Journal of Algebra 228, 633᎐642 Ž2000.
doi:10.1006rjabr.2000.8290, available online at http:rrwww.idealibrary.com on
A Characterization of the Common Multiples of the
Degrees of the Absolutely Irreducible Representations
of a Semisimple Algebra and Applications
Manfred Leitz
Fachbereich Informatik und Mathematik, Fachhochschule Regensburg,
Postfach 12 03 27, D-93025 Regensburg, Germany
E-mail: manfred.leitz@mathematik.fh-regensburg.de
Communicated by Walter Feit
Received June 29, 1998
1. INTRODUCTION
In the following, the term semisimple algebra always means a semisimple finite dimensional algebra over the field C of the complex numbers.
The aim of this paper is to give a proof of the following theorem which
characterizes the set of the common multiples of the degrees of the
irreducible characters of a semisimple algebra. Moreover, concrete applications of this theorem shall be demonstrated.
1.1. THEOREM. Let A be a finite dimensional semisimple algebra o¨ er the
field C of the complex numbers, and let 1 be the unit element of A. The set of
the irreducible characters of A is denoted by IrrŽ A., and ␳ denotes the regular
character of A. Furthermore, let bi 4 and bXi 4 be dual bases of A with respect
to ␳ such that ␳ Ž bXi bj . s ␦ i j Ž Kronecker delta.. Define the element s␳ g A by
s␳ [
Ý bXi bXj bi bj .
i, j
If m g N is any fixed positi¨ e integer, then
m2 ny1 ⭈ ␳ Ž s␳n . g alg.int. Ž C . ᭙ n g N m ␹ Ž 1 . ¬ m ᭙␹ g Irr Ž A . .
Ž Here alg.int.ŽC. denotes the ring of all algebraic integers..
Proof. See Section 3.
633
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634
MANFRED LEITZ
Remarks. Ži. Note that by the above equivalence the set of the common
multiples of the set c.d.Ž A. [ ␹ Ž1. : ␹ g IrrŽ A.4 is on the whole independently of terms of representation theory characterized. ŽThe term
regular character may be associated with linear algebra as well as with
representation theory..
Žii. It is always true that ␳ Ž s␳n . is a positive rational number Žsee
Proposition 3.5Žii.., and hence in Theorem 1.1 the set alg.int.ŽC. could be
replaced by the set N of the positive integers. Since we are mainly
interested in applications of the direction « , it will be better to retain
the formulation such as given.
2. APPLICATIONS OF THEOREM 1.1
Throughout this section, let G be a finite group with the identity
element e. A familiar result says that the degree of each irreducible
complex projecti¨ e representation of G divides the order of G Žsee w1,
Theorem 53.16; 6, Corollary 11.18x.. The main point of the standard proof
of this result is the reduction of the problem to ordinary irreducible
representations by introduction of so-called representation groups, due to
Schur Žsee w1, Theorem 53.7; 11x., which requires considerable effort.
Another proof which avoids representation groups has been given by
Fossum w4, Sect. 4x, but this proof contains a lot of algebraic number
theory. We will give a new proof by means of Theorem 1.1.
Let ␣ : G = G ª C_ 04 be a so-called factor set Žsee w6, Definition
11.4x., thus ␣ Ž xy, z . ⭈ ␣ Ž x, y . s ␣ Ž x, yz . ⭈ ␣ Ž y, z . for all x, y, z g G. Furthermore, let A [ Cw G, ␣ x s [g g G C ⭈ bg , bx ⭈ by s ␣ Ž x, y . ⭈ bx y Ž x, y g
G . be the so-called twisted group algebra with respect to the factor set ␣ .
As is well known, A s Cw G, ␣ x is semisimple because the regular character induces on A a nondegenerate symmetric associative bilinear form Žsee
w5, V Hilfssatz 2.6x.. The following theorem is the equivalent of the
theorem on the degrees of the irreducible projective representations of G
mentioned above Žsee w6, p. 177x..
2.1. THEOREM Žm w1, Theorem 53.16x.. Let A s Cw G, ␣ x be the twisted
group algebra of the group G relati¨ e to the factor set ␣ , and let 1 be the unit
element of A. Then the degree ␹ Ž1. of each irreducible character ␹ of A
di¨ ides the order of G.
Before giving the proof of Theorem 2.1, we state some elementary facts
Žsee w3, pp. 136᎐137; 5, p. 632x.. ŽWe omit the evident proofs..
