Frobenius reciprocity on the space of functions invariant under a group action
FFrobenius reciprocity on the space of functionsinvariant under a group action
Teerapong Suksumran ∗ and Tanakorn UdomworaratResearch Center in Mathematics and Applied MathematicsDepartment of MathematicsFaculty of Science, Chiang Mai UniversityChiang Mai 50200, Thailand [email protected] (T. Suksumran) [email protected] (T. Udomworarat) Abstract
This article studies connections between group actions and their correspondingvector spaces. Given an action of a group G on a nonempty set X , we examine thespace L ( X ) of scalar-valued functions on X and its fixed subspace: L G ( X ) = { f ∈ L ( X ) : f ( a · x ) = f ( x ) for all a ∈ G , x ∈ X } . In particular, we show that L G ( X ) is an invariant of the action of G on X . Inthe case when the action is finite, we compute the dimension of L G ( X ) in terms offixed points of X and prove several prominent results for L G ( X ) , including Bessel’sinequality and Frobenius reciprocity. Keywords:
Bessel’s inequality, free action, Frobenius reciprocity, function space,group action.
Primary 20C15; Secondary 05E18, 05E15, 05E10, 46C99.
Let G be a finite group and let H be a subgroup of G . Denote by C ( G ) and C ( H ) the spaces of complex-valued class functions on G and on H , respectively. Frobeniusreciprocity for class functions on G states that Res GH and Ind GH are Hermitian adjointwith respect to the Hermitian inner product defined by (cid:104) f , g (cid:105) = | G | ∑ x ∈ G f ( x ) g ( x ) and (cid:104) h , k (cid:105) H = | H | ∑ x ∈ H h ( x ) k ( x ) (1.1) ∗ Corresponding author. a r X i v : . [ m a t h . G R ] J un robenius reciprocity on the space of functions invariant under a group action 2 for all f , g ∈ C ( G ) , h , k ∈ C ( H ) . In other words, if f is a class function on H and if g isa class function on G , then (cid:104) Ind GH f , g (cid:105) H = (cid:104) f , Res GH g (cid:105) , (1.2)where Res GH is a linear transformation from C ( G ) to C ( H ) and Ind GH is a linear trans-formation from C ( H ) to C ( G ) . This result is crucial and plays fundamental rolesin proving well-known results in the representation theory of finite groups such asMackey’s irreducibility criterion; see, for instance, [9, Theorem 8.3.6].It is well known that the conjugation relation in any group G may be viewed as agroup action of G on itself by the formula g · x = gxg − for all g , x ∈ G . This suggestsstudying Frobenius reciprocity in the setting of group actions. In the present article,we generalize Frobenius reciprocity to the family of functions that are invariant undera given group action. We remark that there are other versions of Frobenius reciprocity;see, for instance, [2, 4].Let F be a field and let X be a nonempty set with an action of a group G (that is, X is a G -set). Define L ( X ) = { f : f is a function from X to F } . (1.3)Recall that L ( X ) is a vector space under the following vector addition and scalar mul-tiplication: ( f + g )( x ) = f ( x ) + g ( x )( α f )( x ) = α f ( x ) for all f , g ∈ L ( X ) , α ∈ F , x ∈ X . Furthermore, G acts linearly on L ( X ) by the formula ( a (cid:63) f )( x ) = f ( a − · x ) , x ∈ X , (1.4)for all a ∈ G , f ∈ L ( X ) , where (cid:63) is the induced G -action on L ( X ) and · is the given G -action on X . Therefore, we can speak of the fixed subspace of L ( X ) :Fix ( L ( X )) = { f ∈ L ( X ) : a · f = f for all a ∈ G } . It is not difficult to check that a · f = f for all a ∈ G if and only if f ( a · x ) = f ( x ) forall a ∈ G , x ∈ X . The fixed subspace of L ( X ) is so important that we give a separatedefinition. Definition 1.1.
Let X be a G -set. The fixed subspace of L ( X ) associated with the actiongiven by (1.4) is defined as L G ( X ) = { f ∈ L ( X ) : f ( a · x ) = f ( x ) for all a ∈ G , x ∈ X } . (1.5)In the case when X is a finite-dimensional vector space, (1.4) induces an action of G on the space F [ X ] of polynomial functions on X . The study of this action along with thecorresponding fixed subspace is a fundamental topic in invariant theory [3, 5–7]. Thefollowing examples indicate that several familiar families of functions in the literaturemay be viewed as L G ( X ) with appropriate group actions. robenius reciprocity on the space of functions invariant under a group action 3 Example 1.2.
Let X be a nonempty set. If G = { id X } is the trivial subgroup ofSym ( X ) , then G acts on X by evaluation: σ · x = σ ( x ) for all σ ∈ G , x ∈ X and inthis case, L G ( X ) = L ( X ) . If G = Sym ( X ) , then L G ( X ) becomes the space of constantfunctions: L G ( X ) = { f α : α ∈ F } , where f α ( x ) = α for all x ∈ X . Example 1.3 (Class functions).
Let G be a group and let F = C . Recall that G acts onitself by conjugation: a · x = axa − for all a , x ∈ G . In this case, L G ( G ) = { f : G → C : f ( axa − ) = f ( x ) for all a , x ∈ G } , which is the family of complex-valued class functions defined on G . Example 1.4 (Periodic functions).
