Symmetry classes of tensors as group modules

Journal of Algebra 393 (2013) 30–40

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Symmetry classes of tensors as group modules ✩ M.H. Jafari, A.R. Madadi ∗ Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 22 November 2012 Available online 26 July 2013 Communicated by Ronald Solomon MSC: primary 20C15, 15A69 secondary 11A07, 11A25 Keywords: Characters of finite groups Symmetry classes of tensors Arithmetic functions

Let V be a finite dimensional vector space over C, H a subgroup of S m , λ an irreducible character of H, and V λ ( H ) the symmetry class of tensors associated with H and λ. If the vector space V is a C[G ]-module, where G is a finite group, then the corresponding symmetry class of tensors naturally becomes a C[G ]-module. In this paper, we study V λ ( H ) as a C[G ]-module and then obtain the corresponding character. Finally we will obtain some interesting applications of the character of this C[G ]-module to the character theory of finite groups and the theory of numbers. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Let us begin with some definitions and preliminaries. Our notations are standard and follow [5] and [6]. Let V be an n-dimensional vector space over C, H a subgroup of the full symmetric group S m , and V ⊗m the tensor product of m copies of V . For any σ ∈ H , define the operator

P σ : V ⊗m → V ⊗m by

P σ ( v 1 ⊗ · · · ⊗ v m ) = v σ −1 (1) ⊗ · · · ⊗ v σ −1 (m) . ✩ This paper is published as part of a research project supported by the University of Tabriz Research Affaires Office (S/27/2545). Corresponding author. E-mail addresses: [email protected] (M.H. Jafari), [email protected] (A.R. Madadi).

*

0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.06.033

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

31

Now if λ is any complex irreducible character of H , then the symmetry class of tensors associated with H and λ, denoted by V λ ( H ), is defined as the image of the following symmetrizer

Sλ =

λ(1)  λ(σ ) P σ . |H | σ ∈H

For simplicity the tensor S λ ( v 1 ⊗ · · · ⊗ v m ) is denoted by v 1 ∗ · · · ∗ v m and is called a symmetrized decomposable tensor. Note that if H = S m and λ = 1 H , the principal character of H , then V λ ( H ) is V (m) , them-th m V, symmetric power of V , and if H = S m and λ =  , the alternating character of H , then V λ ( H ) is the m-th Grassman space of V . Now let Γnm be the set of all sequences of natural numbers

α = (α1 , . . . , αm ) with 1  αi  n. It is easy to see that H acts on Γnm via

α σ = (ασ −1 (1) , . . . , ασ −1 (m) ). Let  be a system of representatives of the orbits of this action,

  Ω = α ∈ Γnm : [λ, 1 H α ] = 0 , and  =  ∩ Ω . Suppose that {e 1 , . . . , en } is a basis for V and denote e α (1) ∗ · · · ∗ e α (m) by e ∗α , for . For any α ∈ , we define the orbital subspace of α any α ∈ Γnm . It is known that e ∗α = 0 iff α ∈ Ω as V α∗ = e ∗α σ : σ ∈ H . It follows that V λ ( H ) = V α∗ , where α runs over . Obviously there exists a  } is a subset α of the orbit of α such that {e ∗β : β ∈ α } is a basis for V α∗ . Thus, the set {e ∗β : β ∈ 

= basis for V λ ( H ), where  It is proved that



α ∈ α .

sα := dim V α∗ = λ(1)[λ, 1 H α ], and

dim V λ ( H ) =

λ(1)  λ(σ )nc(σ ) , |H | σ ∈H

where c (σ ) is the number disjoint cycles (including cycles of length one) in the disjoint cycle decomposition of σ . We refer the reader to [5], chapter two, and [6], chapter six, for these results and some other information about symmetry classes of tensors. Our purpose in the sequel is to decompose V λ ( H ) into “special” subspaces. Let A(σ ) = (ai j (σ )) be any complex irreducible representation of H which affords λ. For any 1  i , j  λ(1), we define the ij subspace V A ( H ) of V ⊗m as the image of the following operator

Tij =

λ(1)  ai j (σ ) P σ . |H | σ ∈H

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M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

It is proved that T i j T rs = δ jr T is and S λ = ii VA ( H ) and

λ(1)

ij

T ii . We deduce from these relations that V A ( H ) =

i =1

V λ(H ) =

λ(1)

ii VA ( H ).

