Nontrivial irreducible component of permutation representation of projective general linear group of degree two on projective line

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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