A Decomposition of twisted equivariant K-theory

For $G$ a finite group, a normalized 2-cocycle $\alpha\in Z^{2}(G,\mathbb{S}^{1})$ and $X$ a $G$-space on which a normal subgroup $A$ acts trivially, we show that the $\alpha$-twisted $G$-equivariant $K$-theory of $X$ decomposes as a direct sum of twisted equivariant $K$-theories of $X$ parametrized by the orbits of an action of $G$ on the set of irreducible $\alpha$-projective representations of $A$. This generalizes the decomposition obtained by G\'omez and Uribe for equivariant $K$-theory. We also explore some examples of this decomposition for the particular case of the dihedral groups $D_{2n}$ with $n\ge 1$ an even integer.


Introduction
In the past few years there has been a growing interest in studying twisted K-theory motivated by its appearance in string theory and also due to the celebrated theorem of Freed, Hokpins and Teleman (see [3,Theorem 1]). In this article we study G-equivariant twisted K-theory when G is a finite group. Our main goal is to show that, under suitable hypothesis, the canonical decomposition theorem for projective representations can be used to obtain a decomposition for twisted G-equivariant K-theory as a direct sum of other twisted equivariant K-theories, thus generalizing the work in [4] where a similar decomposition was obtained for equivariant K-theory.
Suppose that we have a short exact sequence of finite groups Let X be a compact and Hausdorff G-space on which A acts trivially and fix α a normalized 2-cocycle on G with values in S 1 . Associated to the cocycle α we have a central extension and the action of G on X can be extended to an action of G α on X in such a way that the central factor S 1 acts trivially. The α-twisted G-equivariant K-theory of X, α K * G (X), is constructed using G α -equivariant vector bundles on which the central factor S 1 acts by multiplication of scalars. The main object of study in this work are the twisted K-groups α K * G (X). Representations of the group G α on which the central factor S 1 acts by multiplication of scalars are in a one to one correspondence with α-projective representations of G. Via this correspondence we can use the classical tools of projective representations to study the twisted K-groups α K * G (X). To this end we show in Section 2 that, if Irr α (A) denotes the set The first author acknowledges and thanks the financial support provided by COLCIENCIAS through grant number FP44842-013-2018 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación. The second author acknowledges and thanks the financial support provided by COLCIEN-CIAS through grant number 727 of the program Doctorados nacionales 2015 of the Fondo Nacional de Financiamiento para la Ciencia, la Tecnología y la Innovación. of isomorphism classes of α-projective representations of A, then there is an action of G on Irr α (A). and this action factors through an action of Q on Irr α (A). Given an isomorphism class [τ ] ∈ Irr α (A) let Q [τ ] denote the isotropy subgroup at [τ ]. Using Lemma 2.7 we show that we can associate to each [τ ] ∈ Irr α (A) a normalized 2-cocycle defined on Q [τ ] with values in S 1 . This is precisely the data needed to construct a twisted version of Q [τ ] -equivariant K-theory. With this in mind, the main result of this article is the following theorem. Theorem 1.1. Suppose that A is a normal subgroup of a finite group G. Let α be a normalized 2-cocycle on G with values in S 1 and X a compact G-space on which A acts trivially. Then there is a natural isomorphism Hom Aα (V τ , E) .
When α is the trivial cocycle α K * G (X) agrees with K * G (X) and the previous decomposition agrees with [4,Theorem 3.2] in this case. Therefore Theorem 1.1 generalizes [4,Theorem 3.2]. In fact, Theorem 1.1 is proved using ideas similar to the ones used to prove [4,Theorem 3.2].
We remark that Theorem 1.1 could be proved using the work of Freed, Hopkins and Teleman [3]. However, we have chosen to prove it directly as to obtain an explicit description of this decomposition. Also, we chose to work with finite groups to obtain explicit formulas for the cocycles used to twist equivariant K-theory in this decomposition. Theorem 1.1 also holds in general for compact Lie groups, a proof in this context can be obtained generalizing the work in [2].
The outline of this article is as follows. In Section 2 we review some basic definitions of projective representations that we use throughout this article. Section 3 is the main part of the article, Theorem 1.1 is proved there. Finally, in Section 4 we explore some examples of Theorem 1.1 for the particular case of the dihedral group D 2n with n ≥ 1 an even integer.

Projective representations
In this section we recall some basic definitions and properties of projective representations that will be used throughout this article.

Basic definitions.