i =1

For any α ∈ , we now subdivide V α∗ into subspaces. First let e ⊗ α denote e α (1) ⊗ · · · ⊗ e α (m) . Now for any 1  i  λ(1), let V αi =  T i j (e ⊗ α ): 1  j  λ(1) . It is proved that

V α∗ =

λ(1)

V αi ,

i =1



ii VA (H ) =

V αi ,

α ∈ 1 ⊗ and if { T 1 j (e ⊗ α ): j ∈ J α } is a basis for V α for some subset J α of {1, . . . , λ(1)}, then { T i j (e α ): j ∈ J α } is a basis for V αi , and hence the set

Bi =





T i j e⊗ α : j ∈ Jα



α ∈ ii is a basis for V A ( H ). This shows that sα = λ(1) dim V αi and

ii dim V A (H ) =

1

λ(1)



sα =

α ∈

dim V λ ( H )

λ(1)

,

for any 1  i  λ(1) and α ∈ . We refer the reader to [6], chapter eight, and [8] for the abovementioned results. In the continuation, let x1 , . . . , xn be independent indeterminates over C. Then the weight of α ∈ Γnm is defined by

w (α ) =

m 

xα (t ) .

t =1

Also the character weighted pattern inventory is defined by λ WH (x1 , . . . , xn ) =



[λ, 1 H α ] w (α ).

α ∈

For example, if λ = 1 H , then we obtain the traditional pattern inventory 1

W HH (x1 , . . . , xn ) =

 α ∈

w (α ),

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

and if H = S m and λ is the irreducible character of S m associated to a partition the well-known Schur polynomial

λ WH (x1 , . . . , xn ) =



K π ,ν (α )

α ∈G m,n

m 

33

π of m, then we have

xα (t ) ,

t =1

where G m,n is the set of all nondecreasing sequences in Γnm , ν (α ) is the multiplicity partition of α , and K π ,ν (α ) are the so-called Kostka coefficients. Now let y 1 , . . . , ym be independent indeterminates over C. Then the character weighted cycle index polynomial afforded by H and λ is defined by

Z λH ( y 1 , . . . , ym ) =

m  λ(1)  c (σ ) λ(σ ) yt t , |H |

σ ∈H

t =1

where ct (σ ) is the number of cycles of length t in the disjoint cycle decomposition of Finally for any 1  t  m, the t-th power sum polynomial is defined as follows:

σ.

P t := P t (x1 , . . . , xn ) = xt1 + · · · + xnt . It is proved that

λ(1) W Hλ (x1 , . . . , xn ) = Z λH ( P 1 , . . . , P m ). The reader may consult the references [6,7,9,10] for these definitions and their applications to the theory of enumeration. 2. A C [ G ]-module structure on V λ ( H ) and its character Symmetry classes of tensors have been studied extensively as vector spaces. Now it is natural to ask the following question: If a vector space has some structure which induces on its corresponding symmetry class of tensors, then what can be obtained from the induced structure? As far as the authors are aware of, for the first time the second author of this paper and his colleague in [4] studied the symmetry classes of tensors from Lie algebra module point of view. In fact, they have shown that if the vector space V is an L-module, where L is a Lie algebra, then this induces an L-module structure on the corresponding symmetry class of tensors. In particular, they have been able to determine explicitly the irreducible constituents of the symmetry classes of tensors as sln (C)-modules. Now we want to look at the symmetry classes of tensors from a different angle. Our purposes here are indeed first giving a group module structure to the symmetry classes of tensors and then obtaining the corresponding characters. Throughout let G be any finite group and V be a C[G ]-module of dimension n with representation X which affords the character χ . Also let V λ ( H ) be the symmetry class of tensors associated with H and λ, where H is a subgroup of S m and λ is an irreducible character of H which is afforded by some representation A of H . Note that V λ ( H ) can be zero for some H and λ, and so we will assume that V λ ( H ) is nonzero. It is known that V ⊗m is a C[G ]-module as follows:

g(v 1 ⊗ · · · ⊗ vm) = g v 1 ⊗ · · · ⊗ g vm, where g ∈ G and v i ∈ V . Hence if Y is the corresponding representation, then we have

Y ( g )( v 1 ⊗ · · · ⊗ v m ) = X ( g ) v 1 ⊗ · · · ⊗ X ( g ) v m .

