- Email: [email protected]
On the Orthogonal Basis of Symmetry Classes of Tensors
Journal of Algebra 237, 637᎐646 Ž2001. doi:10.1006rjabr.2000.8332, available online at http:rrwww.idealibrary.com on
On the Orthogonal Basis of Symmetry Classes of Tensors M. A. Shahabi, K. Azizi, and M. H. Jafari Department of Pure Mathematics, Tabriz Uni¨ ersity, Tabriz, Iran E-mail: [email protected] Communicated by Walter Feit Received September 13, 1999
A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki Žin press, Linear and Multilinear Algebra. and R. R. Holmes and T. Y. Tam Ž1992, Linear and Multilinear Algebra 32, 21᎐31.. These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups. 䊚 2001 Academic Press Key Words: symmetry classes of tensors; orthogonal basis; dihedral group; dicyclic group; 2-adic valuation.
1. INTRODUCTION Let V be an n-dimensional complex inner product space and G be a permutation group on m elements. Let be any irreducible character of G. For any g g G, define the operator
Pg :
m
m
1
1
mV ª mV
by Pg Ž ¨ 1 m ⭈⭈⭈ m ¨ m . s ¨ gy1 m ⭈⭈⭈ m ¨ gy1 Ž m. .
Ž 1.
637 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
638
SHAHABI, AZIZI, AND JAFARI
The symmetry classes of tensors associated with G and is the image of the symmetry operator T Ž G, . s
Ž 1.
Ý Ž g . Pg ,
Ž 2.
ggG
and it is denoted by V Ž G .. We say that the tensor T Ž G, . Ž ¨ 1 m ⭈⭈⭈ m ¨ m .
Ž 3.
is a decomposable symmetrized tensor, and it is denoted by ¨ 1 ) ⭈⭈⭈ ) ¨ m . The inner product on V induces an inner product on V Ž G . which satisfies ² ¨ 1 ) ⭈⭈⭈ ) ¨ m , u1 ) ⭈⭈⭈ ) u m : s
Ž 1. G d Ž A. ,
Ž 4.
where A s w² ¨ i , u j :x, and dG is the generalized matrix function, m
dG Ž A . s
Ý Ž g . Ł ai , g Ž i. .
Ž 5.
is1
ggG
Let ⌫nm be the set of all sequences ␣ s Ž ␣ 1 , . . . , ␣ m ., with 1 F ␣ i F n. Define the action of G on ⌫nm by g . ␣ s Ž ␣ gy1 Ž1. , . . . , ␣ gy1 Ž m. . .
Ž 6.
We write ␣ ;  if ␣ and  belong to the same orbit in ⌫nm. Let ⌬ be a system of distinct representatives of the orbits. We denoted by G␣ the stabilizer subgroup of ␣ . Define ⍀ s ␣ g ⌫nm N
½
Ý
Ž g. / 0 ,
ggG␣
5
Ž 7.
and put ⌬ s ⌬ l ⍀. Let e1 , . . . , e n4 be an orthonormal basis of V. Denote by eU␣ the tensor e␣ Ž1. ) ⭈⭈⭈ ) e␣ Ž m. . We have
¡0
² eU␣ ,
eU :
s
~ Ž 1.
¢
if ␣ ¤ 
Ý ggG
Ž ghy1 .
if ␣ s h ⭈  .
Ž 8.
SYMMETRY CLASSES OF TENSORS
639
In particular, for g 1 , g 2 g G and ␥ g ⌬ we obtain ² eUg ⭈␥ , eUg ⭈␥ : s 1
2
Ž 1.
Ý
Ž x. .
Ž 9.
xgg 2 G␥ gy1 1
Moreover, eU␣ / 0 if and only if ␣ g ⍀. For ␣ g ⌬, V␣U s ² eUg⭈␣ : g g G : is called the orbital subspace of V Ž G .. It follows that V Ž G . s
Ý ␣g⌬
V␣U
Ž 10 .
is an orthogonal direct sum. In w3x it is proved that dim V␣U s
Ž 1. < G␣ <
Ý ggG␣
Ž g . s Ž 1 . , 1 G␣ .
Ž 11 .
From Ž11. we deduce that if is a linear character, then dim V␣U s 1 and in this case the set
eU␣ N ␣ g ⌬ 4
Ž 12 .
is an orthogonal basis of V Ž G .. A basis which consists of the decomposable symmetrized tensors eU␣ is called an orthogonal )-basis. If is not linear, it is possible that V Ž G . has no orthogonal )-basis. For further information about the symmetry classes of tensors we refer to w6x. In w1x Cummings noted that if permutation groups which are isomorphic as abstract groups have different permutation structure they may yield quite different symmetry classes of tensors; e.g., G s ²Ž12.: and H s ²Ž12.Ž34.: are isomorphic subgroups of S4 but dim V 0 Ž G . s n3 Ž n q 1.r2, dim V 0 Ž H . s n2 Ž n2 q 1.r2, where 0 is the principal character. In w2, 4x the proof of the necessary condition of Theorem 3 and Theorem 3.1, respectively, depends on a certain permutation structure of the generators of dicyclic and dihedral groups. So, if one changes the permutation structure of the generators, then he may obtain different results. We prove that if G is any subgroup of Sm which is isomorphic to the dicyclic group in w2x or to the dihedral group in w4x, then the necessary condition of Theorem 3 in w2x and Theorem 3.1 in w4x is independent of the permutation structure of these groups.
640
SHAHABI, AZIZI, AND JAFARI
2. GENERALITIES 2.1. Dihedral Group The dihedral group of order 2 k Ž k G 3. is defined by D 2 k s ² a, b N a k s 1 s b 2 , bab s ay1 : . In particular, D 2 k s a i, a i b N 0 F i F k y 14 . For each integer 0 - h - kr2, D 2 k has an irreducible non-linear character h of degree 2 given by
h Ž a r . s 2 cos
2 rh k
h Ž a r b . s 0, 0 F r F k y 1.
