# tailieunhanh - Đề tài " Unique decomposition of tensor products of irreducible representations of simple algebraic groups "

## We show that a tensor product of irreducible, ﬁnite dimensional representations of a simple Lie algebra over a ﬁeld of characteristic zero determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers. 1. Introduction . Let g be a simple Lie algebra over C. The main aim of this paper is to prove the following unique factorisation of tensor products of irreducible, ﬁnite dimensional representations of g: | Annals of Mathematics Unique decomposition of tensor products of irreducible representations of simple algebraic groups By C. S. Rajan Annals of Mathematics 160 2004 683 704 Unique decomposition of tensor products of irreducible representations of simple algebraic groups By C. S. Rajan Abstract We show that a tensor product of irreducible finite dimensional representations of a simple Lie algebra over a field of characteristic zero determines the individual constituents uniquely. This is analogous to the uniqueness of prime factorisation of natural numbers. 1. Introduction . Let g be a simple Lie algebra over C. The main aim of this paper is to prove the following unique factorisation of tensor products of irreducible finite dimensional representations of g Theorem 1. Let g be a simple Lie algebra over C. Let Vi . Vn and Wi . Wm be nontrivial irreducible finite dimensional g-modules. Assume that there is an isomorphism of the tensor products Vi Vn Wi o o Wm as g-modules. Then m n and there is a permutation T of the set 1 . n such that Vi Wt i as g-modules. The particular case which motivated the above theorem is the following corollary Corollary 1. Let V W be irreducible g-modules. Assume that End V End W as g-modules. Then V is either isomorphic to W or to the dual g-module W . When g sl2 and the number of components is at most two the theorem follows by comparing the highest and lowest weights that occur in the tensor 684 C. S. RAJAN product. However this proof seems difficult to generalize see Subsection . The first main step towards a proof of the theorem is to recast the hypothesis as an equality of the corresponding products of characters of the individual representations occurring in the tensor product. A pleasant arithmetical proof for sl2 see Proposition 4 indicates that we are on a right route. The proof in the general case depends on the fact that the Dynkin diagram of a simple Lie algebra is connected and proceeds by induction on the rank of g by

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