I am reading the file. I have a question on pae 28. How to prove that $[X_{\{2,4\}}] = S_{(1)} = x_1 + x_2 + \cdots$ and $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$? I tried to verify $S_{(1)}^4 = 2 S_{(2,2)} + S_{(3,1)} + S_{(2,1,1)}$ directly. But on the left hand side we have a term $x_1^4$ and it seems that on the right hand side we don't have $x_1^4$. Thank you very much.
2026-03-25 06:00:18.1774418418
Two questions about Schubert calculus and Schur functions.
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That formula is not correct. The correct decomposition is $$S_{(1)}^4 = S_{(4)} + 3 S_{(3,1)} + 2 S_{(2,2)} + 3 S_{(2,1,1)} + S_{(1,1,1,1)}.$$ This can be calculated by the formula $$S_\lambda \cdot S_{(1)} = \sum_{\mu} S_{\mu}$$ where $\mu$ ranges over all partitions obtained from $\lambda$ by adding a single box. (Pieri's formula)