,
Ž 13 .
ŽSee w5, p. 183x.. The other characters of D 2 k are linear. For convenience of reference, we summarize here several results that will be needed later. LEMMA 1. Suppose k G 3, 0 - h - kr2, 0 F r F k y 1, and l s krŽ k, r ., where Ž k, r . s g.c.d. of k and r. Then l
Ý ts1
cos
2 trh k
s
½
l 0
if k N rh if k ¦ rh.
Ž 14 .
Proof. See w2x. LEMMA 2. Assume that 0 - h - kr2 Ž k G 3.. Then there exist t 1 , t 2 , 0 F t 1 , t 2 F k y 1 such that cosŽ2Ž t 1 y t 2 . hrk . s 0 if and only if 2 Ž2 hrk . - 0, where 2 is the 2-adic ¨ aluation. Proof. See w2x. LEMMA 3. Let G s D 2 k Ž k G 3., and let s h Ž0 - h - kr2. and ␣ g ⌬. Then G␣ s ² a r : or ² a r : - G␣ , for some 0 F r F k y 1. In both cases G␣ l ² a: s ² a r :, k N rh, <² a r :< s l, and 5 eU␣ 5 2 s 2 lrk, where l s krŽ k, r .. Proof. See w2x. Now we state our first theorem. THEOREM 1. Let G s D 2 k Ž k G 3. be a subgroup of Sm , let s h Ž0 - h - kr2., and assume n s dim V G 2. Then V Ž G . has an orthogonal )-basis if and only if 2 Ž2 hrk . - 0. Proof. It is enough to prove that for any ␣ g ⌬ the orbital subspace V␣U has orthogonal )-basis if and only if 2 Ž2 hrk . - 0. Let 2 Ž2 hrk . - 0 and assume ␣ g ⌬. By Lemma 3, G␣ s ² a r : or ² a r : - G␣ Žin both cases
641
SYMMETRY CLASSES OF TENSORS
² a r : s G␣ l ² a:. and k N rh. Put oŽ a r . s l s krŽ r, k .. Now we consider two cases. Case 1. If G␣ s ² a r : s a r , a2 r , . . . , a l r s 14 , then by Ž11., dim V␣U s 2 Ž Ž1.r< G␣ <.Ý g g G Ž g . s Ž2 l . s 4. For any g 1 , g 2 g G, we have ␣ l
¡ a
g 2 G␣ gy1 1 s
~ a
¢ a
rqiyj
, a2 rqiyj , . . . , a l rqiyj 4
if g 1 s a j , g 2 s a i
rqiqj
b, a2 rqiqj b, . . . , a l rqiqj b 4
if g 1 s a j b, g 2 s a i
iyryj
, a iy2 ryj , . . . , a iyl ryj 4
if g 1 s a j b, g 2 s a i b.
Ž 15 . If g 1 s a j, g 2 s a i, by Ž9. we have ² eUg ⭈␣ , eUg ⭈␣ : s 1
2
s s s
Ž 1.
xgg 2 G␣ gy1 1
l
2
Ý
k
Ž x. s
Ý
cos
Ý cos k Ý
2 trh
ž
k
cos
ts1
2k
ts1
k q
k
2 Ž i y j . h k
2 Ž i y j . h
l
2
Ý Ž at rqiyj .
2 Ž tr q i y j . h
ts1
2
l
2
ž
k
/
s
2l k
/
cos
2 Ž i y j . h k
, Ž 16 .
where the equality of the one before last is due to Lemma 3. Similarly, for g 1 s a j b, g 2 s a i b, we have ² eUg ⭈␣ , eUg ⭈␣ : s 1
2
2l k
cos
2 Ž i y j . h k
,
Ž 17 .
and for g 1 s a j b, g 2 s a i, we have ² eUg ⭈␣ , eUg ⭈␣ : s 0. 1
Therefore
¡21 cos 2Ž i y j . h
k ² eUg ⭈␣ , eUg ⭈␣ : s 0 1 2
k
~
¢k
2l
Ž 18 .
2
if g 1 s a j , g 2 s a i if g 1 s a j b, g 2 s a i
cos
2 Ž i y j . h k
if g 1 s a j b, g 2 s a i b.
Ž 19 .
642
SHAHABI, AZIZI, AND JAFARI
Since 2 Ž2 hrk . - 0, hence by Lemma 2 there exist t 1 , t 2 , 0 F t 1 , t 2 F k y 1 such that cosŽ2Ž t 1 y t 2 . hrk . s 0. Put S s a t 1 ⭈ ␣ , a t 2 ⭈ ␣ , a t 1 b ⭈ ␣ , a t 2 b ⭈ ␣ 4 : ⌫nm . Then by Ž19. for every ␥ ,  g S and ␥ /  we have ² e␥U , eU : s 0. But dim V␣U s 4; hence eU N  g S4 is an orthogonal )-basis for V␣U . Case 2. If ² a r : - G␣ , then ² a r : s ² a: l G␣ and we have G␣ l ² a: s a r , a2 r , . . . , a l r s 14 and < G␣ < G 2 l. Therefore by Ž11., dim V␣U s
Ž 1. < G␣ <
Ý
Ž g. F
ggG␣
2 2l
Ž 2 l . s 2.