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M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

It is easy to see that Y ( g ) P σ = P σ Y ( g ) for any σ ∈ H . So Y ( g ) S λ = S λ Y ( g ). This shows that V λ ( H ) remains invariant under Y ( g ) and therefore V λ ( H ) becomes a C[G ]-module as follows:

g(v 1 ∗ · · · ∗ vm) = g v 1 ∗ · · · ∗ g vm. ii Similarly, Y ( g ) T ii = T ii Y ( g ), for any g ∈ G and 1  i  λ(1), and so V A ( H ) is a C[G ]-submodule of ii V λ ( H ). Thus, the mapping Di : G −→ End( V A ( H )) defined by Di ( g ) = Y ( g ) ↓ V ii ( H ) is a representaA

ii ( H ). tion of V A ii ( H ), 1  i  λ(1), have the same charThe following theorem shows that the C[G ]-modules V A ii ( H ), acter and this character is independent of the representation A. Hence, the C[G ]-modules V A 1  i  λ(1), are all isomorphic. ii ( H ) is given by Theorem 2.1. Let 1  i  λ(1). Then the character of V A

θ( g ) =

1 

|H |

λ(σ )

σ ∈H

m  t ct (σ )

χ g

,

t =1

for any g ∈ G. Proof. Let g ∈ G be arbitrary. Then, by Lemma 2.15 of [1], there exists a basis {e 1 , . . . , en } for V such that the matrix representation of X ( g ) with respect to this basis is the diagonal matrix diag(x1 , . . . , xn ), where x1 , . . . , xn are the eigenvalues of X ( g ). So for any 1  t  m,





χ g t = tr X g t = xt1 + · · · + xnt = P t . We know that Bi = have



ii ⊗ ⊗ α ∈ { T i j (e α ): j ∈ J α } is a basis for V A ( H ), and so for any T i j (e α ) ∈ Bi we





Di ( g ) T i j e ⊗ α = Y ( g ) T i j (e α (1) ⊗ · · · ⊗ e α (m) )

= T i j Y ( g )(e α (1) ⊗ · · · ⊗ e α (m) )

= T i j X ( g )e α (1) ⊗ · · · ⊗ X ( g )e α (m)  m    = Tij xα (t ) e ⊗ α t =1

= w (α ) T i j e ⊗ α . Therefore

θ( g ) = tr Di ( g )  w (α ) = α ∈ j ∈ J α

=

 sα w (α ) λ(1)

α ∈

= W Hλ (x1 , . . . , xn )

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

= = and the proof is complete.

1

λ(1)

Z λH ( P 1 , . . . , P m )

1 

|H |

35

λ(σ )

σ ∈H

m  t ct (σ )

χ g

,

t =1

2

Remark. Note that by the definition of Z λH ( y 1 , . . . , ym ), in the formula of θ we put ct (σ ) = 0.

χ ( g t )ct (σ ) = 1 if

In the following corollary we obtain ψ , the corresponding character of the C[G ]-module V λ ( H ). Corollary 2.2. For any g ∈ G we have

ψ( g ) =

m  c (σ ) λ(1)  λ(σ ) χ gt t . |H |

σ ∈H

t =1

In particular, ker χ ⊆ ker ψ . Proof. We know that

V λ(H ) =

λ(1)

V Aii ( H ).

i =1

Also by the above theorem, the C[G ]-modules V Aii ( H ), 1  i  λ(1), have the same character. Hence ψ( g ) = λ(1)θ( g ), for any g ∈ G. This completes the proof. 2 Remark. It should be noted that if λ is nonlinear, then the C[G ]-module V λ ( H ) is not irreducible, for ψ = λ(1)θ . Now let us give an example of the corresponding character of the C[G ]-module V λ ( H ) for a particular H and λ. Example. Let λπ be the irreducible character of S m associated with the partition π of m. By Corollary 6.38 of [6], we know that V λπ ( S m ) is nonzero iff l(π )  dim V , where l(π ) is the length of π . Therefore, if l(π )  dim V , then the corresponding character of V λπ ( S m ) is given by

ψ( g ) =

l (ν )  ν λπ (1)  |Kν |λπ (ν ) χ gt t , m!

