Chapter 4 – Schur Multiplier, Projective Representations and Characters
Corollary 4.2.7. Let H be a central extension ofG with A, X and α be as in Lemma 4.2.6. Let T be an ordinary representation of H such that the restriction TA is the scalar representation λI for some λ∈Hom(A,C∗),that is
T(a) =
λ(a)
λ(a) . ..
λ(a)
n×n
∀a∈A,
where n= deg(T). Define P(g) =T(xg) for g ∈G. Then (P, λ(α)) is a projective representation of G, where λ(α)(g, h) = λ(α(g, h)). Furthermore, P is irreducible if and only if T is and the equivalence class ofP is independent of the choice of coset representatives X.
PROOF. See Isaacs [38].
Remark 4.2.5. Note that ifT is an ordinary irreducible representation of H then the condition that TA be scalar representation is satisfied by the Schur’s lemma (see Moori [49]), since A lies in the center of H.
Thus it makes sense to say a conjugacy class of Gto be α-regular if every element in this class is α-regular.
We recall from Proposition 3.1.1 that all the ordinary characters of a finite group G are class functions. However, this no longer true for projective characters. For projective characters we have the following proposition.
Proposition 4.3.1. Let IrrProj(G, α−1) ={ξ1, ξ2,· · ·, ξt} be set of all the projective characters of G with associated factor set α and suppose that ξ ∈ IrrProj(G, α−1). If for any α-regular element x in G and for any y in G(not necessarily α-regular), the relation α(x, y) =α(y, y−1xy) satisfied, thenξ is a class function.
PROOF. Refer to Proposition 2.2(iii) of Karpilovsky [43].
The following theorem gives a criterion to see whether two projective representations P1 and P2
with factor set α are linearly equivalent or not.
Theorem 4.3.2. Two projective representationsP1 andP2 with factor setαare linearly equivalent if and only if they have the same projective character.
PROOF. See Theorem 4.4 of Morris [51].
If P is a projective representation of G with projective character ξ then there is an ordinary representation T of C(G) such that P(g) = T(xg) for g ∈ G. Let χ be the character of C(G) afforded byT, thenξ(g) =χ(xg) for all g∈G.
We have seen that in Theorem 3.2.3 that the ordinary characters of a group G satisfy certain row and column orthogonality relations. The Projective characters of G also satisfy the usual orthogonality relations.
Theorem 4.3.3. Let IrrProj(G, α−1) be as in Proposition 4.3.1 and let {C1, C2,· · · , Cm} be the collection of the α-regular conjugacy classes of G with gi being a representative of Ci, ∀i ∈ {1,2,· · · , m}. Then
(i) |IrrProj(G, α−1)|= the number of α-regular conjugacy classes of G (i.e., t=m).
(ii)
t
X
i=1
ξi(gj)ξi(gk) =δjk|CG(gj)| for j, k∈ {1,2, . . . , t}.
Chapter 4 – Schur Multiplier, Projective Representations and Characters
(iii) An elementg of G isα-regular if and only if there is an irreducible projective character ξ of G with factor setα such thatξ(g)6= 0.
PROOF. See Haggarty and Humphreys [32].
Now let G0 be the set of all α-regular elements of the groupG. We have the following theorem.
Theorem 4.3.4. Let IrrProj(G, α−1) = {ξ1, ξ2,· · · , ξt} be as before and let {C1, C2,· · · , Ct} be the collection of the α-regular conjugacy classes of G with gi being a representative of Ci, ∀i ∈ {1,2,· · · , t}.Then
X
g∈G0
ξi(g)ξj(g) =|G|δij.
PROOF. See Karpilovsky [43].