Hence dim V␣U s 1 or 2. If dim V␣U s 1, then it is obvious that we have an orthogonal )-basis. If dim V␣U s 2, then by Lemma 2 there exist i, j, 0 F i, j F k y 1 such that cosŽ2Ž i y j . hrk . s 0. Set g 1 s a j, g 2 s a i. Then rqiyj ² : g 2 G␣ gy1 , . . . , a l rqiyj 4 . 1 l a s a
Hence by Ž9. and Ž16. we have ² eUg ⭈␣ , eUg ⭈␣ : s 0. 1
eUa j⭈␣ ,
2
eUa i ⭈␣ 4
Thus is an orthogonal )-basis for V␣U . Conversely, let V␣U have an orthogonal )-basis for every ␣ g ⌬. Now we prove that 2 Ž2 hrk . - 0. Let a s a1 a2 ⭈⭈⭈ a r be the disjoint cycle decomposition of a in Sm and let oŽ a i . s m i , for every 1 F i F r. Since oŽ a. s k, so k s w m1 , . . . , m r x, the least common multiple of m1 , . . . , m r . Set a i s Ž i1 , i 2 , . . . , i m i . and define ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ m . g ⌫nm in such a way that ␣ i1 s 1, ␣ i 2 s ␣ i 3 s ⭈⭈⭈ s ␣ i m s 2, for every 1 F i F r, and for j g 1, 2, . . . , m4 _ i r D is1 i1 , . . . , i m i 4 , ␣ j s 1. EXAMPLE. Let a s Ž567.Ž1234. g S8 , where a1 s Ž567., a2 s Ž1234.. Then 11 s 5, 1 2 s 6, 1 3 s 7, 2 1 s 1, 2 2 s 2, 2 3 s 3, 2 4 s 4. Hence j s 8 and ␣ 8 s 1. If ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ 8 . g ⌫n8 , then ␣ 11 s ␣ 5 s 1, ␣ 1 2 s ␣ 6 s 2, ␣ 1 3 s ␣ 7 s 2, ␣ 2 1 s ␣ 1 s 1, ␣ 2 2 ss ␣ 2 s 2, ␣ 2 3 s ␣ 3 s 2, ␣ 2 4 s ␣ 4 s 2. Hence ␣ s Ž1, 2, 2, 2, 1, 2, 2, 1.. Now we show that a t f G␣ , for every 1 F t F k y 1. Since t - k s w m1 , . . . , m r x, hence m s ¦ t, for some 1 F s F r. Let t s m s q q r 1 , where q
SYMMETRY CLASSES OF TENSORS
643
and r 1 are integers and 0 - r 1 - m s . Then a t s a1t ⭈⭈⭈ a ts ⭈⭈⭈ a tr s a1t ⭈⭈⭈ a sr 1 ⭈⭈⭈ a rt . Since a sr 1 moves the position s1 in ␣ , and all other a it Ž i / s . fix the elements of s1 , s2 , . . . , sm s 4 , then a t does not fix the position s1 in ␣ ; therefore a t f G␣ . Now we show that G␣ s 14 or G␣ s 1, a t b4 , for some 0 F t F k y 1. If G␣ / 14 and a t 1 b, a t 2 b g G␣ , then a t 1 b ⭈ a t 2 b s a t 1yt 2 g G␣ . This implies that t 1 s t 2 . In both cases we have 5 eU␣ 5 2 s
2 2k
Ý
Ž g. s
ggG␣
2 k
/ 0.
Ž 20 .
Therefore ␣ g ⌬. Now we prove that in both cases 2 Ž2 hrk . - 0. Case 1. Let G␣ s 14 . By Ž11. dim V␣U s
2 1
Ž 2 . s 4.
For every g 1 , g 2 g G, we have
¡ a
g 2 G␣ gy1 1 s
iyj
~ a ba
¢ a
if g 1 s a j , g 2 s a i
4
i
yj
iyj
4
if g 1 s a j , g 2 s a i b
4
if g 1 s a j b, g 2 s a i b.
Hence
¡2 cos 2Ž i y j . h
k ² eUg ⭈␣ , eUg ⭈␣ : s 0 1 2
~
¢k cos 2
k
if g 1 s a j , g 2 s a i if g 1 s a j , g 2 s a i b
2 Ž i y j . h k
if g 1 s a j b, g 2 s a i b.
Since dim V␣U s 4, hence there exist 0 F i, j F k y 1 such that cosŽ2Ž i y j . hrk . s 0. Therefore by Lemma 2, 2 Ž2 hrk . - 0. Case 2. If G␣ s 1, a t b4 , for some 0 F t F k y 1, then by using a similar argument as w4, p. 25x we prove that 2 Ž2 hrk . - 0. Remark 1. The condition of Theorem 1 is equivalent to the condition of Theorem 3.1 in w4x. If 0 - h - kr2 and h s h 2 h 2 ⬘ with h 2 a power of 2
644
SHAHABI, AZIZI, AND JAFARI
and h 2 ⬘ odd, then 2 Ž2 hrk . - 0 if and only if 2 Ž2 h 2 h 2 ⬘rk . - 0 if and only if 4 h 2 N k. COROLLARY 1. Let G s D 2 k F Sm and assume that n s dim V G 2. Then m m 1 V has an orthogonal )-basis if and only if k is a power of 2. Proof. Let k s k 2 k 2 ⬘ with k 2 a power of 2 and k 2 ⬘ odd. Assume that k 2 - k. Then 0 - k 2 - kr2 and 4 k 2 ¦ k. Therefore, if s k 2 , then Theorem 1 implies that V Ž G . has no an orthogonal )-basis. Conversely, assume k is a power of 2. If 0 - h - kr2, then h 2 F kr4, and so 4 h 2 N k, where h s h 2 h 2 ⬘ with h 2 a power of 2 and h 2 ⬘ odd. Then Theorem 1 implies that V hŽ G . has an orthogonal )-basis, and so m 1m V. 2.2. Dicyclic Group The dicyclic group of order 4 k Ž k G 2. is defined by T4 k s ² a, b N a2 k s 1, a k s b 2 , by1ab s ay1 : . In particular, T4 k s a i, a i b N 0 F i F 2 k y 14 . For each integer 0 - h - k, T4 k has an irreducible non-linear character h of degree 2 given by
h Ž a r . s 2 cos
rh k
h Ž a r b . s 0, 0 F r F 2 k y 1.
