On a Tensor Category for the Exceptional Lie Groups

Abstract

The construction of representations of Lie groups is intertwined with combinatorics. For instance the combinatorial Littlewood-Richardson rule tells us how the tensor product of two irreducible GL _n -representations W ₁ and W ₂ decomposes into irreducibles, provided n is big with respect to the size of the Young diagrams corresponding to W ₁ and W ₂. Moreover, each irreducible GL _n -representation occurs in a tensor power V ^⊗dof the natural representation V and can be isolated by use of combinatorics of the symmetric group S _d acting on V ^⊗d by permuting the d factors. The intriguing part of the latter two combinatorial involvements is that they hardly depend on the series parameter n. For instance, once the primitive idempotents of the enveloping algebra of S _d inside End(V ^⊗d) are figured out, the irreducible components of V ^⊗d for GL _n follow (again, at least for n sufficiently large).