Definition 2.1. Let G be a finite group and V a finite dimensional complex vector space. A map ρ : for all g, h ∈ G and ρ(1) = Id V .
To stress the dependence of ρ on V and α, we shall often refer to ρ as an α-representation of G on the space V or, simply as an α-representation of G, if V is not pertinent to the discussion.
Remark 2.2. If α is the trivial cocycle; that is, if α(g, h) = 1 for all g, h ∈ G, then α-representations of G are simply ordinary representations of G.
Just as in the case of the ordinary theory of representations of groups we have similar notions for projective representations such as irreducible representations and unitary representations. Given a C * -valued normalized 2-cocycle α on G we denote by Irr α (G) the set of isomorphism classes of complex irreducible α-representations of G. If ρ : G → GL(V ) is an irreducible α-representation of G then [ρ] ∈ Irr α (G) denotes the corresponding isomorphism class. We remark that the classical results of representation theory such as complete reducibility and Schur's lemma also hold for the case of projective representations. We refer the reader to [5] for the basic theory of projective representations.
Example 2.4. Consider the dihedral group D 2n of order 2n defined by For such groups we have if n is odd, Z/2 if n is even.
In this example we only consider the case where n is even as otherwise we will obtain usual representations. Let n ≥ 2 be an even integer, ǫ a primitive n-th root of unity in C and let α : D 2n × D 2n → S 1 be the function defined by α(a j , a k b l ) = 1 and α(a j b, a k b l ) = ǫ k for 0 ≤ j, k ≤ n − 1 and l = 0, 1.
The function α defines a normalized 2-cocycle on D 2n with values in S 1 whose corresponding cohomology class is a generator in H 2 (D 2n , S 1 ) ∼ = Z/2. For each i ∈ {1, 2, . . . , n/2} put Consider the map for 0 ≤ k ≤ n − 1 and l = 0, 1. These assignments determine the irreducible, non-equivalent α-representations of D 2n so that A key feature of projective representations is that we also have the following canonical decomposition whose proof can be obtained in a similar way as in the case of regular representations.
Theorem 2.5. (Canonical Decomposition) Suppose that α is a normalized 2-cocycle of a finite group G with values in S 1 . Let W be a finite-dimensional α-representation. Then the assignment γ : defines an isomorphism of α-representations.
Suppose now that α is a normalized 2-cocycle on G with values in S 1 . We can associate to α a central extension of G by S 1 in the following way. As a set define The product structure in G α is given by the assignment This way G α a compact Lie group that fits into a central extension Let ρ : G → GL(V ) be an α-representation of G. If we defineρ : G α → GL(V ) bỹ ρ(g, z) = zρ(g) thenρ defines a representation of G α on which the central factor S 1 acts by multiplication of scalars. Conversely, ifρ : G α → GL(V ) is a representation of G α on which the central factor S 1 acts by multiplication of scalars then the function ρ : G → GL(V ) given by ρ(g) =ρ(g, 1) defines an α-representation of G. The above assignment defines a one to one correspondence between α-representations of G and representations of G α on which the central factor S 1 acts by multiplication of scalars. Via this correspondence we will switch back and forth between α-representations of G and representations of G α on which the central factor S 1 acts by multiplication of scalars without explicitly mentioning it.

2.2.
Cocycles and projective representations. Suppose now that A is a normal subgroup of a finite group G so that we have a short exact sequence Assume that α is a normalized 2-cocycle of G with values in S 1 , by restriction we can also see α as a cocycle defined on A. Let us define an action of G on the set Irr α (A) in the following way. Given ρ : A → U(V ρ ) an α-representation and g ∈ G we define g · ρ : A → U(V ρ ) so that if a ∈ A we have: Proposition 2.6. The above assignment defines a left action of G on Irr α (A). Furthermore, for all b ∈ A, we have that b · ρ ∼ = ρ so that the action of G on Irr α (A) factors to an action of Q = G/A on Irr α (A).
Proof. First we show that g · ρ is an α-representation of A. Indeed, for all g ∈ G and a, b ∈ A we have The above equalities are obtained using the 2-cocycle equation for α. Now, we show that this definition satisfies the axioms of an action. If ρ : Moreover, given g, h ∈ G and a ∈ A, we have Manipulating the cocycle equation for α it can be proved that This implies that for all a ∈ A g · (h · ρ)(a) = (gh) · ρ(a).
Finally, for a, b ∈ A expanding and using the cocycle equation we obtain and thus b · ρ ∼ = ρ as α-representations.