,
ŽSee w5x, p. 187x.. The other irreducible characters of T4 k are linear. THEOREM 2. Let G s T4 k Ž k G 2. be a subgroup of Sm , let s h Ž0 - h - k ., and assume n s dim V G 2. Then V Ž G . has an orthogonal )-basis if and only if 2 Ž hrk . - 0. Proof. If 2 Ž hrk . - 0, the proof of the existence of the orthogonal )-basis is similar to Theorem 1. ŽFor more details see w2x.. Conversely, let V␣U have an orthogonal )-basis for every ␣ g ⌬. Now we prove that 2 Ž hrk . - 0. As Theorem 1 we take a s a1 a2 ⭈⭈⭈ a r to be the disjoint cycle decomposition of a in Sm . By taking the notation as Theorem 1 you may prove that a t f G␣ , for every 0 F t F 2 k y 1. Now we show that G␣ s 14 . If G␣ / 14 , then a t b g G␣ for some 0 F t F 2 k y 1. Now Ž a t b . 2 g G␣ . But Ž a t b . 2 s a t b ⭈ a t b s b 2 s a k g G␣ , and this is impossible. Therefore G␣ s 14 . In this case we have 5 eU␣ 5 2 s
2 4k
Ý ggG␣
Ž g. s
4 4k
s
1 k
/ 0.
SYMMETRY CLASSES OF TENSORS
645
Therefore ␣ g ⌬. By using a similar argument as w2x we can prove that 2 Ž hrk . - 0. Remark 2. The condition 2 Ž hrk . - 0 is equivalent to 2 h 2 N k, where h s h 2 h 2 ⬘ with h 2 a power of 2 and h 2 ⬘ odd. COROLLARY 2. Let G s T4 k F Sm and assume n s dim V G 2. Then m m1 V has an orthogonal )-basis if and only if k is a power of 2. Proof. Let k s k 2 k 2 ⬘ with k 2 a power of 2 and k 2 ⬘ odd. Assume that 0 - k 2 - k; then 2 k 2 ¦ k. By Theorem 2, if s k 2 , then V Ž G . has no orthogonal )-basis. Conversely, assume k is a power of 2. If 0 - h - k, then h 2 - kr2 and so 2 h 2 N k. Then Theorem 2 implies that V hŽ G . has an orthogonal )-basis, and so m m 1 V. In w7x M. Shahryari proved the following theorem. THEOREM. Let be a non-linear irreducible character of G and suppose that there is ␣ g ⌬ such that < G : G␣ < 2 Ž 1.
- , 1 G␣ -
< G : G␣ <
Ž 1.
.
Then V Ž G . has no orthogonal )-basis. It is obvious that the condition of this theorem is equivalent with 1
- 5 eU␣ 5 2 - 1.
2
Using Theorem 1, we show that 1r2 is the best lower bound for 5 eU␣ 5 2 for which one can obtain. Also we give a counter-example that the converse of this theorem is not true. Remark 3. Let G s D 8 F S8 Ž k s 4.. By Ž13., G has only one non-linear irreducible character h Ž0 - h - 2.. Now it is immediate from Theorem 1 that V hŽ G . has an orthogonal )-basis. Now let ␣ be an arbitrary element of ⌬. Then Lemma 3 implies that k N rh Ž0 F r F 3, k s 4, h s 1.. So 4 N r, and consequently r s 0, l s 1. Thus 5 eU␣ 5 2 s 2r4 s 1r2. Remark 4. Counter-example. Let p G 5 be a prime and G s D 2 p s ² a, b N a p s 1 s b 2 , bab s ay1 : and let G be a subgroup of S2 p . Then the non-linear characters of D 2 p are h , where 0 - h - pr2 and
h Ž a r . s 2 cos
2 hr p
,
h Ž a r b . s 0, 0 F r F p y 1.
The other irreducible characters of D 2 p are of degree 1.
646
SHAHABI, AZIZI, AND JAFARI
It is easy to see that ⌬ is non-empty. If we take ␣ as in the converse of Theorem 1, then by Ž20., 5 e␣ 5 2 s 2rp / 0. Hence ␣ g ⌬. Therefore V Ž G . / 0. Since 2 Ž2 hrp . ) 0, so by the Theorem 1, V hŽ G . has no orthogonal )-basis. Now we show that 5 eU␣ 5 2 - 1r2 for every ␣ g ⌬. Let ␣ g ⌬. By Lemma 3, we have G␣ l ² a: s ² a r : , for some 0 F r F p y 1, p N rh. Since 0 - h - pr2, hence p N r and so r s 0. But l s < G␣ l ² a:< s <² a r :<. Hence l s 1 and 5 eU␣ 5 2 s Ž2 = 1.rp s 2rp - 1r2. Remark 5. Using Theorems 1 and 2, one can see that the existence of an orthogonal )-basis for the abelian, dihedral, dicyclic Žin particular, generalized quaternion. groups is independent of permutation structure of the given group. This may give rise to the following question. Question. Is it true that the existence of an orthogonal )-basis for V ŽG. is independent of the permutation structure of group G? REFERENCES 1. L. J. Cummings, Cyclic symmetry classes, J. Algebra 40 Ž1976., 401᎐405. 2. M. R. Darafsheh and M. R. Pournaki, On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group, Linear and Multilinear Algebra, in press. 3. R. Freese, Inequalities for generalized matrix functions based on arbitrary characters, Linear Algebra Appl. 7 Ž1973., 337᎐395. 4. R. R. Holmes and T. Y. Tam, Symmetry classes of tensors associated with certain groups, Linear and Multilinear Algebra 32 Ž1992., 21᎐31. 5. G. James and M. Liebeck, ‘‘Representations and Characters of Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1993. 6. R. Merris, ‘‘Multilinear Algebra,’’ Gordon & Breach, New York, 1997. 7. M. Shahryari, On the orthogonal bases of symmetry classes, J. Algebra, in press.
