zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI arXiv
[1] Belkale, P., Local systems on \(\mathbb{P}^1 - S\) for S a finite set, Compos. math., 129, 1, 67-86, (2001) · Zbl 1042.14031
[2] Belkale, P., Quantum generalization of the Horn conjecture, preprint · Zbl 1134.14029
[3] P. Belkale, Extremal unitary representations of \(\pi_1(\mathbb{P}^1 - \{p_1, \ldots, p_s \})\), in: Proceedings of the International Colloquium on Algebraic Groups and Homogeneous Spaces, 2004, TIFR, Mumbai, 23 pages
[4] Belkale, P., Invariant theory of \(\operatorname{GL}(n)\) and intersection theory of Grassmannians, Int. math. res. not., 2004, 69, 3709-3721, (2004) · Zbl 1082.14050
[5] Belkale, P., Geometric proofs of Horn and saturation conjectures, J. algebraic geom., 15, 133-173, (2006) · Zbl 1090.14014
[6] H. Derksen, J. Weyman, On the sigma-stable decomposition of quiver representations, preprint · Zbl 1016.16007
[7] Derksen, H.; Weyman, J., The combinatorics of quiver representations · Zbl 1271.16016
[8] Fulton, W., Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. amer. math. soc. (NS), 37, 3, 209-249, (2000) · Zbl 0994.15021
[9] Klyachko, A., Stable bundles, representation theory and Hermitian operators, Selecta math., 4, 419-445, (1998) · Zbl 0915.14010
[10] Knutson, A.; Tao, T., The honeycomb model of \(\operatorname{GL}_n(C)\) tensor products. I. proof of the saturation conjecture, J. amer. math. soc., 12, 4, 1055-1090, (1999) · Zbl 0944.05097
[11] Knutson, A.; Tao, T.; Woodward, C., The honeycomb model of \(\operatorname{GL}_n(C)\) tensor products. II. puzzles determine facets of the Littlewood-Richardson cone, J. amer. math. soc., 17, 19-48, (2004) · Zbl 1043.05111
[12] Narasimhan, M.S.; Ramadas, T.R., Factorisation of generalised theta functions I, Invent. math., 114, 565-624, (1993) · Zbl 0815.14014
[13] Pauly, C., Espaces de modules de fibrĂ©s paraboliques et blocs conformes, Duke math. J., 84, 1, 217-235, (1996) · Zbl 0877.14031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.