Young symmetrizer ($h_t=c_t{r}_t$)

For a Young tableau $t$, let $r_t$ sum its row permutations and $c_t$ be the signed sum of its column permutations in the group algebra of the symmetric group. The product $h_t=c_t{r}_t$ is a Young symmetrizer. The other order is another conventional realization. It satisfies $h_t^2=H_{\lambda }{h}_t$, where $H_{\lambda }$ is the hook product of a partition, so division by that scalar gives a primitive idempotent in characteristic zero. Acting on a tensor power constructs a Schur module.

If the rows of a Young tableau of shape $\lambda$ have no repeated intersection with the columns of one of shape $\mu$, then ${\sum }_{i\le k}{\lambda }_i\le {\sum }_{i\le k}{\mu }_i$ for every $k$. If also $\lambda \ge \mu$ in dictionary order on integer partitions, the shapes are equal. Saturation of the prefix bounds places one entry of each eligible row in each column, giving $u=rct$ with $r\in R(t)$ and $c\in C(t)$. A collision instead gives a transposition that makes row symmetrization followed by the relevant column antisymmetrization vanish.