Let F be a field. Suppose that A is an abeliangroup. Fix t ∈ A and set G = (cid:104) t (cid:105) = { nt : n ∈ Z } . Then G acts on A by addition and L G ( A ) = { f : A → F : f ( x + t ) = f ( x ) for all x ∈ A } , which is the family of periodic functions defined on A with period t . Example 1.5 (Modular functions).
Let C ∞ = C ∪{ ∞ } be the extended complex plane.Recall that a modular function f : C ∞ → C ∞ must satisfy the condition that f (cid:18) az + bcz + d (cid:19) = f ( z ) , z ∈ C ∞ , where a , b , c , d ∈ Z and ad − bc = L G ( C ∞ ) , where G is the modular group consisting of matrices of the form (cid:20) a bc d (cid:21) with a , b , c , d ∈ Z and ad − bc = G acts on C ∞ by the formula (cid:20) a bc d (cid:21) · z = az + bcz + d . Example 1.6.
A gyrogroup is a nonassociative group-like structure that is not, ingeneral, a group [13]. Nevertheless, it generalizes the notion of a group and sharesseveral properties with groups. The present article stems from the study of left regularrepresentation of a finite gyrogroup in a series of articles [10–12]. More precisely, let ( K , ⊕ ) be a gyrogroup. As in Section 4 of [10], the space L gyr ( K ) = { f ∈ L ( K ) : f ( a ⊕ gyr [ x , y ] z ) = f ( a ⊕ z ) for all a , x , y , z ∈ K } (1.6)arises as a representation space of K associated with a gyrogroup version of left regularrepresentation. Using the change of variable, w = a ⊕ z , we obtain that L gyr ( K ) = { f ∈ L ( K ) : f ( a ⊕ gyr [ x , y ]( (cid:9) a ⊕ z )) = f ( z ) for all a , x , y , z ∈ K } . robenius reciprocity on the space of functions invariant under a group action 4 Let G be the subgroup of Sym ( K ) generated by the set { L a ◦ gyr [ x , y ] ◦ L − a : a , x , y ∈ K } ,where L a is the left gyrotranslation by a defined by L a ( z ) = a ⊕ z for all z ∈ K andgyr [ x , y ] is the gyroautomorphism generated by x and y . It is clear that G acts on K byevaluation. Furthermore, L gyr ( K ) = L G ( K ) .In Section 2, we study basic properties of arbitrary group actions related to theircorresponding fixed subspaces. In Section 3, we reduce to the case of a finite action(that is, an action of a finite group on a finite set) and compute the dimension of thefixed subspace. This leads to some remarkable properties of fixed-point-free actions.Once the usual Hermitian inner product on L ( X ) is introduced, where X is a finite G -set,an orthogonal decomposition of L ( X ) is obtained and several interesting results suchas Fourier expansion, Bessel’s inequality, and Frobenius reciprocity are established forthe case of functions invariant under the action of G on X . Let G be a group and let X be a G -set. Recall that the action of G on X induces anequivalence relation ∼ given by x ∼ y if and only if y = a · x for some a ∈ G (2.1)for all x , y ∈ X . Also, the orbit of x ∈ X under the action of G is defined asorb x = { y ∈ X : y ∼ x } = { a · x : a ∈ G } . Hence, the collection { orb x : x ∈ X } forms a partition of X . This partition leads to acharacterization of elements in L G ( X ) , as shown in the next theorem, and eventually toa standard basis for L G ( X ) if there are only finitely many orbits of G on X . Theorem 2.1.
Suppose that X is a G-set and let P be the partition of X determined bythe equivalence relation (2.1) . If f ∈ L ( X ) , then f ∈ L G ( X ) if and only if f is constanton C for all C ∈ P .Proof. Suppose that f ∈ L G ( X ) and let C ∈ P . Let x , y ∈ C . Then y = a · x for some a ∈ G . Thus, f ( y ) = f ( a · x ) = f ( x ) . This proves that f is constant on C .Suppose conversely that f is constant on C for all C ∈ P . Let x ∈ X and let a ∈ G .Since x ∼ a · x , it follows that x and a · x belong to the same orbit C in P . By assumption, f is constant on C and so f ( x ) = f ( a · x ) . Since x and a are arbitrary, we obtain that f ∈ L G ( X ) .In view of Theorem 2.1, a natural question arises: Can a space of functions on a setendowed with a partition be viewed as L G ( X ) for a suitable group action? The answerto this question is affirmative. In fact, by Corollary 1.17 of [5], if X is a nonempty setand if P = { X i : i ∈ I } is a partition of X , then the following permutation group S P = { σ ∈ Sym ( X ) : σ ( X i ) = X i for all i ∈ I } (2.2) robenius reciprocity on the space of functions invariant under a group action 5 acts on X by evaluation and induces its orbits on X as the cells of the partition. Thefollowing theorem shows that the group S P may be replaced by its subgroup G P and,among other things, describes a concrete method to construct the group G P . We remarkthat if X is finite, then G P and S P coincide. Theorem 2.2.
