Permutation representation

For a finite $G$-set $X$, the character of $\mathbb{C}[X]$ at $g$ is $|X^g|$. Its invariant dimension is the number of $G$-orbits.

The character of a permutation representation counts fixed points: $\chi (g)=|\{\omega :g\omega =\omega \}|$. Its self-inner-product counts orbits on ordered pairs, while its inner product with the trivial character counts orbits on the original set.

Subgroups $H_1,H_2$ of a finite group $G$ are Gassmann equivalent if $|H_1\cap C|=|H_2\cap C|$ for every conjugacy class $C$. Equivalently their coset permutation representations have the same character of a representation. They need not be conjugate subgroups. The Sunada theorem turns this equality into equality of Laplacian eigenfunction multiplicities.

For an odd prime $p$, embed the elementary abelian group $(C_p{)}^3$ and the Heisenberg group over a prime field regularly in $S_{p^3}$. Every nonidentity element has order $p$ and hence regular permutation cycle type $p^{p^2}$. The two subgroups have equal intersection counts with every symmetric-group conjugacy class, so they are Gassmann equivalent, but they are not isomorphic because one is abelian and the other is not. Using a compact simply connected isometric cover with ambient deck group $S_{p^3}$ gives quotients with these nonisomorphic fundamental groups; Sunada theorem makes them isospectral while their topology differs.

For a subgroup $H\le G$, the permutation character of the coset action on $G/H$ is the induced trivial characterand

For a finite $G$-set $X$, the coefficient-sum map $\epsilon :\mathbb{C}[X]\to \mathbb{C}$ is equivariant. Its kernelis the augmentation subrepresentation. If $X$ is nonempty, then

For a transitive finite $G$-set $X$ with at least two points, the augmentation subrepresentation of a permutation representation is irreducible if and only if the action is two-transitive. Indeed, writing the permutation character as $\pi =1+{\sum }_i{m}_i{\chi }_i$ givesThe augmentation subrepresentation is irreducible exactly when this norm is two. By the character norm of a permutation representation, that means that the diagonal and the off-diagonal are the only two orbits on $X\times X$, which is exactly two-transitivity.

If $\pi$ is the character of $\mathbb{C}[X]$, then Burnside lemma applied to the diagonal action on $X\times X$ givesFor a transitive action $X\simeq G/H$, this also equals the number of double cosets in $H\backslash G/H$.

Let $S_n$ act on the $r$-element subsets of $\{1,\dots ,n\}$, with permutation character ${\pi }_r$. If $0\le l\le k\le n/2$, thenbecause orbits of pairs are classified by intersection size $0,\dots ,l$.

For $r\le n/2$, the equivariant map from $(r-1)$-subsets to $r$-subsets that sends each subset to the sum of the $r$-subsets containing it is injective over $\mathbb{C}$. Hence ${\pi }_r-{\pi }_{r-1}$ is the character of a representation.

The permutation representation of $S_n$ on ${\mathbb{C}}^n$ decomposes asThe first summand is the trivial representation and the second is the $(n-1)$-dimensional standard representation.

For the sum-zero standard representation of the symmetric group, the vectors $v_j=e_1+\cdots +e_j-j{e}_{j+1}$ form a Gelfand–Tsetlin basis. The Young–Jucys–Murphy element $X_i$ has eigenvalue $i-1$ for $i\le j$, $-1$ for $i=j+1$, and $i-2$ for $i>j+1$. Normalizing by $\sqrt{j(j+1)}$ gives an orthonormal basis corresponding to tableaux of shape $(n-1,1)$ whose second-row entry is $j+1$.

The standard representation of the symmetric group $S_4$ restricts irreducibly to $A_4$. Its character has value $3$ at the identity and $-1$ on each nonidentity element of the normal Klein four-group. Restricted to that subgroup, it is the sum of the three nontrivial linear characters.