ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXIV (1984)
M. B. W
a n k e-J
akubow skaand M. E. W
a n k e-J
erie(Wroclaw)
On representations of n-groups
1. Introduction. O ur aim here is to describe the notion of a representation of
«-group, which is analogous to the notion of a group representation. We define a diagonal «-group (Section 2) and its representation g in the diagonal «-group of the reversible linear transformations of a complex vector space (Section 3).
The invariant subspaces of g, sequences of ^-invariant subspaces, the covering representation g and the relations between the reducibility of g and g are also discussed.
Throughout this paper we shall use the same notation as in [1].
2. Diagonal «-group. Let G be a group, ^ = G" 1, « ^ 2. We define an
«-ary operation (p : -* as follows : Let ^ э a, = (an , ai2, ..., ain_ t ),
i =
1, 2 ,..., n.
(!) (p{ax, a 2, a n) = (bu b2, ..., h„_ j),
where b, = ani, . + l t i .. .a2i+ j au . It may described also by the following scheme:
l l2 1
41 2
l 2 2
a\n - l «11 a21
a l 2 a 2 2
« 1 я - 1
*2 n - 1
a n - 11 a n - 1 2
a n l a n 2
N 1
a n n - 1
a n - 1 2
an1 Vfln2
b t b 2
a n - 1 n - 1 a n n - 1 b n - 1
i.e., bt is a product of « elements of G taken from the i-th diagonal. equipped with the «-ary operation (p is a «-group. We call (^ , (p) a diagonal n-group.
T
heorem1. Suppose that |G| > 1; then the diagonal n-group (У,(р) is irreducible.
P ro o f. An «-group sé = (A, f ) is reducible to к -group (« = 1 +s(/c — 1»
(see [2]) if and only if for every c e A there exists d e A such that
(k — 2 )s s
(i) for every x e A , f ( c , d, x) — x,
(ii) for every х 19х 2, ...,X j ( / = (s—1)(/с—1 ) + 1)
f { c 2, d , x i) = f ( d , c ,x[) ■ f(xu c ,d , x l2) = f ( x l ,d,
k - 2 k - 2C , X 2 ) =. . . = f ( x [ , d , kc 2) = f ( x \ , kc 2,d).
We will show that condition (ii) above is not satisfied by the n-group ((S, (p).
Assume to the contrary that (У, (p) is reducible to a к-group. Hence for c
= ( e , ..., e) there exists a d = (dlt . . . , d n- 1), such that for xf = { e , ..., e)
n — 1
(i = 1, 2 , . . 0 (ii) is satisfied. So
k - 2
Ф ( 0. i d, x i) (djç — i •> d^, dfr +1, . . dn_ j , d±,..., d^_ 2)
k - 2
= (p(d, ç , x \ ) = {du d2, . . . , d n- 1)
k - 2 .
= 1 5 £ 9 ( ^ 9 2^2) ( * ^ f c 9 ^ f c + 1 5 • • • 9 1 9 9 ' * • 9 ^k~ l ) ' * *
k - 2
... (p(xj c , d, Xf + ^) (dfr + i — 1 ) dfc J.i , . . .5 dn — d±,.. ., dk+i_ 2)
fc- 2
<jp(xj,d, ç ,x,-+|) (d,- + j , . . . , _ j ,d j ,...,d j ) ...
k - 2
... <jp(xj) ç , d) - (d^, d2, ..., dn_ ^)
k - 2
= ф(*
1>d, c ) = (d/+1 , . . d„_ 1, d i ,..., dj).
We obtain dt — d2 = ... = dn- 1 = d, i.e., d = (d, ..., d). Now take xf = (x, e , ..., e), i = 1 ,..., /, where x Ф e,
k - 2 , k - 2
Ф( £ , d, x \ ) = (xd, x d , ..., x d , d , . . d), = (p(x\, d, c )
= (dx,4, ■■■,d,, d x, . . . , dx). V 1 ' ^ ' V * '
n—l— 2 l
From this equality we get xd = d; hence x = e, which is impossible.
We introduce an operation on У, which is analogous to the inner automorphism on a group. Let ae& . For x e f we put
а л (x) = ( a ï 1 Xj !, a j 1 x 2 Ui, . . . , a~}2 x„_ 2 a„_ 3, a ^ \ x„_ t a„_ 2).
L emma 1. 77ie map а: has the following properties:
(i) a A: У -> ^ is an automorphism of an n-group У, (ii) ( a A)-1 = (a*)A, where g* = (af *, a 2 1, ..., а^Л),
(iii) ( a A)(b A) = (Ь о а )л, w/iere a o b = (ax bl9 a 2b2, ..., a„_j bn- t), (iv) ç A = 1^, w/шге e = (g, ..., e).