On the Orthogonal Basis of Symmetry Classes of Tensors M. A. Shahabi, K. Azizi, and M. H. Jafari Department of Pure Mathematics, Tabriz Uni¨ ersity, Tabriz, Iran E-mail: [email protected] Communicated by Walter Feit Received September 13, 1999
A necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki Žin press, Linear and Multilinear Algebra. and R. R. Holmes and T. Y. Tam Ž1992, Linear and Multilinear Algebra 32, 21᎐31.. These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups. 䊚 2001 Academic Press Key Words: symmetry classes of tensors; orthogonal basis; dihedral group; dicyclic group; 2-adic valuation.
1. INTRODUCTION Let V be an n-dimensional complex inner product space and G be a permutation group on m elements. Let be any irreducible character of G. For any g g G, define the operator
Pg :
m
m
1
1
mV ª mV
by Pg Ž ¨ 1 m ⭈⭈⭈ m ¨ m . s ¨ gy1 m ⭈⭈⭈ m ¨ gy1 Ž m. .
Ž 1.
637 0021-8693r01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
638
SHAHABI, AZIZI, AND JAFARI
The symmetry classes of tensors associated with G and is the image of the symmetry operator T Ž G, . s
Ž 1.
Ý Ž g . Pg ,
Ž 2.
ggG
and it is denoted by V Ž G .. We say that the tensor T Ž G, . Ž ¨ 1 m ⭈⭈⭈ m ¨ m .
Ž 3.
is a decomposable symmetrized tensor, and it is denoted by ¨ 1 ) ⭈⭈⭈ ) ¨ m . The inner product on V induces an inner product on V Ž G . which satisfies ² ¨ 1 ) ⭈⭈⭈ ) ¨ m , u1 ) ⭈⭈⭈ ) u m : s
Ž 1. G d Ž A. ,
Ž 4.
where A s w² ¨ i , u j :x, and dG is the generalized matrix function, m
dG Ž A . s
Ý Ž g . Ł ai , g Ž i. .
Ž 5.
is1
ggG
Let ⌫nm be the set of all sequences ␣ s Ž ␣ 1 , . . . , ␣ m ., with 1 F ␣ i F n. Define the action of G on ⌫nm by g . ␣ s Ž ␣ gy1 Ž1. , . . . , ␣ gy1 Ž m. . .
Ž 6.
We write ␣ ;  if ␣ and  belong to the same orbit in ⌫nm. Let ⌬ be a system of distinct representatives of the orbits. We denoted by G␣ the stabilizer subgroup of ␣ . Define ⍀ s ␣ g ⌫nm N
½
Ý
Ž g. / 0 ,
ggG␣
5
Ž 7.
and put ⌬ s ⌬ l ⍀. Let e1 , . . . , e n4 be an orthonormal basis of V. Denote by eU␣ the tensor e␣ Ž1. ) ⭈⭈⭈ ) e␣ Ž m. . We have
¡0
² eU␣ ,
eU :
s
~ Ž 1.
¢
if ␣ ¤ 
Ý ggG
Ž ghy1 .
if ␣ s h ⭈  .
Ž 8.
SYMMETRY CLASSES OF TENSORS
639
In particular, for g 1 , g 2 g G and ␥ g ⌬ we obtain ² eUg ⭈␥ , eUg ⭈␥ : s 1
2
Ž 1.
Ý
Ž x. .
Ž 9.
xgg 2 G␥ gy1 1
Moreover, eU␣ / 0 if and only if ␣ g ⍀. For ␣ g ⌬, V␣U s ² eUg⭈␣ : g g G : is called the orbital subspace of V Ž G .. It follows that V Ž G . s
Ý ␣g⌬
V␣U
Ž 10 .
is an orthogonal direct sum. In w3x it is proved that dim V␣U s
Ž 1. < G␣ <
Ý ggG␣
Ž g . s Ž 1 . , 1 G␣ .
Ž 11 .
From Ž11. we deduce that if is a linear character, then dim V␣U s 1 and in this case the set
eU␣ N ␣ g ⌬ 4
Ž 12 .
is an orthogonal basis of V Ž G .. A basis which consists of the decomposable symmetrized tensors eU␣ is called an orthogonal )-basis. If is not linear, it is possible that V Ž G . has no orthogonal )-basis. For further information about the symmetry classes of tensors we refer to w6x. In w1x Cummings noted that if permutation groups which are isomorphic as abstract groups have different permutation structure they may yield quite different symmetry classes of tensors; e.g., G s ²Ž12.: and H s ²Ž12.Ž34.: are isomorphic subgroups of S4 but dim V 0 Ž G . s n3 Ž n q 1.r2, dim V 0 Ž H . s n2 Ž n2 q 1.r2, where 0 is the principal character. In w2, 4x the proof of the necessary condition of Theorem 3 and Theorem 3.1, respectively, depends on a certain permutation structure of the generators of dicyclic and dihedral groups. So, if one changes the permutation structure of the generators, then he may obtain different results. We prove that if G is any subgroup of Sm which is isomorphic to the dicyclic group in w2x or to the dihedral group in w4x, then the necessary condition of Theorem 3 in w2x and Theorem 3.1 in w4x is independent of the permutation structure of these groups.
640
SHAHABI, AZIZI, AND JAFARI
2. GENERALITIES 2.1. Dihedral Group The dihedral group of order 2 k Ž k G 3. is defined by D 2 k s ² a, b N a k s 1 s b 2 , bab s ay1 : . In particular, D 2 k s a i, a i b N 0 F i F k y 14 . For each integer 0 - h - kr2, D 2 k has an irreducible non-linear character h of degree 2 given by
h Ž a r . s 2 cos
2 rh k
h Ž a r b . s 0, 0 F r F k y 1.
,
Ž 13 .
