Let $\mathbf V$ be a representation of the symmetric group $S_n$, say irreducible. Pieri's formula describes the induced representation $$ \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}} \mathbf V \boxtimes \mathbf 1 $$ in terms of Young diagrams: it decomposes as a direct sum of representations obtained by adding $k$ boxes to the Young diagram of $\mathbf V$, no two in the same column. On the other hand one can also write $$ \mathrm{Ind}_{S_n \times S_k}^{S_{n+k}}\mathbf V \boxtimes \mathbf 1 \cong \bigoplus_{\binom{n+k}n} \mathbf V.$$ To make this canonical one needs to choose, for each $n$-element subset $S$ of $[n+k]$, a permutation $\pi \in S_{n+k}$ carrying $\{1,\ldots,n\}$ to $S$. We could say that we choose the lexicographically first such permutation, for instance, or anything else that's more convenient.

Can one make Pieri's formula explicit in this latter formula, by directly writing down a decomposition of the right hand side into irreducible representations?

  • $\begingroup$ It seems to be asking a lot to have an explicit decomposition written down in this generality. Aside from that, it might help (or not) to provide a little more in the way of background and references for "Pieri's formula". This usually seems to come up in connection with the combinatorics of Schubert varieties or the like, for instance in Howard Hiller's old papers, though I don't know the exact origins. $\endgroup$ – Jim Humphreys Jan 26 '12 at 15:50

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