2.2. LEMMA. Let A s Cw G, ␣ x be the twisted group algebra of G relati¨ e
to the factor set ␣ with the basis bg : g g G4 Ž bx ⭈ by s ␣ Ž x, y . ⭈ bx y for all
635
DEGREES OF REPRESENTATIONS
x, y g G .. Note 1 denotes the unit element of A, and e denotes the identity
element of G. Then:
Ži. ␣ Ž g, e . s ␣ Ž e, g . s ␣ Ž e, e . Ž g g G ..
Žii. ␣ Ž gy1 , g . s ␣ Ž g, gy1 . Ž g g G ..
Žiii. 1 s ␣ Ž e, e .y1 ⭈ be .
Živ. by1
s Ž ␣ Ž gy1 , g . ⭈ ␣ Ž e, e ..y1 ⭈ bgy1 Ž g g G ..
g
2.3. LEMMA. Let notations be as in Lemma 2.2, and let ␳ denote the
regular character of A.
Ži. Then
␳ Ž bg . s ␦ g , e ⭈ ␣ Ž e, e . ⭈ < G < Ž g g G . ,
Žii.
Then
␳ Ž 1. s < G < .
If g g G, let bXg [ < G <y1 ⭈ by1
s Ž< G < ⭈ ␣ Ž gy1 , g . ⭈ ␣ Ž e, e ..y1 ⭈ bgy1 .
g
␳ Ž bXx ⭈ by . s ␦ x , y
Ž x, y g G . ,
i.e., bg 4 and bXg 4 are dual bases with respect to ␳ .
Proof of Theorem 2.1. In accordance with Theorem 1.1, we regard in
A s Cw G, ␣ x the element
s␳ [
bXx bXy bx by .
Ý
Ž x , y .gG=G
Then by Lemma 2.3Žii., we have
w␳ [ < G < 2 ⭈ s␳ s
Ý
y1
by1
x b y bx b y .
Ž x , y .gG=G
According to Theorem 1.1, we have to show that the number
< G < 2 ny1 ⭈ ␳ Ž s␳n . s < G <y1 ⭈ ␳ Ž w␳n . s
Ý
Žx , y.gG n=G n
< G <y1 ⭈ ␳
n
ž
y1
Ł by1
x b y bx b y
is1
i
i
i
i
/
is an algebraic integer, for all n g N.
From now on, we regard an arbitrary fixed n g N and an arbitrary fixed
y1
. Ž y1 y1
.
pair Žx, y. g G n = G n. If Ž xy1
1 y 1 x 1 y 1 ⭈⭈⭈ x n yn x n yn / e, then, by
n
y1
.
Lemma 2.2Živ. and Lemma 2.3Ži., it follows that ␳ ŽŁ is1 bx i by1
y i bx i b y i s 0.
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MANFRED LEITZ
y1
. Ž y1 y1
.
Ž .
Ž .
If Ž xy1
1 y 1 x 1 y 1 ⭈⭈⭈ x n yn x n yn s e, then, by Lemma 2.2 iii and iv , we
have
n
␬x, y [
y1
Ł by1
x b y bx b y
is1
i
i
i
i
s c ⭈ 1,
Ž ).
where c is a complex number, and by Lemma 2.3Ži., < G <y1 ⭈ ␳ Ž ␬ x, y . s c. It
suffices to show that c is an algebraic integer. Let D␳ : A ª Hom C Ž A, A.
be the left regular representation of A, and let Det : Hom C Ž A, A. ª C be
the determinant. Application of the composed mapping Det(D␳ to Eq. Ž).
yields c <G < s 1.
Remark. Note that in the above proof, it was not necessary to know
that ␣ is equivalent to a factor set ␤ whose values are roots of unity Žsee
w2, Lemma 11.38; 3, Theorem 25.3x..
Still let G be a finite group with the identity e. Up to the end of this
section, let A s Cw G x be the ordinary group algebra, and let IrrŽ G . be the
set of all irreducible complex characters of G. By the following theorem,
the common divisors of the co-degrees < G <r␹ Ž e . Ž ␹ g IrrŽ G .. are independently of terms of representation theory characterized.
2.4. THEOREM Žcompare w8, Satz 1x.. For x, y g G let w x, y x [ xy1 yy1 xy
be the commutator of x and y, and for positi¨ e integers n g N let L nŽ e . denote
the set of solutions of the equation Ž within the set G n = G n .
w x 1 , y 1 x ⭈ w x 2 , y 2 x ⭈⭈⭈ w x n , yn x s e.
Let d g N di¨ ide the order of G. Then
d 2 ny1 L n Ž e . ᭙ n g N m d
<G<
␹ Ž e.
᭙␹ g Irr Ž G . .