As above assume that A is a normal subgroup of a group G and Q = G/A. Fix an assignment σ : Q → G such that π(σ(q)) = q for all q ∈ Q with σ(1) = 1. We remark that the map σ is only a set theoretical map so in particular it does not necessarily have to be a group homomorphism.
Suppose that ρ : A → U(V ρ ) is a complex irreducible α-representation with the property that g · ρ is isomorphic to ρ for every g ∈ G (under the action defined in Proposition 3). Under this assumption, as σ(q) · ρ ∼ = ρ we can find an element M q ∈ U(V ρ ) for each q ∈ Q such that for all a ∈ A. We can choose M 1 = 1 as σ(1) = 1 and σ(1) · ρ = ρ. Let A α be the central extension of A by S 1 associated to the cocycle α andρ : A α → U(V ρ ) the corresponding representation. Remember thatρ(a, z) = zρ(a) so we can define σ(q) ·ρ(a, z) := z(σ(q) · ρ(a)). Therefore Define χ : Q × Q → A by the equation Note that χ(q 1 , q 2 ) belongs to A since π(χ(q 1 , q 2 )) = 1 and the map χ is normalized in the sense that χ(q 1 , q 2 ) = 1 whenever q 1 = 1 or q 2 = 1. In addition, define τ : Now, for q 1 , q 2 in Q we notice that the element is a complex irreducible α-representation such that g · ρ ∼ = ρ for every g ∈ G then the map β ρ,α : Q × Q → S 1 defines a normalized 2-cocycle on Q with values in S 1 .
Proof. If either q 1 = 1 or q 2 = 1 we have that τ (q 1 , q 2 ) = 1 as α is normalized. Also, as χ is normalized we have χ(q 1 , q 2 ) = 1. Since we are choosing σ(1) = 1 and M 1 = 1 it follows that β ρ,α (q 1 , q 2 ) = 1 if either q 1 = 1 or q 2 = 1. Therefore either β ρ,α is normalized. To finish we need to prove that for every q 1 , q 2 , q 3 ∈ Q we have To see this note that as β ρ,α (q 1 , q 2 ) belongs to S 1 we have that Therefore, for q 1 , q 2 and q 3 in Q we have, In a similar way, writing M q 1 q 2 q 3 = M (q 1 q 2 )q 3 and expanding we obtain This implies that as we wanted to prove.

Decomposition of twisted equivariant K-theory
In this section we use the canonical decomposition of vector bundles to show that under some hypothesis the α-twisted equivariant K-theory α K * G (X) of a G-space X can be decomposed as a direct sum of twisted equivariant K-theories parametrized by the orbits of the action of G on Irr α (A) constructed on the previous section.
We start by recalling the definition of α-twisted equivariant K-theory that we will use. We follow the treatment used in [1,Section 7.2]. Assume that G is a finite group acting on a compact and Hausdoff space X. Let α be a normalized 2-cocycle on G with values in S 1 .
Observe that the action of G on X can be extended to an action of G α in such a way that the central factor S 1 acts trivially.
, is defined as the Grothendieck group of the set of isomorphism classes of G α -equivariant vector bundles over X on which S 1 acts by multiplication of scalars on the fibers. For n > 0 the twisted groups α K n G (X) are defined as αK 0 G (Σ n X + ), where as usual X + denotes the space X with an added base point.
To obtain the desired decomposition of twisted equivariant we are going to construct an equivalent formulation for such twisted K-groups following the work in [4]. For this suppose that we have a short exact sequence of finite groups Assume that G acts on a compact and Hausdorff space X in such a way that A acts trivially. Fix α ∈ Z 2 (G, S 1 ) a normalized 2-cocycle. Notice that by restriction we can see any G αequivariant vector bundle over X as an A α -equivariant vector bundle, where A α denotes the central extension associated to the cocycle α seen as a cocycle defined on A. Also, recall that associated to an α-representation ρ : ρ)-equivariant vector bundle over X is a G α -vector bundle on which the central factor S 1 acts by multiplication of scalars on E and the map is an isomorphism of A α -vector bundles on which S 1 acts by multiplication of scalars.
