---- Denote by $S_{n}$ the [[symmetric group]] on $n$ letters. > [!definition] Definition. ([[standard representation of the symmetric group]]) > The **standard representation** of $S_{n}$ is an [[irreducible group representation]] of [[group representation|dimension]] $n-1$ obtained as follows: > - Take the usual $n$-dimensional [[permutation matrix|permutation action]] of $S_{n}$ acting on [[basis]] vectors of $\mathbb{R}^{n}$. The 1-dimensional [[linear subspace|subspace]] spanned by the sum of these basis vectors ($\b 1_{n}$ if they're the standard basis) is and $G$-[[group-invariant subspace|invariant]]. It has a $G$-[[group-invariant subspace admits group-invariant complement over C|invariant]] [[complement of a linear subspace|complement]], the restriction of the permutation representation to which is what we call the **standard representation**. > - Said $G$-invariant complement has as [[basis]] $(e_{1}-e_{2}, e_{2}-e_{3},\dots, e_{n-1}-e_{n})$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```