Defining relations of fusion products and Schur positivity

In this note we give defining relations of an $\mathfrak{sl}_{n+1}[t]$-module defined by the fusion product of simple $\mathfrak{sl}_{n+1}$-modules whose highest weights are multiples of a given fundamental weight. From this result we obtain a surjective homomorphism between two fusion products, which can be considered as a current algebra analog of Schur positivity.

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