In the above definition, if ρ is an α-representation of A then V ρ denotes the trivial A αvector bundle over π 1 : X × V ρ → X. Observe that if p : E → X is a G α -vector bundle on which the central factor S 1 acts by multiplication then, as we are assuming that A acts trivially on X, it follows that for every x ∈ X the fiber E x can be seen as a representation of A α on which the central factor acts by scalar multiplication. Thus for every x ∈ X the fiber E x can be seen as an α-representation of A. With this point of view, a ( G α , ρ)-equivariant vector bundle is a G α -equivariant vector bundle p : E → X such that for every x ∈ X the fiber E x is an α-representation of A isomorphic to a direct sum of the α-representation ρ. Let Vect Gα,ρ (X) denote the set of isomorphism classes of ( G α , ρ)-equivariant vector bundles, where two ( G α , ρ)-equivariant vector bundles are isomorphic if they are isomorphic as G αvector bundles. Notice that if E 1 and E 2 are two ( G α , ρ)-equivariant vector bundles then so is E 1 ⊕ E 2 . Therefore Vect Gα,ρ (X) is a semigroup. Following [4, Definition 2.2] we have the next definition. Definition 3.3. Assume that G acts on a compact space X in such a way that A acts trivially on X and let α be a normalized 2-cocycle α ∈ Z 2 (G, S 1 ). We define K 0 Gα,ρ (X), the ( G α , ρ)-equivariant K-theory of X, as the Grothendieck construction applied to Vect Gα,ρ (X). For n > 0 the group K n Gα,ρ (X) is defined asK 0 Gα,ρ (Σ n X + ). As our next step we show that the previous definition can be described using the usual definition of twisted equivariant K-theory provided in Definition 3.1. For this suppose that α is a normalized 2-cocycle of G with values in S 1 . As above assume that A is a normal subgroup of G and let Q = G/A. Let ρ be an irreducible α-representation such that g · ρ ∼ = ρ for all g ∈ G. Fix a set theoretical section σ : Q → G such that σ(1) = 1 as in the previous section. We can extend σ to obtain a mapσ : Q → G α by definingσ(q) = (σ(q), 1) ∈ G α . Let β ρ,α be the 2-cocycle defined on Q with values in S 1 constructed in Equation (8). With this cocycle we can consider the central extension With this in mind we have the following generalization of [4, Theorem 2.1].
Theorem 3.4. Suppose that ρ is an irreducible α-representation such that g · ρ ∼ = ρ for all g ∈ G. Let X be a G-space such that A acts trivially on X. If p : E → X is a ( G α , ρ)equivariant vector bundle, then Hom Aα (V ρ , E) has the structure of a Q βρ,α -vector bundle on which the central factor S 1 acts by multiplication of scalars. Moreover, the assignment is a natural one to one correspondence between isomorphism classes of ( G α , ρ)-equivariant vector bundles over X and isomorphism classes of Q βρ,α -equivariant vector bundles over X for which the central factor S 1 acts by multiplication of scalars.
Proof. We are only going to provide a sketch of the proof as it follows the same steps used in the proof of [4, Theorem 2.1].
Suppose that p : E → X is a G α -vector bundle. Then Hom Aα (V ρ , E) is a non-equivariant vector bundle over X. Next we give Hom Aα (V ρ , E) an action of Q βρ,α . Suppose that f ∈ Hom Aα (V ρ , E) x . If q ∈ Q we define q • f ∈ Hom Aα (V ρ , E) q·x by where M q ∈ U(V ρ ) is the element chosen in the Section 2.2. It is easy to see that q We conclude that The last equation allows us to define an action of Q βρ,α on Hom Aα (V ρ , E) as follows. If (q, z) ∈ Q βρ,α and f ∈ Hom Unraveling the definitions, it is easy to see that this way Hom Aα (V ρ , E) has the structure of a Q βρ,α -equivariant vector bundle such that the central factor S 1 acts by multiplication of scalars. Suppose now that p : F → X is a Q βρ,α -equivariant vector bundle over X for which the central S 1 acts by multiplication of scalars. Given (g, z) ∈ G α , f ∈ F x and v ∈ V ρ define (9) (g, z) · (v ⊗ f ) := M π(g)ρ (σ(π(g)) −1 (g, z))v ⊗ ((π(g), 1) · f ) ∈ (V ρ ⊗ F ) π(g)·x .
We conclude that the inverse map of the assignment [E] → [Hom Aα (V ρ , E)] is precisely the map defined by the assignment As an immediate corollary of Theorem 3.4 we obtain the following identification of the ( G α , ρ)-equivariant K-theory groups of Definition 3.3 with the β ρ,α -twisted Q-equivariant K-theory groups provided in Definition 3.1.