P ro o f, (i) We will show that a: <S -* is a homomorphism of an n-group into itself. Let x, = (xfl, x i2, ..., x,„_ j) (г = 1 ,2 ,..., ri), g A (x)f = bt
= (b n ,b i2, ... ,b in_ 1), where for j > 1
On representations o f n-groups
337
b f j Cl j X i j Q j _ j , b j i Cl i Х ц 1 .
Take (p(QA{x1) , . . . , a A(xn)) = (p(b1, b 2, . . . , b n) = (hl , h 2, ...,h „ _ l), where for j > 1
hj b nj ... b„_ j j bn_j„_ j .. ■b2j +1 hij
= (ar1xnjaj_1)...(a1'1xn_ -j+ 1 1 an- l) • ••ЦЛ ^
2j
+ 1aj)(aj 1 xl7. j ) Uj Xnj . . . X2 j +
2Xij üj — i
On the other hand, cp(xi , x 2, ... , ï j =(Ti, •• ■>Уп- 1 ) = У, where yk = Xnk...
• • • Xn — k-i-
11Xn — k
1•••^2k+l Xiki а л (y) = (ci, where, for j > 1,
Cj = dj 1 y j d j - и cx = ax 1 у г a„_ i .
For j > 1, Cj = aj x nj . . . X 2 j + i X ij aj _i — hj and
^1 = ^nl b n -ln -l • • -^22 ^11
= ( a f 1 x A a„_i)(a~-i x„_ j a„_2) . . .(a2 1 *2 2 1 *i 1 1)
= а Г 1^ ^ . . . x2 2 = « Г1 an- 1 = C i.
Thus а л (<p(x i, . . x„)) = cp (a A ( x ^ , . . а л (x„)), which proves that a A : ^ -> ^ is homomorphism.
For arbitrary, there exists
Z_ (dx Xi i,U 2X2Uj , . . ., — 2 X„_ 2 Un_ 3, j X„_ j &n — 2 ) £ ^ such that a A(z) = x, that is a A maps ^ onto the whole of
If QA(x) = a A{y), then ( a f 1 Xi a„_ j , a 2“ 1 x 2« j, . . a ; . \ x„_t a „ .2)
= ( a f 1 У 1 a „ - 1, a f 1 y2 ax, . . y„ - 1 a„- 2), then a f 1 Xj a„_ ! = a f 1 u„_ x and for 1 < i < и—1, = a f 1yi ai_ 1. Hence for every / = 1,2, ...
— 1, x ( = yh that is а л is 1-1.
(ii) ( а л) (a*)A (x) = q A (ax x x af} x, u2 x2 a f 1,.. ., an. x x„_ ! а 'Л ) (Xi, X2, .. ., X„_ j).
(iii) (а л) (Ь л)(x) = a A ( b f 1 Xi b„_ x, b f 1 x2 bx,.. ., b".1! x„_ ! b„_ 2)
= ( a f 1 ЬГ1 Xi b„_ j u„_ j , a2 1 b2 1 x2 b: Ui,.. ., « “Л bf—i x„_ x b„_ 2 a„_ 2)
= ((hi Я 1Г 1 Xi (b„_ ! a„_ 1 ), (b2 a2) ~1 x2 (bj a j ) ,..., (b„_ j a„_ j)“ 1 x„_ j (b„_ 2 a„_ 2)).
So (aA)(b A) = (boa) A.
(iv) е л (х) = (e~ 1 Xi e, e~ 1 x2 e, ..., e- 1 x„_ 1 c) = x, thus e A = L*.
3. Representations of n-groups. Let F be a complex linear space, G L(F) the general linear group, and let G L(F, n) denote the diagonal n-group of G L (F ).
R e m a rk . For n = 2, GL(F, 2) = GL (F).
D
e f in it io n1. A homomorphism g: G -> GL(F, n) is called a representation of the n-group G on F.
D
e f in it io n2. A sequence of subspace Vlt V2, ..., K1-1 of F is called A- invariant, where A — (Ax, ..., A„_1)eGL(F, n), if А Х(УХ) a V2, A 2(V2) c= F3, ...
. . . , A n- i ( V „ - i ) c : V i .
L emma 2. (i) I f Vx, V2, ..., Vn_ l is an A = ( A 1, A 2, ...,A „ _ ^-invariant sequence o f subspaces, then dim V1 = dim V2 = . . . — d im V„_ j .
(ii) I f Vlt V2, ..., V„-i is invariant for A t , A 2, ..., A neGL( V, n), then it is (p(di, A 2, ..., Ad-invariant.