We conclude this chapter by remarking that to find the set IrrProj(G, α−1), it is not necessarily to find the full covering group. Rather another group L can be constructed, where every ξ ∈ IrrProj(G, α−1) can be lifted to an ordinary character of L. This is has been shown in Haggarty and Humphreys [32]. To construct L, suppose that α is a factor set of G, such that o([α]) = r in the Schur multiplierM(G).Letω be anrthroot of unity and letα0be a representative of [α] whose values are powers ofω.For all g, h∈G, we defineα0(g, h) byα0(g, h) =ωα(g,h). Now for allg∈G, let
L= D
x, xg|o(x) =r, xixgxjxh=xi+jxα(g,h)xgh
E .
Then L is a quotient of the representation group H and any ξ ∈IrrProj(G, α−1) can be lifted to an ordinary representation ofL.Thus the set IrrProj(G, α−1) can be determined from the ordinary character table ofL.
The Theory of Clifford-Fischer Matrices
The theory of Clifford-Fischer matrices, which is based on Clifford Theory (see Clifford [18]), was developed by B. Fischer ([25], [26] and [27]). This technique, which is used to construct the character tables of group extensions, has also been discussed and applied to both split and non-split extensions in several publications, for example see Ali and Moori [2], [3], Barraclough [6], Basheer and Moori [8], [9], [10], [11], [12], [13], [14], Fischer [27], Moori [47], Moori and Mpono [50], Pahlings [57], Rodrigues [62], Seretlo [69], Whitely [70] and in a recent book by K. Lux and H. Pahlings [58].
5.1. The Clifford Theory
Definition 5.1.1. Let N be any normal subgroup of G (not necessarily abelian) and let θ be any character of N. For g ∈ G, define another character θg of N by θg(n) =θ(gng−1), ∀n∈N. The character θg is called a conjugate character of θ.
Remark 5.1.1. From the above definition, it is not difficult to see that θg ∈Irr(N) if and only if θ ∈ Irr(N). Also it follows that G acts on Irr(N) by conjugation and since N acts trivially on Irr(N), Irr(N) is permuted by G/N,by gN :θ7→θg.
The groupGhas dual actions on the conjugacy classes ofN and on Irr(N).The following important result by Brauer relates the number of orbits on the two actions.
Theorem 5.1.1(Brauer’s Theorem). The number ofG−orbits onIrr(N)is same as the number of G−orbits on the conjugacy classes ofN.
Chapter 5 – The Theory of Clifford-Fischer Matrices
PROOF. See Lemma 4.5.2 of Gorenstein [31] or Theorem 5.1.5 of Mpono [55].
Remark 5.1.2. Brauer Theorem affirms that when the group G act on Irr(N), it produces the same number of orbits as whenGact on the conjugacy classes ofN,but the orbit lengths may be different. Indeed ifN is non-abelian, then
t
X
k=1
|θkG|=|Irr(N)| 6=|N|=
t
X
k=1
|[nk]GN|, where
• tis the number of orbits for the action of Gon the conjugacy classes ofN or on Irr(N),
• θk and nk are respective representative character and element of a conjugacy class of N,
• θkGand [nk]GN are orbits ofN containingθkandnk,respectively, for the action ofGon Irr(N) and on the conjugacy classes of N respectively.
Definition 5.1.2. Let G be a group and K ≤G be any subgroup. Then for a character χ of K, we define
IG(χ) ={g∈NG(K)|χg=χ},
where χg is defined in a similar manner to Definition 5.1.1. We call IG(χ) the inertia group of χ in G. If K is normal inG, then
IG(χ) ={g∈G|χg =χ}.
Now if N EG and Hk is the inertia group of a character θk ∈ Irr(N), it is easy to show that N EHk ≤ G. The quotient Hk ∼= Hk/N is referred to as the inertia factor group. The group Hk can be regarded as the inertia group of θk in the factor group G/N ∼= G. That is Hθk =Hk={g∈G|θkg=θk}.
The following theorem, due to Clifford [18], is very important.