Let X be a nonempty set and let P = { [ x ] : x ∈ X } be a partition of X.Then there exists a subgroup G P of S P that acts on X such that orb x = [ x ] for all x ∈ X.Proof.
Define an equivalence relation ∼ P on X by y ∼ P z if and only if y ∈ [ x ] and z ∈ [ x ] for some x ∈ X . For all x , y ∈ X , define τ ( x , y ) = (cid:40) id X if x = y or x (cid:28) P y ; ( x y ) otherwise . Then τ ( x , y ) ∈ Sym ( X ) . Set G P = (cid:104) τ ( x , y ) : x , y ∈ X (cid:105) . Then G P acts on X by evaluation.Next, we prove that orb x = [ x ] for all x ∈ X . Let x ∈ X . By definition, [ x ] ⊆ orb x . Toprove the reverse inclusion, suppose that y ∈ orb x . Then y = τ · x for some τ ∈ G . Weclaim that y ∼ P x . If x = y , then we are done. We may therefore assume that x (cid:54) = y . Byconstruction, τ = τ ( y , y ) ◦ τ ( y , y ) ◦ · · · ◦ τ ( y m − , y m ) . Moreover, we can assumethat τ ( y i − , y i ) (cid:54) = id X for all i = , , . . . , m . Therefore, τ = ( y y )( y y ) · · · ( y m − y m ) . Since y = τ ( x ) , there is a maximum value j ∈ { , , . . . , m } such that y j (cid:54) = x and ( y j x ) is a transposition factor in τ . So y j ∼ P x . If y j = y , then y ∼ P x . If y j (cid:54) = y ,then there is a maximum value j ∈ { , , . . . , m } such that 1 ≤ j < j and ( y j y j ) is a transposition factor in τ . So y j ∼ P y j ∼ P x . Continuing this procedure, weobtain that y ∼ P y j k ∼ P y j k − ∼ P · · · ∼ P y j ∼ P y j ∼ P x . Hence, y ∈ [ x ] . This provesorb x ⊆ [ x ] and so equality holds.Next, we prove that τ ( x , y ) ∈ S P for all x , y ∈ X . Hence, G P ⊆ S P by the minimalityof G P . Let x , y ∈ X . If x = y or x (cid:28) P y , then τ ( x , y ) = id X and hence τ ( x , y )( X i ) = X i for all X i ∈ P . We may therefore assume that x (cid:54) = y and x ∼ P y . Let X i ∈ P and let z ∈ X i . Note that τ ( x , y )( z ) = z if z (cid:54) = x and z (cid:54) = y ; y if z = x ; x if z = y . Since x ∼ P y , x and y are in the same cell of P . It follows that τ ( x , y )( z ) ∈ X i . Thisimplies that τ ( x , y )( X i ) = X i , which completes the proof.Let X be a G -set and let P be the partition of X determined by the equivalencerelation (2.1). For each C ∈ P , the indicator function δ C is defined by δ C ( x ) = (cid:40) x ∈ C ;0 if x ∈ X \ C . (2.3)By Theorem 2.1, δ C belongs to L G ( X ) for all C ∈ P . In fact, we obtain the followingtheorem. robenius reciprocity on the space of functions invariant under a group action 6 Theorem 2.3.
Suppose that X is a G-set and let P be the partition of X determined bythe equivalence relation (2.1) . Then B = { δ C : C ∈ P} is a linearly independent set inL G ( X ) . Furthermore, P is finite if and only if B forms a basis for L G ( X ) . In particular, dim ( L G ( X )) ≥ |P| and equality holds if P is finite.Proof. By definition, B is linearly independent. Assume that P is finite, say P = { C , C , . . . , C n } . Fix c i ∈ C i for all i = , , . . . , n . Then f = f ( c ) δ C + f ( c ) δ C + · · · + f ( c n ) δ C n for all f ∈ L G ( X ) and so B spans L G ( X ) . To prove the converse, suppose that P isinfinite. Assume to the contrary that B forms a basis for L G ( X ) . Define f by f ( x ) = x ∈ X . Then f ∈ L G ( X ) and so f = a δ C + a δ C + · · · + a n δ C n for some C , C , . . . , C n ∈ P . Since P is infinite, there is an orbit C ∈ P\{ C , C , . . . , C n } . Choose c ∈ C . Then f ( c ) =
1, whereas ( a δ C + a δ C + · · · + a n δ C n )( c ) = a δ C ( c ) + a δ C ( c ) + · · · + a n δ C n ( c ) = . Hence, f (cid:54) = a δ C + a δ C + · · · + a n δ C n , a contradiction. This shows that B is not abasis for L G ( X ) .Since B is linearly independent, it follows that dim ( L G ( X )) ≥ |B| = |P| . Moreover,if P is finite, then B is a basis for L G ( X ) and so dim ( L G ( X )) = |P| .According to Theorem 2.3, { δ C : C ∈ P} does not form a basis for L G ( X ) in thecase when P is infinite. It turns out that { δ C : C ∈ P} forms a basis for the followingsubspace of L G ( X ) : L G fs ( X ) = { f ∈ L G ( X ) : f is nonzero on finitely many orbits in X } (2.4)so that the dimension of L G fs ( X ) equals |P| . Next, we mention some related propertiesbetween group actions and their corresponding spaces. Theorem 2.4.