P ro o f, (i) dim Vx ^ dim l1 ^ ^ ... ^ dim ^ ^ dim Kj 9 hcncc dim
= dim V2 = ... = dim x.
(ii) Let A i = ( A n , A i2, . . . , A in_ 1) for i = l , 2 , . . . , n , <p{Au A 2, . . A n)
= (Bl , B 2, . . . , B n. lf where
... A„_i + i i A n- in_ i ••• A2,+1 A u , ВАЦ) = А Ы... A n—i +11 i„ - i . . . A 2i + i A u (Vi)
cz A ni.. • An-i +11 A n-in - 1 ■.. . A 2i+1{Vi + l) cz c A ni... Л - . + i i T O c : . . • ^ A ^ m c z Vi+l
D efinition 3. A sequence of subspaces Vt , V2, ..., F„_ i of Fis invariant for the representation
qif for every g e G the sequence Vlt V2, ..., Vn- 1 is q(g)- invariant.
Let
q1 ,q2be representations of an n-group G on F, (2) ei - Q2o 3 A e G L { V , n ) V g c G А л (д1{д)) = Q2{g).
T heorem 2. (2) is an equivalence relation.
Proof is an immediate consequence of Lemma 1.
D efinition 4. We say that the representations Qi and g2 are equivalent if
@i ~~ Q2-
T heorem 3. I f Vl f ..., F„_ j is a sequence o f Q-invariant subspaces of V, then there exist an equivalent representation
q' = {q\ , q2 ,• • -,
q' „ -i) and a subspace V o f V such that Ql(g)(V') с V for every g e G, i — 1, 2, n — 1, and dim V \
= dim Vi.
P ro o f. Let A = ( A i , . . . , A „ - i ) e G L ( V , n ) , A,(Pi) c Vi+1 for i = 1,2, ...
..., n — 2, A„- i (Fj) <= Vi. We observe that
q'=
A a(q)and V = F1 is a desirable representation.
We will call V an invariant subspace for the representation
q,if the sequence V , V , . . . , V is invariant for
q.D efinition 5. A representation
q: G -> G L(F, n) is called irreducible if its only invariant subspaces are {0} and F.
4. Covering representation. Let (G ,/) be a n-group, g : G -+G L(F, n) its representation on F. Let т : G -+ G(2)(„_ 1} be a homomorphism of the n-group G into the free covering group G (2). V — F © ... © F- Define
j: G L (K n )-> G L (F ) by "_1
On representations o f n-groups
339
(3) j ( A u ..., A„_ i) ( x i, x2, ..., j) = (A„_ ! (x„_ i), A l (xt) , ..., 2 (*„- 2)) for A e G L (V , n), A = (At , ..., A„^1).
We will show that j is a homomorphism of GL(K, n) into G L (F)(„_1),
j { 4 > ( A u . . . . , 4 „ ) ) ( * 1 , •
3 3 1 3-1 • • • А ц , . • ч A „ i A n _ i ,• — i . . . A i f , . . . , A-n n - 1 А , - ■ l n - 2 ’ - - A i „ ~ i ) X
cX
£X
- i ) = ( A „ n i b* 3 1 3 1
>
3 1 I?3 1 i)> A nl A „ - l /i 1 ••• А ц ( х 1) , . .4 > ( } { A i ) , . . ■ J ( A n) ) = ^ { { A i n - i •> A 1 1 , A 1 2 , . . . » ^ 1 n - 2)» ■
= ( A „ n - l ^ n - й - ! . . . A 1 „ 1 3 3“ i n i - A n , . . . )
Hence there exists a unique homomorphism g: G(2) -> GL(K, n) such that gz(g)
=je(g),
G ± G(2\n- U
e i
; é
G L (V,n) Л G L ( V i )
We call the above representation g a covering representation for
q.
L
emma3. Let W
<V be an invariant subspace for a representation
q :G -+ G L (V,ri). Then W — И7® , ^ . ® W is an invariant subspace for the covering
representation. n~ 1
P ro o f. We must show that \/g e G (2 ) g(g)(W) c= W. Take x e W , gez(G).
Let g' = t ~ l {g). Then gz(g'){x) = jg(g')(x),
jQ{g'){x) = j(ei{9% - • вп-Л9')){хи
= •••, в»- 2 (д’)(Хп- 2 ))-
Also jg (g )(x )e W, by the ^-invariance of W.
Since t (G) generates G (2), for geG (2)\x(G ) we have g = g f .. .g™, К = 1 , - 1 , gt s r(G), i = 1, 2 ,..., m,
Я(в)(х) = ê ( 9 Î ‘ 9г ■ ■ • 9 » " )(ï) = ê te i) * 1 • • • J * " ( ï ) 6 Ж L
em m a4 . I f W
<V is g-invariant, then W is g-invariant.