ŽSee w5, p. 183x.. The other characters of D 2 k are linear. For convenience of reference, we summarize here several results that will be needed later. LEMMA 1. Suppose k G 3, 0 - h - kr2, 0 F r F k y 1, and l s krŽ k, r ., where Ž k, r . s g.c.d. of k and r. Then l
Ý ts1
cos
2 trh k
s
½
l 0
if k N rh if k ¦ rh.
Ž 14 .
Proof. See w2x. LEMMA 2. Assume that 0 - h - kr2 Ž k G 3.. Then there exist t 1 , t 2 , 0 F t 1 , t 2 F k y 1 such that cosŽ2Ž t 1 y t 2 . hrk . s 0 if and only if 2 Ž2 hrk . - 0, where 2 is the 2-adic ¨ aluation. Proof. See w2x. LEMMA 3. Let G s D 2 k Ž k G 3., and let s h Ž0 - h - kr2. and ␣ g ⌬. Then G␣ s ² a r : or ² a r : - G␣ , for some 0 F r F k y 1. In both cases G␣ l ² a: s ² a r :, k N rh, <² a r :< s l, and 5 eU␣ 5 2 s 2 lrk, where l s krŽ k, r .. Proof. See w2x. Now we state our first theorem. THEOREM 1. Let G s D 2 k Ž k G 3. be a subgroup of Sm , let s h Ž0 - h - kr2., and assume n s dim V G 2. Then V Ž G . has an orthogonal )-basis if and only if 2 Ž2 hrk . - 0. Proof. It is enough to prove that for any ␣ g ⌬ the orbital subspace V␣U has orthogonal )-basis if and only if 2 Ž2 hrk . - 0. Let 2 Ž2 hrk . - 0 and assume ␣ g ⌬. By Lemma 3, G␣ s ² a r : or ² a r : - G␣ Žin both cases
641
SYMMETRY CLASSES OF TENSORS
² a r : s G␣ l ² a:. and k N rh. Put oŽ a r . s l s krŽ r, k .. Now we consider two cases. Case 1. If G␣ s ² a r : s a r , a2 r , . . . , a l r s 14 , then by Ž11., dim V␣U s 2 Ž Ž1.r< G␣ <.Ý g g G Ž g . s Ž2 l . s 4. For any g 1 , g 2 g G, we have ␣ l
¡ a
g 2 G␣ gy1 1 s
~ a
¢ a
rqiyj
, a2 rqiyj , . . . , a l rqiyj 4
if g 1 s a j , g 2 s a i
rqiqj
b, a2 rqiqj b, . . . , a l rqiqj b 4
if g 1 s a j b, g 2 s a i
iyryj
, a iy2 ryj , . . . , a iyl ryj 4
if g 1 s a j b, g 2 s a i b.
Ž 15 . If g 1 s a j, g 2 s a i, by Ž9. we have ² eUg ⭈␣ , eUg ⭈␣ : s 1
2
s s s
Ž 1.
xgg 2 G␣ gy1 1
l
2
Ý
k
Ž x. s
Ý
cos
Ý cos k Ý
2 trh
ž
k
cos
ts1
2k
ts1
k q
k
2 Ž i y j . h k
2 Ž i y j . h
l
2
Ý Ž at rqiyj .
2 Ž tr q i y j . h
ts1
2
l
2
ž
k
/
s
2l k
/
cos
2 Ž i y j . h k
, Ž 16 .
where the equality of the one before last is due to Lemma 3. Similarly, for g 1 s a j b, g 2 s a i b, we have ² eUg ⭈␣ , eUg ⭈␣ : s 1
2
2l k
cos
2 Ž i y j . h k
,
Ž 17 .
and for g 1 s a j b, g 2 s a i, we have ² eUg ⭈␣ , eUg ⭈␣ : s 0. 1
Therefore
¡21 cos 2Ž i y j . h
k ² eUg ⭈␣ , eUg ⭈␣ : s 0 1 2
k
~
¢k
2l
Ž 18 .
2
if g 1 s a j , g 2 s a i if g 1 s a j b, g 2 s a i
cos
2 Ž i y j . h k
if g 1 s a j b, g 2 s a i b.
Ž 19 .
642
SHAHABI, AZIZI, AND JAFARI
Since 2 Ž2 hrk . - 0, hence by Lemma 2 there exist t 1 , t 2 , 0 F t 1 , t 2 F k y 1 such that cosŽ2Ž t 1 y t 2 . hrk . s 0. Put S s a t 1 ⭈ ␣ , a t 2 ⭈ ␣ , a t 1 b ⭈ ␣ , a t 2 b ⭈ ␣ 4 : ⌫nm . Then by Ž19. for every ␥ ,  g S and ␥ /  we have ² e␥U , eU : s 0. But dim V␣U s 4; hence eU N  g S4 is an orthogonal )-basis for V␣U . Case 2. If ² a r : - G␣ , then ² a r : s ² a: l G␣ and we have G␣ l ² a: s a r , a2 r , . . . , a l r s 14 and < G␣ < G 2 l. Therefore by Ž11., dim V␣U s
Ž 1. < G␣ <
Ý
Ž g. F
ggG␣
2 2l
Ž 2 l . s 2.