Let G be a group and let X be a G-set. Then the following are equi-valent: (1) L ( X ) = L G ( X ) ; (2) | orb x | = for all x ∈ X; (3) G acts trivially on X.Proof.
To prove the equivalence (1) ⇔ (2), suppose that L ( X ) = L G ( X ) . Let x ∈ X .Define δ x by δ x ( z ) = (cid:40) z = x ;0 otherwisefor all z ∈ X . By assumption, δ x ∈ L G ( X ) . For each y ∈ orb x , δ x ( y ) = δ x ( x ) = y = x . Thus, orb x = { x } and so | orb x | =
1. Conversely,suppose that | orb x | = x ∈ X . Let f ∈ L ( X ) . Then f ( y ) = f ( z ) for all y , z ∈ orb x .By Theorem 2.1, f ∈ L G ( X ) . This proves L ( X ) ⊆ L G ( X ) and so equality holds. robenius reciprocity on the space of functions invariant under a group action 7 To prove the equivalence (2) ⇔ (3), suppose that | orb x | = x ∈ X . Since { x } = orb x = { a · x : a ∈ G } , we obtain that a · x = x for all a ∈ G . Hence, G actstrivially on X . Conversely, suppose that G acts trivially on X . Let x ∈ X . Then a · x = x for all a ∈ G . This implies orb x = { x } and so | orb x | = Theorem 2.5.
Let F be a field and let X be a G-set. Then the following are equivalent: (1) The action of G on X is transitive; (2) dim ( L G ( X )) = ; (3) L G ( X ) = { f α : α ∈ F } = span f , where f α ( x ) = α for all x ∈ X, α ∈ F .Proof. Let x , y ∈ X . By assumption, there is an element a ∈ G such that y = a · x ; thatis, x ∼ y . Thus, x and y are in the same orbit. This shows that X has only one orbit. ByTheorem 2.3, dim ( L G ( X )) =
1. This proves that (1) implies (2).It is clear that f α ∈ L G ( X ) for all α ∈ F . Thus, { f α : α ∈ F } ⊆ L G ( X ) . Let f ∈ L G ( X ) and let x , y ∈ X . Since dim ( L G ( X )) = X has only one orbit and so y ∼ x .Hence, there is an element a ∈ G such that y = a · x . It follows that f ( y ) = f ( a · x ) = f ( x ) and so f is constant. Therefore, L G ( X ) ⊆ { f α : α ∈ F } . This proves that (2) implies(3).Let x , y ∈ X . Since L G ( X ) = span f , dim L G ( X ) =
1. By Theorem 2.3, X has onlyone orbit. Hence, x ∼ y and so there is an element a ∈ G such that y = a · x . Therefore,the action of G on X is transitive. This proves that (3) implies (1).We close this section with the following result, which indicates that L G ( X ) is an invariant of the action of G on X . Therefore, in certain circumstances, one can use thenotion of L G ( X ) to distinguish inequivalent group actions. Proposition 2.6.
Let X and Y be G-sets. If Φ : X → Y is an equivalence, then the map τ defined by τ ( f ) = f ◦ Φ − , f ∈ L ( X ) , (2.5) is a linear isomorphism from L ( X ) to L ( Y ) that restricts to a linear isomorphism fromL H ( X ) to L H ( Y ) for any subgroup H of G. Consequently, if X and Y are equivalent,then L ( X ) ∼ = L ( Y ) and L H ( X ) ∼ = L H ( Y ) as vector spaces for any subgroup H of G.Proof. The proof that τ is a linear isomorphism is straightforward. Let H be a subgroupof G and let f ∈ L H ( X ) . We claim that τ ( f ) ∈ L H ( Y ) . Let a ∈ H and let y ∈ Y .By surjectivity, there is an element x ∈ X such that y = Φ ( x ) . Thus, τ ( f )( a · y ) = τ ( f )( a · Φ ( x )) = τ ( f )( Φ ( a · x )) = f ( Φ − ( Φ ( a · x ))) = f ( a · x ) = f ( x ) = f ( Φ − ( y )) =( f ◦ Φ − )( y ) = τ ( f )( y ) . Hence, τ ( f ) ∈ L H ( Y ) and so τ maps L H ( X ) to L H ( Y ) .Let g ∈ L H ( Y ) and set f = g ◦ Φ . Note that f is a map from X to F and that f ( a · x ) = g ( Φ ( a · x )) = g ( a · Φ ( x )) = g ( Φ ( x )) = f ( x ) for all a ∈ H and x ∈ X . Hence, f ∈ L H ( X ) . Furthermore, τ ( f ) = f ◦ Φ − = ( g ◦ Φ ) ◦ Φ − = g . This proves that τ issurjective. Therefore, the restriction τ : L H ( X ) → L H ( Y ) is a linear isomorphism. robenius reciprocity on the space of functions invariant under a group action 8 The converse to Proposition 2.6 is not, in general, true. That is, the conditionthat “ L H ( X ) ∼ = L H ( Y ) as vector spaces for some subgroup H of G ” does not implythat “ X ∼ = Y as G -sets”. In fact, let X be a set having at least two distinct elements,namely that x , y ∈ X and x (cid:54) = y . Then Sym ( X ) acts transitively on { x } and on { x , y } byevaluation. By Theorem 2.5, dim ( L Sym ( X ) ( { x } )) = = dim ( L Sym ( X ) ( { x , y } )) and so L Sym ( X ) ( { x } ) ∼ = L Sym ( X ) ( { x , y } ) . However, { x } and { x , y } are not equivalent. If G is a finite group and if X is a finite G -set (that is, if the action is finite), wemay use the Cauchy–Frobenius lemma (also called the Burnside lemma) to computethe dimension of L G ( X ) . Moreover, the space L ( X ) (and hence also L G ( X ) ) possessesa standard Hermitian inner product (the base field is assumed to be the field of complexnumbers). This allows us to prove further related properties between group actions andtheir corresponding spaces, including Bessel’s inequality and Frobenius reciprocity. With Theorem 2.3 in hand, we give a formula for computing the dimension of L G ( X ) , where G and X are finite, in terms of fixed points of X . As a consequence ofthis result, we obtain a few remarkable properties of free (also called fixed-point-free)actions. Lemma 3.1.