P ro o f. Our aim is to show that \ / g e G V x e l k V I < i < n - 1 Qi (g) (x)eW. T ake g e G, z{g)eG(2). Let xe W , x =v( x ,. , x). Clearly,
я- i _
ё ( т ( з ) ) ( х ) е ^ ё(т(0))(*) =jQ{g){x) =(Qn-i{g){x),--->en- 2 (g){x))eW. Hence VI < i < и- l eti9)(x)eW .
Let ( , ) be a scalar product on V. Then for x = (xb ..., x ^ ) ,
n — 1
У = Oh > • • •> Уп-1) e К (2?» У) = I (xh yt) is a scalar product on V.
L emma 5. I f H е GL(F, n) is a sub n-group, and H( 2) is a free covering group of H, then H ( 2) <= G1(F) and for h e H ( 2), x = (xb we have
h(x) = (hk(xk), hk+l{xk+l), ..., /!„-!(%„_!), M * l), ..., /lk_ !(**_!)).
P ro o f. This is because j is a homomorphism, j(H) generates H ( 2), and j ( h ) ( x i x „ _ J = (h„_ ! (x„_ t), hx (xk) , ..., Л„_ 2 (x„_ 2)).
Next we shall consider only the finite и-groups. Suppose that there exists a scalar product in the space V.
D efinition 7. For x j e F w e put
(4) <*,У> 1
Z (M*), My)),
heH(2)
then <x, y> is а H(2)-invariant scalar product on V, i.e., (h(x), h(y)) = <x, y>
for heH (2), x , y e V .
Le m m a 6. 7 / x = ( 0 , . . . , 0 , 0 , . . . , 0 ) , y = ( 0 , . . . , 0 ,
yjf
0 , . . . , 0 ) ,i
f / i e «<х*У> = 0.
P ro o f. By Lemma 5 the action of h e H ( 2) on Fis given by a superposition of the action of hk on the k-th coordinate (к = 1 , 2 , ...,« —1) and a cyclic permutation of the coordinates. Since hk(0) = 0, к = 1, 2 , . . . , « —1, only one coordinate of h(x) and only one coordinate of h(y) is different of 0, and, moreover, the numbers of these coordinates are different. Hence Lemma 6 follows.
Th e o r e m 4 .
I f W
<V is a g-invariant subspace, then there exists a g- invariant scalar product < , > on V such that :
(i) the orthogonal complement W 1 of W is g-invariant,
(ii) there is a sequence < , >s, s = 1, 2 ,..., n — 1 of scalar products on V, such that the projection Щ of W 1 on i-th coordinate subspace is an orthogonal complement to W relative to < , >f,
(iii) the sequence of subspaces Wt is g-invariant.
P ro o f. Applying Lemma 5 with H = ^(G), H (2) = ^(G(2)), we obtain a g- invariant scalar product < , ) . By Lemma 3. W < V is a ^-invariant.
(i) If x e W x, then y ) = 0 for every y e W . Since g{g)~1 (y)e JTfor g in G(2), we have (g(g)(x), y } = <x, g i g ) ' 1 (y)> = 0. Hence g{g)GWx, that is, g m w 1) cz W 1.
(ii) If x , y e V, we put <x, y ) s = <(0,..., 0, x, 0 ,..., 0), (0 ,..., 0, y , ..., 0)>.
s s
n — 1
Obviously, < , ) s is a scalar product on V. Observe that <x, y ) = Z <Х*’ У*>*
s = 1
Let Ps(xl9 x 2, ■ • i) = xs be a projection on the sth coordinate subspace in
V. Now (Pi (x ),..., P„_ J (x)) = X.
On representations o f n-groups
341
Put WS = PS{WX). Let x e W ; then xs = ( 0 , . . 0 , x, 0 , . . 0)e W. So
S
for y e W 1 we have <Ps(y), x>s = £ ( y h х{>,- = (y, xs> = 0 and every element
i
of Ws is orthogonal to W. On the other hand V = PS(V) = PS(W ® W X)
= PS(W) + PS(W X) = W@WS. Hence is an orthogonal complement to W.
(iii) Since for V = W@WS, s = 1, 2 , . . n — 1, ( P ^ x ) , P „ - i ( x ) ) = x for every x in V, W is equal to "©* Ws. Let g e G , g(g) = {g), . . x {g)),
s = 1
and х е Щ . Then x t = ( 0 ,..., 0, x, 0 , .. ., 0)e W 1, and
q( t (^f))(xt) = jg{g)(x,)
i