Proof. We use as a pair of dual bases Žwith respect to ␳ . G and
< G <y1 ⭈ gy1 : g g G4 . Then, in the notation of Theorem 1.1, we have
s␳ s < G <y2 ⭈ ÝŽ x, y.g G=G w x, y x. Apparently, the identity
<G<
ž /
d
2 ny1
⭈ ␳ Ž s␳n . s
Ln Ž e .
d 2 ny1
Ž n g N.
holds, and now we are done by Theorem 1.1.
Remark. If N is a normal abelian subgroup of G, then it can be shown
without use of further representation theory that < N < 2 ny1 ¬ < L nŽ e .< holds
for all n g N Žsee w8, p. 278x., and in consequence of this, we obtain a well
known classical result, due to Ito
ˆ Žsee w6, Theorem 6.15x..
DEGREES OF REPRESENTATIONS
637
3. PROOF OF THEOREM 1.1 AND RELATED RESULTS
In this section, we will prove a generalization ŽTheorem 3.1. of Theorem
1.1 and a related result ŽTheorem 3.7. which can be done without additional effort. First of all, we fix frequently used notations.
Notations.
N, Z, Q, C
R
A
1
bi 4
␳
IrrŽ A.
␺ Ž1.
␻␺
e␺
set of positive rational integers, rational integers, rational
numbers, and complex numbers
R [ alg.int.ŽC., ring of all algebraic integers
semisimple finite dimensional algebra over the field C
unit element of A
basis of A
regular character of A
set of the irreducible characters of A
degree of the character ␺ g IrrŽ A.
␻␺ [ ␺ Ž1.y1 ⭈ ␺ Ž ␺ g IrrŽ A..
central primitive idempotent associated with ␺ g IrrŽ A.
Let ␩ g Hom C Ž A, C. be a symmetric linear function on A, i.e., ␩ Ž xy . s
␩ Ž yx . for all x, y g A. Then, as is well known, ␩ can be expressed in a
unique way as a linear combination of the set IrrŽ A. Žwith coefficients in
C., ␩ s Ý␺ g IrrŽ A. c␺ ⭈ ␺ . The symmetric associative bilinear form Ž x, y . g
A = A ª ␩ Ž xy . g C associated with ␩ is nondegenerate if and only if
c␺ / 0 holds for all ␺ g IrrŽ A. Žsee w2, Lemma 9.7x.. Then the dual basis
bXi 4 Žwith respect to ␩ . associated with the basis bi 4 of A exists such that
␩ Ž bXi bj . s ␦ i j .
3.1. THEOREM. Let ␩ s Ý␺ g IrrŽ A. f␺ ⭈ ␺ be a character of A, where
f␺ g N holds for all ␺ g IrrŽ A.. ŽThus it is f␺ / 0 for all ␺ g IrrŽ A...
Furthermore, let bi 4 and bXi 4 be dual bases of A with respect to ␩ , i.e.,
␩ Ž bXi bj . s ␦ i j . Define the element s␩ g A by
s␩ [
Ý bXi bXj bi bj .
i, j
If m g N is a fixed positi¨ e integer, then
m2 ny1 ⭈ ␩ Ž s␩n . g R ᭙ n g N m f␹ ¬ m ᭙␹ g Irr Ž A . .
The proof of Theorem 3.1 which will be given after the proof of Lemma
3.6 results from a combination of Proposition 3.5Žii. and of Lemma 3.6
638
MANFRED LEITZ
below. Preparatory to the proof of Proposition 3.5, we need Proposition 3.3
and Lemma 3.4.
3.2. DEFINITION Žsee w1, p. 481x.. Let notations and assumptions be as
in Theorem 3.1. The so-called Gaschutz᎐Ikeda
operator Žwith respect to
¨
␩ . is the C-linear mapping ␨␩ : A ª A, where
␨␩ Ž x . [
Ý bXi xbi
Ž x g A. .
i
We are sure that the following proposition is completely known though
we could not find Žiii. and Živ. exactly in full generality in the literature
Žsee w1, p. 481; 10, Satz 5 Ž1., Ž4.x..
3.3. PROPOSITION. Let notations and assumptions be as in Theorem 3.1
and in Definition 3.2. For the Gaschutz᎐Ikeda
operator ␨␩ the following
¨
holds:
Ži.
respect to
Žii.
Žiii.
Živ.
The map ␨␩ does not depend on the choice of dual bases with
␩.
␨␩ Ž A. : centerŽ A..
␳ s ␩ ( ␨␩ Ž composition of mappings..
␨␩ Ž x . s Ý␺ g IrrŽ A. f␺y1 ⭈ ␺ Ž x . ⭈ e␺ Ž x g A..
Proof. For Ži. and Žii., see w1, pp. 481, 482x.