Let G be a finite group and let X be a finite G-set. For any subgroup Hof G, dim ( L H ( X )) = | H | ∑ a ∈ H | Fix a | , (3.1) where Fix a = { x ∈ X : a · x = x } .Proof. Recall that H acts on X by the action inherited from G since H is a subgroupof G . Let orb H x = { a · x : a ∈ H } and let P = { orb H x : x ∈ X } . By Theorem 2.3,dim ( L H ( X )) equals |P| , the number of orbits of H on X . By the famous Cauchy–Frobenius lemma, |P| = | H | ∑ a ∈ H | Fix a | . Lemma 3.2.
Let G be a finite group and let X be a finite G-set. For any subgroup Hof G, | G | dim ( L G ( X )) − | H | dim ( L H ( X )) = ∑ a ∈ G \ H | Fix a | . (3.2) Proof.
Note that Fix H a = Fix G a for all a ∈ H because H acts on X by the actioninherited from G . It follows from Lemma 3.1 that | G | dim ( L G ( X )) − | H | dim ( L H ( X )) = ∑ a ∈ G | Fix a | − ∑ a ∈ H | Fix a | = ∑ a ∈ G \ H | Fix a | . (cid:3) robenius reciprocity on the space of functions invariant under a group action 9 Recall that an action of a group G on a set X is free if stab x = { e } for all x ∈ X ; thatis, if for all a ∈ G , x ∈ X , a · x = x implies a = e . It is clear that an action of G on X isfree if and only if Fix a = /0 for all a ∈ G \ { e } . By Lemma 3.2, the ratio of dim ( L H ( X )) and dim ( L G ( X )) is simply the index of H in G when the action of G on X is free. Itturns out that the family of free actions is rather limited in the sense of Corollary 3.4. Theorem 3.3.
Let G be a finite group with a subgroup H and let X be a finite nonemptyset. If G acts freely on X, then dim ( L H ( X )) dim ( L G ( X )) = [ G : H ] , (3.3) where [ G : H ] denotes the index of H in G.Proof. Since G acts freely on X , Fix a = /0 for all a ∈ G \ H . By Lemma 3.2, | G | dim ( L G ( X )) − | H | dim ( L H ( X )) = ∑ a ∈ G \ H | Fix a | = . Hence, dim ( L H ( X )) dim ( L G ( X )) = | G || H | = [ G : H ] . Corollary 3.4.
If a finite group G acts freely on a finite nonempty set X, thenthe number of orbits of G on X = dim ( L G ( X )) = | X || G | . (3.4) Therefore, if | G | does not divide | X | , then G does not act freely on X.Proof. Note that L { e } ( X ) = L ( X ) . Since X is finite, it follows that dim ( L ( X )) = | X | .By Theorem 3.3, | X | dim ( L G ( X )) = dim ( L { e } ( X )) dim ( L G ( X )) = [ G : { e } ] = | G | and the corollaryfollows.Corollary 3.4 provides a numerical condition that can be used to verify that a givenaction is not free. Hence, in this case, a nontrivial stabilizer subgroup exists and somefixed-point set is nonempty. For instance, by Corollary 3.4, we know that the followingactions are not free without having to find explicit stabilizer subgroups (or fixed-pointsets).(1) The action of Sym ( X ) on X in the case when X is finite and | X | > G be a finite group and let H be a nontrivial subgroup of G . Then G actson the set of left cosets, G / H = { xH : x ∈ G } , by left multiplication: a · ( xH ) =( ax ) H for all a , x ∈ G .(3) Let G be a finite group and let Y be a subset of G with nontrivial normalizer (thatis, the normalizer subgroup N G ( Y ) does not equal { e } ). For example, we maylet Y be a nontrivial subgroup of G . Then G acts on the set X = { xY x − : x ∈ G } by conjugation: a · ( xY x − ) = ( ax ) Y ( ax ) − for all a , x ∈ G . robenius reciprocity on the space of functions invariant under a group action 10 (4) Let G be a finite group whose order is divisible by a prime p . Then G acts on theset of Sylow p -subgroups of G by conjugation.(5) Let G be a finite group whose order is divisible by a prime p . Then G acts on theset X = { x ∈ G : | x | = p } by conjugation.(6) Let F be a finite field and let G be the general linear group GL ( n , F ) with n ≥ G acts on the set X = α α ... α n : α i ∈ F for all i = , , . . . , n by (left) matrix multiplication: A · [ α α · · · α n ] t = A [ α α · · · α n ] t for all A ∈ GL ( n , F ) , [ α α · · · α n ] t ∈ X .(7) Let G be a finite group that is not a 2-group and let X be a G -set. Then G acts onthe collection of subsets of X by letting a · Y = { a · y : y ∈ Y } for all a ∈ G andfor all subsets Y of X .(8) Let G be a finite group. Then G × G acts on G by ( a , b ) · x = axb − for all ( a , b ) ∈ G × G , x ∈ G (cf. Section 1.2.8 of [5]).We close this section with another application of Lemma 3.1. Proposition 3.5.