Corollary 3.5. Let X be a G-space such that A acts trivially on X. Assume that ρ is an α-representation of A such that g · ρ ∼ = ρ for every g ∈ G. Then the assignment defines a natural isomorphism.
Assume that X is a compact and Hausdorff G-space on which A acts trivially. As before we can extend the action of G on X to an action of G α on X in such a way that S 1 acts trivially. Let p : E → X be a G α -equivariant vector bundle on which S 1 acts by scalar multiplication on the fibers. As A α acts trivially on X each fiber of E can be seen as an α-representation of A. Using fiberwise the canonical decomposition theorem for α-representations (Theorem 2.5) we see that the assignment γ : defines an isomorphism of A α -equivariant vector bundles. Using this decomposition we obtain the following theorem.
Theorem 3.6. Under the above assumptions there is a natural isomorphism This isomorphism is functorial on maps X → Y of G-spaces on which A acts trivially.
Proof. The proof of this theorem follows the same lines of the proof of [4, Theorem 3.1] so we only provide an outline of the proof. Let us show first that the map Ψ X is well defined. To see this we have to show that if ρ is an α-representation of A then V ρ ⊗ Hom Aα (V ρ , E) has the structure of a (( G [ρ] ) α , ρ)-vector bundle. Following the notation of Section 2.2 fix a set theoretical section σ : in such a way that σ(1) = 1. Also, for every q ∈ Q [ρ] fix an element M q ∈ U(V ρ ) such that σ(q) · ρ(a) = α(σ(q) −1 a, σ(q))α(σ(q), σ(q) −1 a) −1 ρ(σ(q) −1 aσ(q)) = M −1 q ρ(a)M q . This is possible as σ(q) ∈ G [ρ] so that we have σ(q) · ρ ∼ = ρ. We can choose M 1 = 1 as whereρ andσ are defined in a similar way as in Theorem 3.4. Observe that M (a,z) =ρ(a, z) for all (a, z) ∈ A α . Moreover, given (h, Unraveling the definitions it can be seen that this defines an action of ( G [ρ] ) α on V ρ ⊗Hom Aα (V ρ , E) in such a way that the central factor S 1 acts by multiplication of scalars. This way V ρ ⊗Hom Aα (V ρ , E) has the structure of a ( G [ρ] ) α -vector bundle and A acts by the α-representation ρ on the fibers so that [V ρ ⊗ Hom Aα (V ρ , E)] ∈ K * ( G [ρ] )α,ρ (X). This shows that Ψ X is well defined. Next we show that Ψ X is an isomorphism. For this write Irr α (A) = A 1 ⊔ A 2 ⊔ · · · ⊔ A k , where A 1 , A 2 , . . . , A k are the different G-orbits of the action of G on Irr α (A) defined in the Equation (3). For every 1 ≤ i ≤ k define Note that V τ ⊗ Hom Aα (V τ , E) is an A α -equivariant vector bundle over X, so each E A i is also an A α -equivariant vector bundle over X and the map γ : defines an isomorphism of A α -vector bundles. We are going to show that each E A i is a G α -vector bundle and that the map γ is G α -equivariant. For this fix an index 1 ≤ i ≤ k and an irreducible α-representation ρ : A → U(V ρ ) such that [ρ] ∈ A i . The elements in A i can be written in the form [g 1 · ρ], . . . , [g r i · ρ] for some elements g 1 = 1, g 2 , . . . , g r i ∈ G. Therefore We can give a structure of G α -space on E A i in the following way. Suppose that (g, s) ∈ G α and that v ⊗ f ∈ V ρ ⊗ Hom Aα (V ρ , E) x . Decompose gg j in the form gg j = g l h, where 1 ≤ l ≤ r i and h ∈ G [ρ] . In other words g l is the representative chosen for the coset (gg j )G [ρ] and h = g −1 l gg j . Then (g, s)(g j , 1) = (g l , 1)(h, z) where z = sα(g, g j )α(g l , h) −1 . We define It can be seen that this defines an action of G α ∈ E A i . for each 1 ≤ i ≤ k making the vector bundle E A i into a G α -vector bundle in such a way that the central factor S 1 acts by multiplication of scalars. Furthermore, the map γ : is an isomorphism of A α -vector and the map γ is G α -equivariant so that γ is an isomorphism of G α -vector bundles. Now, the desired map Ψ X can be seen as the direct sum ⊕ k i=1 Ψ i X choosing for each A i a representation [τ i ] ∈ A i . Where each Ψ i X is given by In what follows we will construct the map ζ i : which will be the right inverse of Ψ i X . Take ρ = τ i and consider a vector bundle F ∈ Vect ( G [ρ] )α,ρ (X). Fix g 1 = 1, g 2 , . . . , g r representatives for the different cosets in G/G [ρ] . Let Using ideas similar to the ones used in [4, Theorem 3.1] we can endow L F with the structure of a G α -vector bundle on which the central factor S 1 acts by multiplication of scalars in such a way that the action of ( G [ρ] ) α on [(g 1 , 1) −1 ] * F ∼ = F agrees with the given action of ( G [ρ] ) α on F . We define ) α vector bundles. We obtain at the level of K-theory that ζ i is a right inverse for Ψ i so that ζ = ⊕ r i=1 is a right inverse for Ψ X . In a similar way, using the work given above it can be seen that ζ is also a left inverse so that the map Ψ X is indeed an isomorphism.
