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A generalization of the Littlewood-Richardson rule. (English) Zbl 0704.20033

Let G be a reductive algebraic group over \({\mathbb{C}}\). Let V, W be two finite dimensional, irreducible G-modules. In view of complete reducibility of G, \(V\otimes W\) breaks up as \(\oplus_{\lambda}V_{\lambda}\), a direct sum of irreducible G-modules. For \(G=GL_ n\), the classical Littlewood-Richardson rule gives a combinatorial rule to determine the \(\lambda\) ’s appearing in the decomposition \(V\otimes W=\oplus V_{\lambda}\). The author generalizes this result to simple algebraic groups of type \(A_ n\), \(B_ n\), \(C_ n\), \(D_ n\), \(E_ 6\) and \(G_ 2\). Using the Standard Monomial Theory for semi-simple algebraic groups as developed by Lakshmibai, Musili, and Seshadri, the author describes in an explicit way the decomposition \(V\otimes W=\oplus V_{\lambda}\). He also describes explicitly the decomposition \(V|_ L=\oplus_{\mu}V_{\mu}\) (as L-modules), where L is a Levi subgroup of G, \(V|_ L\) is V considered as an L-module and \(V_{\mu}'s\) are irreducible L-modules. This paper makes an important contribution to the Representation Theory of semi-simple Algebraic Groups.
Reviewer: V.Lakshmibai

MSC:

20G05 Representation theory for linear algebraic groups
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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