Let G be a finite group and let A be a group of automorphisms of G,that is, A ≤ Aut ( G ) . Then ∑ τ ∈ A |{ x ∈ G : τ ( x ) = x }| ≤ | A || G | . (3.5) Proof.
Recall that Aut ( G ) acts on G by τ · a = τ ( a ) for all τ ∈ Aut ( G ) , a ∈ G . Since A is a subgroup of Aut ( G ) , by Lemma 3.1, dim ( L A ( G )) = | A | ∑ τ ∈ A |{ x ∈ G : τ ( x ) = x }| .Since L A ( G ) ⊆ L ( G ) , it follows that dim ( L A ( G )) ≤ dim ( L ( G )) = | G | . Hence, the in-equality follows. In this section, let G be a (finite or infinite) group and let X be a finite G -set unlessotherwise stated. We also suppose that F = C . Thus, L ( X ) = { f : f is a function from X to C } (3.6)and L ( X ) admits the complex inner product structure. In fact, for all f , g ∈ L ( X ) , define (cid:104) f , g (cid:105) = | X | ∑ x ∈ X f ( x ) g ( x ) , (3.7)where ¯ · denotes complex conjugation. Then the following result is obtained. robenius reciprocity on the space of functions invariant under a group action 11 Proposition 3.6.
Equation (3.7) defines a Hermitian inner product on L ( X ) . Hence,L ( X ) forms a complex inner product space. If P = { orb x : x ∈ X } , then B = (cid:40)(cid:115) | X || C | δ C : C ∈ P (cid:41) forms an orthonormal basis for L G ( X ) .Proof. The proof that (3.7) defines a Hermitian inner product on L ( X ) is direct com-putation. By Theorem 2.3, B forms a basis for L G ( X ) . Next, we prove that B isorthonormal. Let (cid:115) | X || C | δ C , (cid:115) | X || D | δ D ∈ B with C , D ∈ P . If C (cid:54) = D , then (cid:104) (cid:115) | X || C | δ C , (cid:115) | X || D | δ D (cid:105) = (cid:112) | C || D | ∑ x ∈ X δ C ( x ) δ D ( x ) = x ∈ X , either x ∈ C or x ∈ D . If C = D , then (cid:104) (cid:115) | X || C | δ C , (cid:115) | X || D | δ D (cid:105) = (cid:112) | C || D | ∑ x ∈ X δ C ( x ) δ D ( x ) = | C | ∑ x ∈ X | δ C ( x ) | = ∑ x ∈ X | δ C ( x ) | = | C | .One advantage of the inner product defined by (3.7) is shown in the followingtheorem, demonstrating that the action of G on X is preserved by this inner product. Proposition 3.7.
The action given by (1.4) is unitary in the sense that (cid:104) a · f , a · g (cid:105) = (cid:104) f , g (cid:105) for all f , g ∈ L ( X ) , a ∈ G. In particular, the map f (cid:55)→ a · f , f ∈ L ( X ) , is a unitaryoperator on L ( X ) for all a ∈ G.Proof.
Let f , g ∈ L ( X ) and let a ∈ G . Then (cid:104) a · f , a · g (cid:105) = | X | ∑ x ∈ X ( a · f )( x )( a · g )( x )= | X | ∑ x ∈ X f ( a − · x ) g ( a − · x )= | X | ∑ x ∈ X f ( x ) g ( x )= (cid:104) f , g (cid:105) . The third equality holds since the map x (cid:55)→ a − · x is a bijection from X to itself.To obtain an orthogonal decomposition of L ( X ) , we define a map σ by σ ( f ) = ∑ x ∈ X f ( x ) , f ∈ L ( X ) . (3.8) robenius reciprocity on the space of functions invariant under a group action 12 Theorem 3.8.