Žiii. For each y g A, we have y s Ý j␩ Ž ybXj . ⭈ bj . Replacing y by xbi
yields xbi s Ý j␩ Ž xbi bXj . ⭈ bj for all i, so by definition of the regular character,
␳Ž x. s
Ý ␩ Ž xbi bXi . s Ý ␩ Ž bXi xbi . s ␩ Ž ␨␩ Ž x . . .
i
i
Živ. In advance, we mention the equations
zs
Ý
␺ gIrr Ž A .
␻␺ Ž z . ⭈ e␺ s
Ý
␺ Ž 1.
␺ gIrr Ž A .
y1
⭈ ␺ Ž z . ⭈ e␺
␺ Ž u . s f␺y1 ⭈ ␩ Ž ue␺ . , ␳ Ž ue␺ . s ␺ Ž 1 . ⭈ ␺ Ž u .
␨␩ Ž u . ⭈ e␺ s ␨␩ Ž ue␺ .
Ž z g center Ž A . . ,
Ž u g A, ␺ g Irr Ž A . . ,
Ž u g A, ␺ g Irr Ž A . . .
ŽThe first of the above equations follows from Schur’s Lemma, the second
from ␹ Ž ue␺ . s 0 if ␺ / ␹ g IrrŽ A., and the third from e␺ g centerŽ A...
From these equations and Proposition 3.3Žiii. which has just been shown
639
DEGREES OF REPRESENTATIONS
follows
␨␩ Ž x . s
Ý
␺ Ž 1.
Ý
␺ Ž 1.
Ý
␺ Ž 1.
Ý
␺ Ž 1.
y1
␺ gIrr Ž A .
s
y1
⭈ f␺y1 ⭈ ␩ Ž ␨␩ Ž x . ⭈ e␺ . ⭈ e␺
y1
⭈ f␺y1 ⭈ ␩ Ž ␨␩ Ž xe␺ . . ⭈ e␺
y1
⭈ f␺y1 ⭈ ␳ Ž xe␺ . ⭈ e␺ s
␺ gIrr Ž A .
s
␺ gIrr Ž A .
s
⭈ ␺ Ž ␨␩ Ž x . . ⭈ e␺
␺ gIrr Ž A .
f␺y1 ⭈ ␺ Ž x . ⭈ e␺ .
Ý
␺ gIrr Ž A .
3.4. LEMMA Žsee w2, Proposition 9.17x.. The following equation holds,
e␺ s f␺ ⭈
Ý ␺ Ž bXi . ⭈ bi
Ž ␺ g Irr Ž A . . .
i
Proof. e␺ s Ý i␩ Ž e␺ bXi . ⭈ bi s Ý i f␺ ⭈ ␺ Ž bXi . ⭈ bi .
Special cases of the following Proposition 3.5Ži. can be found in w7,
Theorem 1; 8, Lemma; 10, p. 42x.
3.5. PROPOSITION. Let notations and assumptions be as in Theorem 3.1.
The element s␩ s Ý i, j bXi bXj bi bj is in the center of the algebra A, and the
following equations hold.
Ž i.
Ž ii .
s␩ s
m
1
Ý
f2
␺ gIrr Ž A . ␺
2 ny1
⭈ ␩Ž
s␩n
⭈ e␺ ,
.s
Ý
s␩n s
1
Ý
␺ gIrr Ž A .
␺ Ž 1. ⭈
␺ gIrr Ž A .
m
f␺2 n
⭈ e␺
Ž n g N. .
2 ny1
Ž m, n g N . .
ž /
f␺
Proof. In the following calculation, we use Definition 3.2, Proposition
3.3Živ., and Lemma 3.4,
s␩ s
Ý bXi bXj bi bj s Ý ␨␩ Ž bXj . ⭈ bj s Ý
i, j
s
j
Ý
␺ gIrr Ž A .
f␺y2 ⭈ f␺ ⭈
ž
j
Ý ␺ Ž bXj . ⭈ bj
j
/
ž
Ý
␺ gIrr Ž A .
⭈ e␺ s
f␺y1 ⭈ ␺ Ž bXj . ⭈ e␺ ⭈ bj
/
Ý
␺ gIrr Ž A .
f␺y2 ⭈ e␺ .
640
MANFRED LEITZ
Taking the nth power yields the equations in Ži.. Especially, we have
s␩ g centerŽ A.. Multiplication of s␩n by m2 ny1 and application of ␩ to the
resulting equation yields the equation in Žii..
3.6. LEMMA Žsee w9, Lemma 1x.. Let qi Ž i g I . be finitely manyᎏnot
necessarily differentᎏpositi¨ e rational numbers. Then all the qi Ž i g I . are
integers if and only if for each n g N the power sum sum n [ Ý i qin is an
integer.
