The rank is the number of orbits on ordered pairs under the diagonal group action. Its fixed-pair count at is , since the permutation character counts fixed points. Burnside's lemma identifies its average with the orbit count, while real-valuedness makes that average the character inner product of with itself. For a transitive action on at least two points, rank two is equivalent to transitivity on distinct ordered pairs.
For a finite group action on a finite set , is the free -module with basis , with the group permuting that basis. For a transitive set it is . Disjoint unions of sets give direct sums of permutation modules.
For even degree the augmentation hyperplane has an alternating dot product whose radical is the all-ones line. Quotienting by that line gives a nondegenerate symplectic space of dimension two less than the permutation degree. Coordinate permutations preserve the form. In degree at least six their quotient action is faithful, since two distinct weight-two coordinate vectors cannot differ by the all-ones vector. In degree six this gives ; in degree four the quotient action instead has Klein-four kernel.
The character of a permutation representation counts fixed points: . Its self-inner-product counts orbits on ordered pairs, while its inner product with the trivial character counts orbits on the original set.
Subgroups of a finite group are Gassmann equivalent if for every conjugacy class . 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.
In a finite general linear group, the permutation representations on nonzero vectors and on nonzero dual vectors have equal characters: an element fixes vectors in each action because a matrix and its transpose have the same rank. In characteristic two the stabilizers of a nonzero vector and a hyperplane are therefore almost conjugate subgroups. In dimension three over , both have order and index in a group of order , but they are not conjugate: a point stabilizer preserves no plane. A triangle cover construction for Sunada surfaces with generator orders gives quotient surfaces of genus three.
Embed and in by their regular representations. Both have one element of order one, three of order two, and twelve of order four. An order- element in a regular action has cycle type . Their intersections with every conjugacy class of consequently have the same size, so they are Gassmann equivalent. The first group is abelian and the second is not, so their underlying groups are not isomorphic. Finite-cover constructions from the Sunada theorem therefore give isospectral manifolds with different fundamental groups.
For an odd prime , embed the elementary abelian group and the Heisenberg group over a prime field regularly in . Every nonidentity element has order and hence regular permutation cycle type . 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 gives quotients with these nonisomorphic fundamental groups; Sunada theorem makes them isospectral while their topology differs.
For a subgroup , the permutation character of the coset action on is the induced trivial characterand
For a finite -set , the coefficient-sum map is equivariant. Its kernelis the augmentation subrepresentation. If is nonempty, then
For a transitive finite -set 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 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 , which is exactly two-transitivity.
If is the character of , then Burnside lemma applied to the diagonal action on givesFor a transitive action , this also equals the number of double cosets in .
Let act on the -element subsets of , with permutation character . If , thenbecause orbits of pairs are classified by intersection size .
For , the equivariant map from -subsets to -subsets that sends each subset to the sum of the -subsets containing it is injective over . Hence is the character of a representation.
This representation is a distinguished example in the representation theory of the symmetric group.
The permutation representation of on decomposes asThe first summand is the trivial representation and the second is the -dimensional standard representation.
For the sum-zero standard representation of the symmetric group, the vectors form a Gelfand–Tsetlin basis. The Young–Jucys–Murphy element has eigenvalue for , for , and for . Normalizing by gives an orthonormal basis corresponding to tableaux of shape whose second-row entry is .
The standard representation of the symmetric group restricts irreducibly to . Its character has value at the identity and on each nonidentity element of the normal Klein four-group. Restricted to that subgroup, it is the sum of the three nontrivial linear characters.
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In the context of group theory, a **permutation representation** is a way of representing a group as a group of permutations. Specifically, if \( G \) is a group, a permutation representation of \( G \) is a homomorphism from \( G \) to the symmetric group \( S_n \), which is the group of all permutations of a set of \( n \) elements.