Let σ be the map defined by (3.8) . Then the following assertions hold: (1) σ is a linear functional from L ( X ) to C . (2) ker σ is an invariant subspace of L ( X ) under the action given by (1.4) . (3) ker σ = ( span f ) ⊥ , where f ( x ) = for all x ∈ X. (4) dim ( ker σ ) = | X | − . (5) ker (cid:16) σ (cid:12)(cid:12) L G ( X ) (cid:17) is an invariant subspace of L G ( X ) and its dimension equals thenumber of orbits on X minus . Here, σ (cid:12)(cid:12) L G ( X ) is the restriction of σ to L G ( X ) . (6) L G ( X ) ⊥ ⊆ ker σ ; equality holds if and only if the action of G on X is transitive.Proof. The proofs of Parts (1), (3), and (5) are immediate. Part (2) holds because themap x (cid:55)→ a − · x is a bijection from X to itself. To prove Part (4), note that span f is afinite-dimensional subspace of L ( X ) . By the projection theorem in linear algebra andPart (3), L ( X ) = span f ⊕ ker σ and so dim ( ker σ ) = | X | −
1. That L G ( X ) ⊥ ⊆ ker σ isclear. By Theorem 2.5 and Part (3), L G ( X ) ⊥ = ker σ if and only if L G ( X ) = span f (since L G ( X ) and span f are finite-dimensional subspaces of L ( X ) ) if and only if theaction of G on X is transitive. This proves Part (6). Corollary 3.9.
Let G be a group and let X be a finite G-set. Then (1) L ( X ) = L G ( X ) ⊥ (cid:13) L G ( X ) ⊥ ; (2) L ( X ) = span f ⊥ (cid:13) ker σ ; (3) L G ( X ) = span f ⊥ (cid:13) ker (cid:16) σ (cid:12)(cid:12) L G ( X ) (cid:17) .Here, ⊥ (cid:13) denotes orthogonal direct sum decomposition.Proof. Part (1) follows from the projection theorem. Part (2) follows as in the proof ofTheorem 3.8 (4). Part (3) holds since span f is a subspace of L G ( X ) .According to Proposition 3.6, we have an orthonormal basis for L G ( X ) , which isan orthonormal set in L ( X ) . Thus, several prominent results in linear algebra can bededuced from this fact. Theorem 3.10.
Let G be a group and let X be a finite G-set. Suppose that P = { orb x : x ∈ X } = { C , C , . . . , C n } . Fix the ordered (orthonormal) basis of L G ( X ) : B = (cid:32)(cid:115) | X || C | δ C , (cid:115) | X || C | δ C , . . . , (cid:115) | X || C n | δ C n (cid:33) .Then the following assertions hold: robenius reciprocity on the space of functions invariant under a group action 13 (1) (Fourier expansion) The Fourier expansion with respect to B of a functionf ∈ L ( X ) is (cid:98) f = (cid:32) | C | ∑ x ∈ C f ( x ) (cid:33) δ C + (cid:32) | C | ∑ x ∈ C f ( x ) (cid:33) δ C + · · · + (cid:32) | C n | ∑ x ∈ C n f ( x ) (cid:33) δ C n ;(3.9) that is, the Fourier coefficients of f are given by (cid:104) f , (cid:115) | X || C i | δ C i (cid:105) = (cid:112) | X || C i | ∑ x ∈ C i f ( x ) (3.10) for all i = , , . . . , n. (2) (Bessel’s inequality) For all f ∈ L ( X ) , n ∑ i = | C i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ x ∈ C i f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∑ x ∈ X | f ( x ) | . (3.11)(3) If G acts nontrivially on X, then there exists a function f ∈ L ( X ) with (cid:107) (cid:98) f (cid:107) < (cid:107) f (cid:107) .That is, the equality in Bessel’s identity is not attained.Proof. Recall that the Fourier coefficients of f are (cid:104) f , (cid:115) | X || C i | δ C i (cid:105) = | X | ∑ x ∈ C i f ( x ) (cid:115) | X || C i | = (cid:112) | X || C i | ∑ x ∈ C i f ( x ) for all i = , , . . . , n . Hence, the Fourier expansion of f with respect to B is (cid:98) f = (cid:104) f , (cid:115) | X || C | δ C (cid:105) (cid:115) | X || C | δ C + · · · + (cid:104) f , (cid:115) | X || C n | δ C n (cid:105) (cid:115) | X || C n | δ C n = (cid:32) (cid:112) | X || C | ∑ x ∈ C f ( x ) (cid:33) (cid:115) | X || C | δ C + · · · + (cid:32) (cid:112) | X || C n | ∑ x ∈ C n f ( x ) (cid:33) (cid:115) | X || C n | δ C n = (cid:32) | C | ∑ x ∈ C f ( x ) (cid:33) δ C + · · · + (cid:32) | C n | ∑ x ∈ C n f ( x ) (cid:33) δ C n . This proves Part (1).Recall that Bessel’s inequality states that (cid:107) (cid:98) f (cid:107) ≤ (cid:107) f (cid:107) . Hence, n ∑ i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) f , (cid:115) | X || C i | δ C i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:104) f , f (cid:105) . Direct computation shows that n ∑ i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) f , (cid:115) | X || C i | δ C i (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | X | n ∑ i = | C i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∑ x ∈ C i f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and that (cid:104) f , f (cid:105) = | X | ∑ x ∈ X | f ( x ) | . Hence, (3.11) follows. This proves Part (2). robenius reciprocity on the space of functions invariant under a group action 14 Suppose that G acts nontrivially on X . By Theorem 2.4, L G ( X ) (cid:40) L ( X ) . Thisimplies that B is not an orthonormal basis for L ( X ) and so there exists a function f in L ( X ) with (cid:107) (cid:98) f (cid:107) < (cid:107) f (cid:107) by Theorem 9.17 of [8]. This proves Part (3).Next, we extend Frobenius reciprocity from the space of class functions to that offunctions invariant under a given group action in a natural way. Let G be a group andlet X be a G -set. Recall that a (nonempty) subset Y of X is invariant if a · y ∈ Y forall a ∈ G , y ∈ Y ; that is, if G · Y = Y . It is not difficult to check that the following areequivalent:(1) Y is an invariant subset of X ;(2) for all a ∈ G , x ∈ X , a · x ∈ Y if and only if x ∈ Y .Let X be a G -set and let Y be an invariant subset of X . Define a map Res XY from L ( X ) to L ( Y ) by Res XY f ( y ) = f ( y ) , y ∈ Y (3.12)for all f ∈ L ( X ) . Also, for each f ∈ L ( Y ) , define ˜ f by˜ f ( x ) = (cid:40) f ( x ) if x ∈ Y ;0 otherwise (3.13)for all x ∈ X . Then ˜ f ∈ L ( X ) . In fact, we have the following lemma. Lemma 3.11.