Proof. Let sum n g N for all n g N. We write the numbers qi as
fractions qi s m irt with suitable numbers m i g N and the least common
denominator t g N. If t ) 1, then there is a prime p, where p ¬ t. We
choose a positive integer r, where p r ) < I < and set n [ Ž p y 1. ⭈ p ry1.
Then we have n G r and t n ⭈ sum n ' 0 Žmod p r .. If I p [ i g I : p ¦ m i 4 ,
we have on the other hand by Euler t n ⭈ sum n s Ý i m ni ' < I p < Žmod p r .. At
all < I p < ' 0 Žmod p r ., and hence I p s ⭋ Žbecause of p r ) < I < G < I p <.. But,
then t cannot be the least common denominator.
Proof of Theorem 3.1. Ž¥. This is evident by Proposition 3.5Žii..
Ž«. Multiplication of the equation in Proposition 3.5Žii. by the positive
integer mX [ l.c.m. f␺ : ␺ g IrrŽ A.4 yields for m, n g N
X
m ⭈m
2 ny1
⭈ ␩Ž
s␩n
.s
␺ Ž 1. ⭈ m ⭈
Ý
␺ gIrr Ž A .
mX
f␺
⭈
m
2
žž / /
f␺
ny 1
.
Ž )).
If m2 ny1 ⭈ ␩ Ž s␩n . g R, then, by the above equation, we have mX ⭈ m2 ny1 ⭈
␩ Ž s␩n . g R l Qqs N. If the last holds for a fixed m g N and all n g N,
then, by Lemma 3.6, Ž mrf␺ . 2 g N. Thus mrf␺ g N for all ␺ g IrrŽ A.. ŽIn
applying Lemma 3.6, note that in Eq. Ž)). the summand corresponding to
the index ␺ of summation again can be written as a sum of ␺ Ž1. ⭈ m ⭈ mXrf␺
equal powers..
It may be of interest to obtain theorems like Theorem 1.1 or Theorem
3.1, where not all the ␺ g IrrŽ A. are under consideration.
3.7. THEOREM. Let ␩ s Ý␺ g IrrŽ A. f␺ ⭈ ␺ be a generalized character of A
Ž i.e., f␺ g Z ᭙␺ ., where f␺ / 0 for all ␺ g IrrŽ A., and let bi 4 and bXi 4
be dual bases of A with respect to ␩ Ž␩ Ž bXi bj . s ␦ i j .. Define the element s␩
g A by
s␩ [
Ý bXi bXj bi bj .
i, j
641
DEGREES OF REPRESENTATIONS
Furthermore, let ␶ s Ý␺ g IrrŽ A. f˜␺ ⭈ ␺ be a character of A and Irr␶ Ž A. [ ␺
g IrrŽ A. : f˜␺ / 04 . If m g N is a fixed positi¨ e integer, then
␶
žŽm
2
⭈ s␩ .
n
/ g R ᭙n g N m f
␹
¬ m ᭙␹ g Irr␶ Ž A . .
Proof. The equation in Proposition 3.5Ži. remains correct with the
more general assumptions concerning ␩. Multiplication of this equation by
m2 n and application of ␶ to the resulting equation yields
␶
žŽm
2
⭈ s␩ .
n
/s
Ý
␺ gIrr Ž A .
␺ Ž 1 . ⭈ f˜␺ ⭈
m
2
n
žž / /
f␺
Ž m, n g N . . Ž ))).
From this, the assertion follows by Lemma 3.6.
The following Corollary 3.8 is related more or less to w12, Theorems 1
and 3x.
3.8. COROLLARY. Let H be a subgroup of the finite group G with the
identity e. For x, y g G, let w x, y x [ xy1 yy1 xy denote the commutator of x
and y, and for positi¨ e integers n g N, let LHn denote the set of solutions Ž in
the set G n = G n = G = H . of the equation
zy1 ⭈ w x 1 , y 1 x ⭈ w x 2 , y 2 x ⭈⭈⭈ w x n , yn x ⭈ z s h.
Let d g N di¨ ide the order of G. Then
< H < ⭈ d 2 n < LHn < ᭙ n g N m d
<G<
␹ Ž e.
᭙␹ g Irr Ž G . , where Ž 1GH , ␹ . G / 0.
Ž Note that 1GH is the character of G induced from the principal character 1 H
of H..
Proof. In Theorem 3.7, choose ␩ s ␳ Žregular character ., ␶ s 1GH , and
m s < G <rd. The carrying out of the technical details is left to the reader.
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642
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