Theorem 5.1.2 (Clifford Theorem (1937)). Let χ ∈Irr(G) and let θ1, θ2,· · · , θt be represen- tatives of orbits ofGonIrr(N).Fork∈ {1,2,· · ·, t},letθGk ={θk=θk1, θk2,· · · , θksk} and letHk be the inertia group in Gof θk.Then for some fixed k∈ {1,2,· · ·, t} we have
χ↓GN =ek
sk
X
u=1
θku, where ek=D
χ↓GN, θkE
. (5.1)
Moreover, for fixed k Irr(Hk, θk) :=
n
ψk∈Irr(Hk)| D
ψk↓HNk, θk E
6= 0o
←→n
χ∈Irr(G)| D
χ↓GN, θk E
6= 0o
:= Irr(G, θk) under the map ψk 7−→ψk↑G
Hk.
PROOF. We only show the first assertion of the theorem, namely Eq. (5.1). For the second part of the theorem, readers can refer to either Theorem 4.1.7 of Ali [1] or Theorem 3.3.2 of Whitely [70]
and the readers have to be careful as there are differences in the notations. To verify Eq. (5.1), we compute (for fixedk∈ {1,2,· · · , t}) the character θk↑GN↓GN.For this, defineθ0k onG by
θk0(x) =
θk(x), ifx∈N, 0 ifx6∈N.
For n∈N, we haveθk↑GN(n) = |N1|X
x∈G
θ0k(xnx−1).Since xnx−1 ∈N, ∀x ∈G, we haveθk↑GN(n) =
1
|N|
X
x∈G
θxk(n).Thus |N|θk↑GN↓GN = X
x∈G
θxk, and if θl ∈Irr(N) such that l 6=k ( i.e., θl 6∈ {θku| 1≤ u≤sk} then 0 =
* X
x∈G
θkx, θl
+ ,so
D
θk↑GN↓GN, θl
E
= 0.Now since χ is an irreducible constituent of θk↑GN by Frobenius reciprocity, it follows thatD
χ↓GN, θlE
= 0.Hence all the irreducible constituents ofχ↓GN are among theθku, 1≤u≤sk.Soχ↓GN =
sk
X
u=1
Dχ↓GN, θkuE
θku.ButD
χ↓GN, θkuE
=D
χ↓GN, θkE
sinceθku and θk are conjugate, so the proof is complete.
Definition 5.1.3. Let N CG. A character θ∈Irr(N) is said to be extendable to a character of its inertia group H =IG(θ) if there existsψ∈Irr(H) such thatψ↓HN =θ.
Theorem 5.1.3. Further to the settings of Theorem 5.1.2, assume that for eachk∈ {1,2,· · ·, t}, there exists ψk∈Irr(Hk, θk) such that ψk↓HNk =θk.Then
Irr(G) =
t˙
[
k=1
n
(ψkinf(ζ))↑GH
k|ζ ∈Irr(Hk/N)o
. (5.2)
PROOF. See Ali [1] or Whitley [70].
Remark 5.1.3. It is by no means necessarily the case that there exists an extension ψk of θk to the inertia group Hk (that is the case Irr(Hk, θk) = ∅,the empty set, or @ ψ ∈Irr(Hk) such that ψ↓HNk =θk, is feasible). However, there is always a projective extensionψek∈IrrProj(Hk, α−1k ) for
Chapter 5 – The Theory of Clifford-Fischer Matrices
some factor setαk of the Schur multiplier ofHk.Thus the more proper formula for Equation (5.2) is (see Remark 4.2.7 of Ali [1])
Irr(G) =
t˙
[
k=1
n
(ψekinf(ζ))↑GH
k|ψek ∈IrrProj(Hk, α−1k ), ζ∈IrrProj(Hk/N, α−1k ) o
, (5.3)
where the factor setαkis obtained fromαkas described in Corollary 7.3.3 of Whitely [70]. Hence the character table ofGis partitioned into tblocks K1,K2,· · · ,Kt,where each blockKk of characters (ordinary or projective) is produced from the inertia subgroupHk.