The map ε : L ( Y ) → L ( X ) given by ε ( f ) = ˜ f is linear and maps L G ( Y ) to L G ( X ) .Proof. The proof that ε is linear is straightforward. By the remark above, ε ( f ) ∈ L G ( X ) for all f ∈ L G ( Y ) . Theorem 3.12.
Let X be a G-set with an invariant subset Y . Then
Res XY : L ( X ) → L ( Y ) is linear and maps L G ( X ) surjectively onto L G ( Y ) .Proof. The proof that Res XY is linear is straightforward. Let f ∈ L G ( Y ) . By Lemma3.11, ˜ f ∈ L G ( X ) and Res XY ˜ f ( y ) = ˜ f ( y ) = f ( y ) for all y ∈ Y . So Res XY ˜ f = f . This provesthat Res XY is surjective.Let G be a finite group, let X be a finite G -set, and let Y be an invariant subset of X . Define a map Ind XY on L ( Y ) byInd XY f ( x ) = | X || G || Y | ∑ b ∈ G ˜ f ( b − · x ) , x ∈ X (3.14)for all f ∈ L ( Y ) . Then Ind XY is a linear transformation from L ( Y ) to L G ( X ) , as shownin the following theorem. Theorem 3.13.
The map
Ind XY defined by (3.14) is a linear transformation from L ( Y ) to L G ( X ) . robenius reciprocity on the space of functions invariant under a group action 15 Proof.
The proof that Ind XY is linear is straightforward. Let f ∈ L ( Y ) . Given a ∈ G and x ∈ X , we have by inspection thatInd XY f ( a · x ) = | X || G || Y | ∑ b ∈ G ˜ f ( b − · ( a · x ))= | X || G || Y | ∑ b ∈ G ˜ f (( b − a ) · x )= | X || G || Y | ∑ c ∈ G ˜ f ( c − · x )= Ind XY f ( x ) . The third equality holds since if b runs over all of G , then so does a − b (that is, thechange of variable c = a − b is permitted). Thus, Ind XY f ∈ L G ( X ) .The following theorem asserts that the linear transformations Res XY and Ind XY areHermitian adjoint with respect to the Hermitian inner product defined earlier. This is agroup-action version of Frobenius reciprocity. Theorem 3.14 (Frobenius reciprocity).
Let G be a finite group, let X be a finite G-set,and let Y be an invariant subset of X. Then (cid:104)
Ind XY f , g (cid:105) = (cid:104) f , Res XY g (cid:105) (3.15) for all f ∈ L G ( Y ) , g ∈ L G ( X ) .Proof. Direct computation shows that (cid:104)
Ind XY f , g (cid:105) = | X | ∑ x ∈ X Ind XY f ( x ) g ( x )= | X | ∑ x ∈ X (cid:32) | X || G || Y | ∑ b ∈ G ˜ f ( b − · x ) (cid:33) g ( x )= | G || Y | ∑ x ∈ X (cid:32) ∑ b ∈ G ˜ f ( b − · x ) (cid:33) g ( x )= | G || Y | ∑ x ∈ Y (cid:32) ∑ b ∈ G f ( b − · x ) (cid:33) g ( x )= | G || Y | ∑ x ∈ Y | G | f ( x ) g ( x )= | Y | ∑ x ∈ Y f ( x ) g ( x )= | Y | ∑ x ∈ Y f ( x ) Res XY g ( x )= (cid:104) f , Res XY g (cid:105) . robenius reciprocity on the space of functions invariant under a group action 16 The fifth equality holds since f ∈ L G ( Y ) , which implies that f ( b − · x ) = f ( x ) for all b ∈ G , x ∈ Y .We remark that the inner product used on the right hand side of (3.15) is computedby the same formula as in (3.7) with Y in place of X . This makes sense because if Y is an invariant subset of X , then the G -action on X restricts to the G -action on Y . Inother words, the restriction of the Hermitian inner product of L ( X ) to L ( Y ) does definean inner product on L ( Y ) . Acknowledgment.
This work was supported by the Research Center in Mathematicsand Applied Mathematics, Chiang Mai University.
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