Nontrivial irreducible component of permutation representation of projective general linear group of degree two on projective line
From Groupprops
Let be a field. Consider the projective general linear group of degree two . This has a natural action on the projective line over , i.e., the collection of one-dimensional subspaces of the two-dimensional vector space . We thus get a permutation representation of on the projective line .
The action could be described in either of these ways:
- For any element of , lift it to an element of , and consider the image of any one-dimensional subspace under the element of . Note that the image subspace does not depend on the choice of the lift, because any two lifts differ multiplicatively by a scalar matrix, which sends every subspace to itself.
- Think of as . For an element of , consider a matrix that is a lift of this element. The permutation induced by this is the map , where the value is taken to be if the denominator becomes , and the image of is taken to be if and to be if .
When is a finite field of size , then this gives a permutation action of a finite group on a finite set of size . View this as a linear representation in any characteristic not dividing the order of . This linear representation splits as a direct sum of a trivial representation and a nontrivial irreducible representation of degree . Our goal here is to discuss this irreducible component.
Summary
Item | Value |
---|---|
Degree of representation | |
Schur index | 1 in all characteristics (because the representation, being a direct summand of a permutation representation, can be realized with integer entries and hence interpreted in any characteristic). |
Kernel of representation | trivial subgroup. In other words, it is a faithful linear representation. |
Quotient on which it descends to a faithful representation | projective general linear group of degree two |
Set of character values | Characteristic zero: Ring generated: -- ring of integers, Ideal within ring generated: whole ring, Field generated: -- [[field of rational |
Ring of realization | Realized over any unital ring, by composing the representation over with the map induced by the natural homomorphism from to that ring. |
Minimal ring of realization (characteristic zero) | -- ring of integers |
Minimal ring of realization in characteristic | The ring of integers mod , |
Minimal field of realization | Prime field in all cases. In characteristic zero, ; in characteristic , the field |
Size of equivalence class under automorphisms | 1 |
Size of equivalence class under Galois automorphisms | 1 |
Size of equivalence class under action of one-dimensional representations by multiplication | 2 if the characteristic of K is not 2, 1 if the characteristic of K is 2. |
Particular cases
Field size | Underlying prime | Group | Order | Information on linear representation theory | Description of the representation |
---|---|---|---|---|---|
2 | 2 | symmetric group:S3 | 6 | linear representation theory of symmetric group:S3 | standard representation of symmetric group:S3 |
3 | 3 | symmetric group:S4 | 24 | linear representation theory of symmetric group:S4 | standard representation of symmetric group:S4 |
4 | 2 | alternating group:A5 | 60 | linear representation theory of alternating group:A5 | standard representation of alternating group:A5 |
5 | 5 | symmetric group:S5 | 120 | linear representation theory of symmetric group:S5 | one of the five-dimensional irreducible representations |
Character
FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma
Character values and interpretations
Nature of conjugacy class upstairs in | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Character value | Explanation (character value = number of fixed subspaces - 1) |
---|---|---|---|---|---|---|---|---|
Diagonalizable over with equal diagonal entries, hence a scalar | where | where | where | 1 | 1 | 1 | Fixes all subspaces, so character is | |
Diagonalizable over , not over , eigenvalues are negatives of each other. | Pair of mutually negative conjugate elements of . All such pairs identified. | , a nonzero non-square | Same as characteristic polynomial | 1 | -1 | No eigenvalues over , so no fixed subspaces, so character value is | ||
Diagonalizable over with mutually negative diagonal entries. | , all such pairs identified. | , all identified | Same as characteristic polynomial | 1 | 1 | Two distinct one-dimensional eigenspaces, so we get . | ||
Diagonalizable over , not over , eigenvalues are not negatives of each other. | Pair of conjugate elements of . Each pair identified with anything obtained by multiplying both elements of it by an element of . | , , irreducible; with identification. | Same as characteristic polynomial | -1 | No eigenspaces, so we get . | |||
Not diagonal, has Jordan block of size two | (multiplicity 2). Each conjugacy class has one representative of each type. | Same as characteristic polynomial | 1 | 0 | Unique one-dimensional eigenspace, so we get | |||
Diagonalizable over with distinct diagonal entries whose sum is not zero. | where and . The pairs and are identified. | , again with identification. | Same as characteristic polynomial. | 1 | Two one-dimensional eigenspaces, so we get . | |||
Total | NA | NA | NA | NA |