ν m

t =1

νt >0

where Kν is the conjugacy class of S m corresponding to the partition ν , λπ (ν ) the value of λπ on Kν , and νt the multiplicity of t in ν . Note that, by Theorem 2.1, (1/λπ (1))ψ is also a character of G which is Corollary 26.17 of [3]. Recall that the quasikernel (or the center) of the character







χ of G is defined as follows: 

Z (χ ) = g ∈ G: χ ( g ) = χ (1) ,

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M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

which is a subgroup of G since by Lemma 2.29 of [1],





Z (χ ) = g ∈ G: ∃ ∈ C, X ( g ) =  I . It is trivial that ker χ ⊆ Z (χ ). The following theorem shows that the relationship between the quasikernels of to that of their kernels.

χ and ψ is similar

Theorem 2.3. If g ∈ Z (χ ), then

ψ( g ) =

χ ( gm ) dim V λ ( H ). χ (1)

In particular, Z (χ ) ⊆ Z (ψ). Proof. We give two proofs for this theorem. The first is character-theoretic and long but the second is module-theoretic and short. 1) First note that for any i , j, there exist 1 , 2 ∈ C such that X ( g i ) = 1 I and X ( g j ) = 2 I . Therefore X ( g i + j ) = 1 2 I and so





χ g i χ g j = χ (1)χ g i+ j .

(∗)

If we let rσ = |{t: 1  t  m, ct (σ ) > 0}|, then we have

ψ( g ) =

m  c (σ ) λ(1)  λ(σ ) χ gt t |H |

σ ∈H

=

t =1 ct (σ )>0

m 

λ(1)  λ(σ ) χ (1)ct (σ )−1 χ g tct (σ ) by (∗) |H |

σ ∈H

t =1 ct (σ )>0

 m  m    tc (σ ) λ(1)  ct (σ )−1 t = λ(σ ) χ (1) χ g |H | σ ∈H

t =1 ct (σ )>0

t =1 ct (σ )>0

λ(1)  λ(σ )χ (1)c(σ )−rσ χ (1)rσ −1 χ g m by (∗) |H | σ ∈H  m  χ ( g ) λ(1)  c (σ ) = λ(σ )χ (1) χ (1) | H | =

σ ∈H

χ ( gm ) dim V λ ( H ). = χ (1) 2) We know that X ( g ) =  I , where  = χ ( g )/χ (1). Hence g v =  v, for any v ∈ V . It follows that g ( v 1 ⊗ · · · ⊗ v m ) =  m ( v 1 ⊗ · · · ⊗ v m ), for any v 1 ⊗ · · · ⊗ v m ∈ V ⊗m , where  m = χ ( g m )/χ (1). Therefore

ψ( g ) =  m ψ(1) = and the proof is complete.

2

χ ( gm ) dim V λ ( H ), χ (1)

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

37

The next theorem says that if V is a direct sum of one-dimensional C[G ]-submodules, then so is V λ ( H ). Theorem 2.4. If χ is a sum of linear characters, then so is ψ . Proof. By Corollary 6.6 of [6], we know that



V ⊗m =

V λ ( H ).

λ∈Irr( H )

Therefore



χm =

ψλ ,

λ∈Irr( H )

where ψλ is the corresponding character of the C[G ]-module V λ ( H ) if V λ ( H ) is nonzero, and zero otherwise. Now if χ is a sum of linear characters, then so is χ m . It follows that ψλ is a sum of linear characters too, for any λ for which V λ ( H ) is nonzero. 2 Remark. Note that the converse of the above theorem is not true. For example, let H = S 2 and G be the Dihedral group of order 8. We know that G has only four linear characters and a unique nonlinear irreducible character of degree two. If we consider the unique nonlinear irreducible character of G as χ , then χ 2 is the sum of four linear characters of G, and so ψλ is a sum of linear characters of G, for any irreducible character λ of H , while χ is not. 3. Applications In this section we give some interesting consequences of our main result of the previous section to the character theory of finite groups as well as the theory of numbers. We will use two famous arithmetic functions: ϕ , the Euler totient function, and μ, the Möbius function. The following theorem gives some new characters of a finite group from a given character of that group. As we will see soon, it is proved that it is really a fruitful result. Theorem 3.1. Let χ be a nonlinear character of a finite group G and m, r be two natural numbers. Then the following function



ψ( g ) =

1   m

d|m

k|d d|kr

μ(k) k

 d



χ gd

md

is also a character of G. Proof. Let H = σ  S m , where σ = (1 2 · · · m) is an m-cycle of S m . Let us first gather together some well-known facts. For each divisor d of m, we define

 A d = k: 1  k  m, (k, m) = Hence

{1, 2, . . . , m } =

◦  d|m

Ad .

m d

 .

