Difference between revisions of "Young symmetrizer"
From Encyclopedia of Mathematics
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| − | An element | + | {{TEX|done}} |
| + | An element $e_d$ of the group ring of the [[Symmetric group|symmetric group]] $S_m$ defined by the [[Young tableau|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 | + | 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 | + | 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==== | ====Comments==== | ||
| − | The ideal | + | 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|Young tableau]] for references and more details. |
Latest revision as of 13:01, 10 August 2014
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.
How to Cite This Entry:
Young symmetrizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_symmetrizer&oldid=12138
Young symmetrizer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_symmetrizer&oldid=12138
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article