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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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