Schur duality for Cartan type Lie algebra $W_n$

Abstract: We decompose tensor products of the defining representation of a Cartan type Lie algebra $W(n)$ in the case where the number of tensoring does not exceed the rank of the Lie algebra. As a result, we get a kind of Schur duality between $W(n)$ and a finite dimensional non-semisimple algebra, which is the semi-group ring of the transformation semigroup ${\frak T}_m$.

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