Note 5.1.1. Observe that if αk ∼ [1] in Equation (5.3), then we get Equation (5.2). That is IrrProj(Hk,1) = Irr(Hk) and IrrProj(Hk,1) = Irr(Hk).
By convention we take θ1 = 1N, the trivial character of N. Thus Hθ1 = H1 = G and thus H1/N ∼=G. Since {1G} ⊆Irr(G,1N) and such that 1G↓GN =1N (this means that the character θ1
is extendable to a character of its inertia group G), the blockK1 will consists only of the ordinary irreducible characters of G.
In the rest of this section, we quote some results on extendability of characters of N to their respective inertia groups. We start with the following two well known results.
Theorem 5.1.4 (Mackey’s Theorem (1951)). Let G =N:G be a split extension and assume thatN is abelian. Ifθ∈Irr(N) is invariant in G(that is θg=θ, ∀ g∈G) then θcan be extended to a linear character of G.
PROOF. We follow the proof given by Whitely [70]. Let G = N:G be a split extension with N abelian. It follows that any g∈G can be expressed uniquely asg =ng,where n∈N and g ∈G.
Define χ on G by χ(ng) = θ(n). Since N is abelian, θ has degree 1 and thus is linear. The fact that θ is invariant in G implies that θ(n) = θ(gng−1), ∀ g ∈ G. Now if g1 =n1g1 and g2 =n2g2
we obtain that
χ(g1g2) = χ(n1g1n2g2) =χ(n1ng21g1g2) =θ(n1ng21)
= θ(n1)θ(ng21) =θ(n1)θ(n2) =χ(g1)χ(g2).
Thereforeχ is a linear character of Gsuch thatχ↓GN =θ.
Theorem 5.1.5 (Gallagher Theorem (1962)). Let G = N·G be any extension such that gcd(|N|,|G|) = 1.Then every G−invariant character of N is extendable to a character of G.
PROOF. See Theorem 6 of Gallagher [29].
The above two results by Mackey and Gallagher are corollaries of a more general result by Karpilovsky [42] which we state without proof.
Theorem 5.1.6. Let the groupGcontain a subgroupGsuch thatG=N GforN normal inGand let θ∈Irr(N) be invariant in G. Then θ extends to an irreducible character of G if the following conditions hold:
• gcd(deg(θ),|G|) = 1,
• NT
G≤N0,where N0 is the derived subgroup of N.
PROOF. See Karpilovsky [42].
Another extension result is given by the following theorem.
Theorem 5.1.7. Let N CG and suppose thatθ∈Irr(N) such thatθ is invariant in G. Thenθ is extendable to an ordinary character ofG if gcd
[G:N],deg(θ)|N|
= 1.
PROOF. See Gagola [28].
The following theorem is very important and will be used later on in Subsection 10.3.3.
Theorem 5.1.8. Suppose that G is a splitting extension of N by some group G. Then any linear character θ∈Irr(N) is extendable to its inertia group IG(θ).
PROOF. We follow the proof given by Mpono [55]. Let G = N:G and θ ∈ Irr(N) be linear. Also let H = IG(θ). Then we obtain that H = N:H, where H = IG(θ). Since H is a split extension, we obtain that NT
H ={1} ≤N0.Also we have that (deg(θ),|H|) = (1,|H|) = 1 and clearlyθ is H−invariant. Now by applications of Theorem 5.1.14 of Mpono [55], it follows that the character
θ is extendable to an ordinary character ofH.
Remark 5.1.4. Note that Mackey Theorem is reinforced by Theorem 5.1.8 since for N abelian, all the irreducible characters are linear and hence are extendable to their inertia groups.
There are many other results for extension of characters from a normal subgroup N of Gto their inertia groups H, for example see Ali [1], Karpilovsky [42], Mpono [55] or Whitely [70].
Chapter 5 – The Theory of Clifford-Fischer Matrices