The Fulton conjecture asserts that
$c_{\lambda,\mu}^\nu =1 \Longleftrightarrow c_{N\lambda,N\mu}^{N\nu} \text{ \quad for\;any \quad } N\in\{1,2,\dots\}.$ This conjecture was proved by A. Knutson, T. Tao and C. Woodward [J. Am. Math. Soc. 17, 19–48 (2004; Zbl 1043.05111)] in a purely combinatorial way. This article provides a proof of Fulton’s conjecture based on a geometric technique. Such a technique was already used in the author’s proof of the Horn and saturation conjectures [J. Algebr. Geom. 15, 133–173 (2006; Zbl 1090.14014)]. The key step of the proof is to use a projective moduli space $$\mathcal M$$ of semistable parabolic vector spaces. The space $$\mathcal M$$ is naturally equipped with an ample line bundle $$\mathcal L$$ such that $$c_{\lambda,\mu}^\nu=\dim H^0(\mathcal M,\mathcal L)$$. Fulton’s conjecture is then equivalent to the rigidity statement that if $$c_{\lambda,\mu}^\nu=1$$, then $$\mathcal M$$ is a point.

##### MSC:
 14L24 Geometric invariant theory 14M15 Grassmannians, Schubert varieties, flag manifolds 22E46 Semisimple Lie groups and their representations 14L35 Classical groups (algebro-geometric aspects) 05E15 Combinatorial aspects of groups and algebras (MSC2010)
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