Hence dim V␣U s 1 or 2. If dim V␣U s 1, then it is obvious that we have an orthogonal )-basis. If dim V␣U s 2, then by Lemma 2 there exist i, j, 0 F i, j F k y 1 such that cosŽ2Ž i y j . hrk . s 0. Set g 1 s a j, g 2 s a i. Then rqiyj ² : g 2 G␣ gy1 , . . . , a l rqiyj 4 . 1 l a s a
Hence by Ž9. and Ž16. we have ² eUg ⭈␣ , eUg ⭈␣ : s 0. 1
eUa j⭈␣ ,
2
eUa i ⭈␣ 4
Thus is an orthogonal )-basis for V␣U . Conversely, let V␣U have an orthogonal )-basis for every ␣ g ⌬. Now we prove that 2 Ž2 hrk . - 0. Let a s a1 a2 ⭈⭈⭈ a r be the disjoint cycle decomposition of a in Sm and let oŽ a i . s m i , for every 1 F i F r. Since oŽ a. s k, so k s w m1 , . . . , m r x, the least common multiple of m1 , . . . , m r . Set a i s Ž i1 , i 2 , . . . , i m i . and define ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ m . g ⌫nm in such a way that ␣ i1 s 1, ␣ i 2 s ␣ i 3 s ⭈⭈⭈ s ␣ i m s 2, for every 1 F i F r, and for j g 1, 2, . . . , m4 _ i r D is1 i1 , . . . , i m i 4 , ␣ j s 1. EXAMPLE. Let a s Ž567.Ž1234. g S8 , where a1 s Ž567., a2 s Ž1234.. Then 11 s 5, 1 2 s 6, 1 3 s 7, 2 1 s 1, 2 2 s 2, 2 3 s 3, 2 4 s 4. Hence j s 8 and ␣ 8 s 1. If ␣ s Ž ␣ 1 , ␣ 2 , . . . , ␣ 8 . g ⌫n8 , then ␣ 11 s ␣ 5 s 1, ␣ 1 2 s ␣ 6 s 2, ␣ 1 3 s ␣ 7 s 2, ␣ 2 1 s ␣ 1 s 1, ␣ 2 2 ss ␣ 2 s 2, ␣ 2 3 s ␣ 3 s 2, ␣ 2 4 s ␣ 4 s 2. Hence ␣ s Ž1, 2, 2, 2, 1, 2, 2, 1.. Now we show that a t f G␣ , for every 1 F t F k y 1. Since t - k s w m1 , . . . , m r x, hence m s ¦ t, for some 1 F s F r. Let t s m s q q r 1 , where q
SYMMETRY CLASSES OF TENSORS
643
and r 1 are integers and 0 - r 1 - m s . Then a t s a1t ⭈⭈⭈ a ts ⭈⭈⭈ a tr s a1t ⭈⭈⭈ a sr 1 ⭈⭈⭈ a rt . Since a sr 1 moves the position s1 in ␣ , and all other a it Ž i / s . fix the elements of s1 , s2 , . . . , sm s 4 , then a t does not fix the position s1 in ␣ ; therefore a t f G␣ . Now we show that G␣ s 14 or G␣ s 1, a t b4 , for some 0 F t F k y 1. If G␣ / 14 and a t 1 b, a t 2 b g G␣ , then a t 1 b ⭈ a t 2 b s a t 1yt 2 g G␣ . This implies that t 1 s t 2 . In both cases we have 5 eU␣ 5 2 s
2 2k
Ý
Ž g. s
ggG␣
2 k
/ 0.
Ž 20 .
Therefore ␣ g ⌬. Now we prove that in both cases 2 Ž2 hrk . - 0. Case 1. Let G␣ s 14 . By Ž11. dim V␣U s
2 1
Ž 2 . s 4.
For every g 1 , g 2 g G, we have
¡ a
g 2 G␣ gy1 1 s
iyj
~ a ba
¢ a
if g 1 s a j , g 2 s a i
4
i
yj
iyj
4
if g 1 s a j , g 2 s a i b
4
if g 1 s a j b, g 2 s a i b.
Hence
¡2 cos 2Ž i y j . h
k ² eUg ⭈␣ , eUg ⭈␣ : s 0 1 2
~
¢k cos 2
k
if g 1 s a j , g 2 s a i if g 1 s a j , g 2 s a i b
2 Ž i y j . h k
if g 1 s a j b, g 2 s a i b.
Since dim V␣U s 4, hence there exist 0 F i, j F k y 1 such that cosŽ2Ž i y j . hrk . s 0. Therefore by Lemma 2, 2 Ž2 hrk . - 0. Case 2. If G␣ s 1, a t b4 , for some 0 F t F k y 1, then by using a similar argument as w4, p. 25x we prove that 2 Ž2 hrk . - 0. Remark 1. The condition of Theorem 1 is equivalent to the condition of Theorem 3.1 in w4x. If 0 - h - kr2 and h s h 2 h 2 ⬘ with h 2 a power of 2
644
SHAHABI, AZIZI, AND JAFARI
and h 2 ⬘ odd, then 2 Ž2 hrk . - 0 if and only if 2 Ž2 h 2 h 2 ⬘rk . - 0 if and only if 4 h 2 N k. COROLLARY 1. Let G s D 2 k F Sm and assume that n s dim V G 2. Then m m 1 V has an orthogonal )-basis if and only if k is a power of 2. Proof. Let k s k 2 k 2 ⬘ with k 2 a power of 2 and k 2 ⬘ odd. Assume that k 2 - k. Then 0 - k 2 - kr2 and 4 k 2 ¦ k. Therefore, if s k 2 , then Theorem 1 implies that V Ž G . has no an orthogonal )-basis. Conversely, assume k is a power of 2. If 0 - h - kr2, then h 2 F kr4, and so 4 h 2 N k, where h s h 2 h 2 ⬘ with h 2 a power of 2 and h 2 ⬘ odd. Then Theorem 1 implies that V hŽ G . has an orthogonal )-basis, and so m 1m V. 2.2. Dicyclic Group The dicyclic group of order 4 k Ž k G 2. is defined by T4 k s ² a, b N a2 k s 1, a k s b 2 , by1ab s ay1 : . In particular, T4 k s a i, a i b N 0 F i F 2 k y 14 . For each integer 0 - h - k, T4 k has an irreducible non-linear character h of degree 2 given by
h Ž a r . s 2 cos
rh k
h Ž a r b . s 0, 0 F r F 2 k y 1.
