Young symmetrizer

An element $e_d$ of the group ring of the symmetric group $S_m$ defined by the Young tableau $d$ of order $m$ by the following rule. Let $R_d$ (respectively, $C_d$) be the subgroup of $S_m$ consisting of all permutations permuting the numbers $1,\ldots,m$ in each row (respectively, column) in $d$. Further, put

$$r_d=\sum_{g\in R_d}g,\quad c_d=\sum_{g\in C_d}\epsilon(g)g,$$

where $\epsilon(g)=\pm1$ is the parity of $g$. Then $e_d=c_dr_d$ (sometimes one defines $e_d=r_dc_d$).

The basic property of a Young symmetrizer is that it is proportional to a primitive idempotent of the group algebra $\mathbf CS_m$. The coefficient of proportionality is equal to the product of the lengths of all hooks of $d$.

Comments

The ideal $\mathbf C[S_m]e_d$ is isomorphic to the Specht module of $S_m$ defined by the Young tableau $d$. Cf. also Young tableau for references and more details.