The saturation conjecture (after A. Knutson and T. Tao). With an appendix by William Fulton
Finite-dimensional irreducible polynomial representations of \(\text{GL}_n(\mathbb{C})\) are determined by partitions of \(n\). Let \(\lambda\) be a partition of \(n\), and let \(V_\lambda\) denote an irreducible representation of \(\text{GL}_n(\mathbb{C})\). Let \(\lambda\), \(\mu\), \(\nu\) be partitions of \(n\). The multiplicity of \(V_\nu\) in the decomposition of \(V_\lambda\otimes V_\mu\) into irreducibles is denoted by \(c^\nu_{\lambda\mu}\) and is called the Littlewood-Richardson coefficient. We set \(T_n=\{(\lambda,\mu,\nu)\mid c^\nu_{\lambda\mu}\neq 0\}\). The saturation conjecture states that if \((\lambda,\mu,\nu)\in\mathbb{Z}^{3n}\) and \(N>0\), then \((\lambda,\mu,\nu)\in T_n\) if and only if \((N\lambda,N\mu,N\nu)\in T_n\). \textit{A. Knutson} and \textit{T. Tao} [in J. Am. Math. Soc. 12, No. 4, 1055-1090 (1999; Zbl 0944.05097)] gave a proof of this conjecture. In the proof a new construction of polytopes called the hive and honeycomb models was used. In the paper under review the goal of the author is to present a simple proof of the conjecture using only the hive model. There is an appendix by W. Fulton showing that the hive model is equivalent to the original Littlewood-Richardson rule.
finite-dimensional irreducible polynomial representations
scientific article; zbMATH DE number 1558754
scientific article