To finish, we observe that functoriality follows from the fact that if τ is an α-representation of A then the bundles V τ ⊗ Hom Aα (V τ , f * E) and f * V τ ⊗ Hom Aα (V τ , E) are canonically isomorphic as (( G [τ ] ) α , τ )-equivariant bundles whenever f : Y → X is a G-equivariant map from spaces on which A acts trivially.
As a result of Theorem 3.6 and Corollary 3.5 we obtain the following theorem that is the main result of this article.
Theorem 3.7. Suppose that A is a normal subgroup of a finite group G. Let α be a normalized 2-cocycle on G with values in S 1 and X a compact G-space on which A acts trivially. Then there is a natural isomorphism This isomorphism is functorial on maps X → Y of G-spaces on which A acts trivially.

Examples
In this section we explore some examples of Theorems 3.6 and 3.7 for the dihedral groups D 2n where n ≥ 1 an even integer.
We start by considering first the particular case where G = D 8 . The group D 8 is generated by the elements a, b subject to the relations a 4 = b 2 = 1 and bab = a 3 . Let α : D 8 × D 8 → S 1 be the 2-cocycle defined by α(a l , a j b k ) = 1 and α(a l b, a j b k ) = i j for 0 ≤ j, l ≤ 3 and k = 0, 1.
Note that α is a nontrivial normalized 2-cocycle such that its corresponding cohomology class defines a generator of H 2 (D 8 ; S 1 ) ∼ = Z/2. By Example 2.4, taking n = 4, we know that up to isomorphism D 8 has two irreducible projective α-representations τ 1 and τ 2 defined by for 0 ≤ j, k ≤ 3 and l = 0, 1.
In the above definition we have With this in mind we are going to explore the following examples of Theorems 3.6 and 3.7.
Example 4.1. Suppose first that G = D 8 and A = Z/4 = a . Therefore Let us take X to be the space with only one point * equipped with the trivial D 8 -action. In this case α K * D 8 ( * ) = R α (D 8 ), where R α (D 8 ) denotes the α-twisted representation ring of D 8 . As pointed out above τ 1 and τ 2 are the only irreducible α-representations of D 8 and thus we have an isomorphism of abelian groups On the other hand, observe that α| A is trivial so that As G [1] = G [ρ 2 ] = A and α restricted to A is trivial we have that ( G [1] (Recall thatρ 2 denotes the representation of ( G [ρ 2 ] ) α on which S 1 acts by multiplication of scalars corresponding to ρ 2 and similarly for1). For the representations τ 1 and τ 2 we have Thus as A-representations τ 1 is isomorphic to 1⊕ρ and τ 2 is isomorphic to ρ 2 ⊕ρ 3 . Moreover, in the isomorphism given by Theorem 3.6 we have On the other hand, by Theorem 3.7 we have an isomorphism is the trivial group. Therefore β 1,α and β ρ 2 ,α are the trivial cocycles and Theorem 3.7 gives us the isomorphism = (−1)(−1) = 1 therefore b · σ = 1. In particular the action of D 8 on Irr α (A) is transitive and we can choose [1] as a representative for the set D 8 \ Irr α (A). For the representation 1 we have G [1] = a ∼ = Z/4. If we take again X = * then Theorem 3.6 gives us an isomorphism Ψ : R α (D 8 ) ∼ = Zτ 1 ⊕ Zτ 2 ∼ = → K * ( G [1] )α,1 ( * ).