,
ŽSee w5x, p. 187x.. The other irreducible characters of T4 k are linear. THEOREM 2. Let G s T4 k Ž k G 2. be a subgroup of Sm , let s h Ž0 - h - k ., and assume n s dim V G 2. Then V Ž G . has an orthogonal )-basis if and only if 2 Ž hrk . - 0. Proof. If 2 Ž hrk . - 0, the proof of the existence of the orthogonal )-basis is similar to Theorem 1. ŽFor more details see w2x.. Conversely, let V␣U have an orthogonal )-basis for every ␣ g ⌬. Now we prove that 2 Ž hrk . - 0. As Theorem 1 we take a s a1 a2 ⭈⭈⭈ a r to be the disjoint cycle decomposition of a in Sm . By taking the notation as Theorem 1 you may prove that a t f G␣ , for every 0 F t F 2 k y 1. Now we show that G␣ s 14 . If G␣ / 14 , then a t b g G␣ for some 0 F t F 2 k y 1. Now Ž a t b . 2 g G␣ . But Ž a t b . 2 s a t b ⭈ a t b s b 2 s a k g G␣ , and this is impossible. Therefore G␣ s 14 . In this case we have 5 eU␣ 5 2 s
2 4k
Ý ggG␣
Ž g. s
4 4k
s
1 k
/ 0.
SYMMETRY CLASSES OF TENSORS
645
Therefore ␣ g ⌬. By using a similar argument as w2x we can prove that 2 Ž hrk . - 0. Remark 2. The condition 2 Ž hrk . - 0 is equivalent to 2 h 2 N k, where h s h 2 h 2 ⬘ with h 2 a power of 2 and h 2 ⬘ odd. COROLLARY 2. Let G s T4 k F Sm and assume n s dim V G 2. Then m m1 V has an orthogonal )-basis if and only if k is a power of 2. Proof. Let k s k 2 k 2 ⬘ with k 2 a power of 2 and k 2 ⬘ odd. Assume that 0 - k 2 - k; then 2 k 2 ¦ k. By Theorem 2, if s k 2 , then V Ž G . has no orthogonal )-basis. Conversely, assume k is a power of 2. If 0 - h - k, then h 2 - kr2 and so 2 h 2 N k. Then Theorem 2 implies that V hŽ G . has an orthogonal )-basis, and so m m 1 V. In w7x M. Shahryari proved the following theorem. THEOREM. Let be a non-linear irreducible character of G and suppose that there is ␣ g ⌬ such that < G : G␣ < 2 Ž 1.
- , 1 G␣ -
< G : G␣ <
Ž 1.
.
Then V Ž G . has no orthogonal )-basis. It is obvious that the condition of this theorem is equivalent with 1
- 5 eU␣ 5 2 - 1.
2
Using Theorem 1, we show that 1r2 is the best lower bound for 5 eU␣ 5 2 for which one can obtain. Also we give a counter-example that the converse of this theorem is not true. Remark 3. Let G s D 8 F S8 Ž k s 4.. By Ž13., G has only one non-linear irreducible character h Ž0 - h - 2.. Now it is immediate from Theorem 1 that V hŽ G . has an orthogonal )-basis. Now let ␣ be an arbitrary element of ⌬. Then Lemma 3 implies that k N rh Ž0 F r F 3, k s 4, h s 1.. So 4 N r, and consequently r s 0, l s 1. Thus 5 eU␣ 5 2 s 2r4 s 1r2. Remark 4. Counter-example. Let p G 5 be a prime and G s D 2 p s ² a, b N a p s 1 s b 2 , bab s ay1 : and let G be a subgroup of S2 p . Then the non-linear characters of D 2 p are h , where 0 - h - pr2 and
h Ž a r . s 2 cos
2 hr p
,
h Ž a r b . s 0, 0 F r F p y 1.
The other irreducible characters of D 2 p are of degree 1.
646
SHAHABI, AZIZI, AND JAFARI
It is easy to see that ⌬ is non-empty. If we take ␣ as in the converse of Theorem 1, then by Ž20., 5 e␣ 5 2 s 2rp / 0. Hence ␣ g ⌬. Therefore V Ž G . / 0. Since 2 Ž2 hrp . ) 0, so by the Theorem 1, V hŽ G . has no orthogonal )-basis. Now we show that 5 eU␣ 5 2 - 1r2 for every ␣ g ⌬. Let ␣ g ⌬. By Lemma 3, we have G␣ l ² a: s ² a r : , for some 0 F r F p y 1, p N rh. Since 0 - h - pr2, hence p N r and so r s 0. But l s < G␣ l ² a:< s <² a r :<. Hence l s 1 and 5 eU␣ 5 2 s Ž2 = 1.rp s 2rp - 1r2. Remark 5. Using Theorems 1 and 2, one can see that the existence of an orthogonal )-basis for the abelian, dihedral, dicyclic Žin particular, generalized quaternion. groups is independent of permutation structure of the given group. This may give rise to the following question. Question. Is it true that the existence of an orthogonal )-basis for V ŽG. is independent of the permutation structure of group G? REFERENCES 1. L. J. Cummings, Cyclic symmetry classes, J. Algebra 40 Ž1976., 401᎐405. 2. M. R. Darafsheh and M. R. Pournaki, On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group, Linear and Multilinear Algebra, in press. 3. R. Freese, Inequalities for generalized matrix functions based on arbitrary characters, Linear Algebra Appl. 7 Ž1973., 337᎐395. 4. R. R. Holmes and T. Y. Tam, Symmetry classes of tensors associated with certain groups, Linear and Multilinear Algebra 32 Ž1992., 21᎐31. 5. G. James and M. Liebeck, ‘‘Representations and Characters of Groups,’’ Cambridge Univ. Press, Cambridge, UK, 1993. 6. R. Merris, ‘‘Multilinear Algebra,’’ Gordon & Breach, New York, 1997. 7. M. Shahryari, On the orthogonal bases of symmetry classes, J. Algebra, in press.