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How to multiply dots with Young idempotents in the degenerate affine Hecke algebra (type A)

Let $\widehat{\cal H}_n$ be the type A degenerate affine Hecke algebra on $n$ strands, and let $x_1,\cdots,x_n$ be the dots. Inside of this algebra lies the algebra $\mathbb C S_n$, and the Young ...