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M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

It can be easily verified that

 Ad =

m d

 k: 1  k  d, (k, d) = 1 ,

and so | A d | = ϕ (d). It is well known that if k ∈ A d , then of length d. Thus

ct



σ

k

=

m

σ k is the product of

m d

disjoint cycles, each

t = d,

d

t = d.

0

From character theory we also know that Irr( H ) = {λs : 1  s  m}, where λs (σ k ) = ωks and ω is a fixed primitive m-th root of unity. We are now ready to complete the proof. First note that there is a unique 1  r0  m such that r ≡ r0 (mod m). Since d|m and d|kr iff d|m and d|kr0 , so we may assume that 1  r  m. Also notice that V λr ( H ) is nonzero. This is because dim V = χ (1)  2 and so we can choose two linearly independent vectors u , v ∈ V . Therefore u ∗ v ∗ · · · ∗ v = S λr (u ⊗ v ⊗ · · · ⊗ v ) = 0. Now if ψ is the corresponding character of V λr ( H ), then by Corollary 2.2 we have

ψ( g ) = = =

=

m 1 

m

m  c (σ k ) λr σ k χ gt t t =1

k =1

1  m

m  c (σ k ) λr σ k χ gt t

m

t =1

d|m k∈ A d

 1   d|m

λr σ

k

d|m

ω

m k d



χ gd

k∈ A d

 d  1  m





r



m d



χ gd

m d

.

k =1 (k,d)=1

It is easy to see that



m

ω d k : 1  k  d, (k, d) = 1



is the set of all primitive d-th roots of unity. If we let

F r (d) =

d 

m

r

ωdk ;

G r (d) =

k =1 (k,d)=1

d  k =1

then obviously

G r (d) =



d d|r , 0 otherwise.

On the other hand, we have

G r (d) =

 k|d

F r (k).

m

r

ωdk ,

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40

39

Thus, by the Möbius inversion formula, we have

F r (d) =



μ(k)G r

  d ,

k|d

k

and therefore

F r (d) =

 μ(k) k

k|d d|kr

This completes the proof.

d.

2

Remark. It should be noted that if

ψ( g ) =

χ is a linear character of G in the above theorem, then 1 m



m χ m ( g )G r (m) = χ ( g ) m|r ,

0

otherwise.

The next corollary gives two interesting characters of a finite group which perhaps have not been appeared in the literature up to now. We feel that these characters both deserve to be more known than they seem to be. Corollary 3.2. Let χ be a nonlinear character of a finite group G and m a natural number. Then the following functions

α ( g ) = m1



m

ϕ (d)χ ( g d ) d , m 1 d d ii) β( g ) = m d|m μ(d)χ ( g ) , i)

d|m

are also characters of G. Proof. Use the previous theorem. For part i), let r = m and use the identity part ii), let r = 1. 2

ter

ϕ (d) =



k|d

μ(k) k

d. For

The following corollary is indeed a generalization of Problem 4.7 of [1]. Recall that for a characχ of a finite group G and a natural number r, the function χ (r ) is defined as χ (r ) ( g ) = χ ( g r ).

Corollary 3.3. Let χ be a nonlinear character of a finite group G, p a prime number, and s a natural number. Then

1 ps

s



χ p − χ ( p)

p s −1

is a character of G. Proof. The result easily follows from part ii) of the above corollary if we choose m = p s .

2

Our final corollary gives two beautiful congruences which both can be viewed as generalizations of Fermat’s Little Theorem. These congruences are the main results of [2]. However, our approach is entirely different than theirs. Corollary 3.4. Let m be a natural number and n an integer. Then

40

i) ii)

M.H. Jafari, A.R. Madadi / Journal of Algebra 393 (2013) 30–40



m

ϕ (d)n d ≡ 0 (mod m), m d d|m μ(d)n ≡ 0 (mod m). d|m



Proof. Obviously the result is true for n = 0. For n  2, if we let G be any finite group with χ = n1G , then, by Corollary 3.2, both α (1) and β(1) are integers, and the result follows. Now if either n is a negative integer or n = 1, then n0 = 3m|n| + n  2, and so the assertion is true for n0 . Since n0 ≡ n (mod m), the result is also true for n. 2 Acknowledgments The authors would like to express their gratitude to Dr. Mohammad Shahryari for suggesting the subject of this paper and also for helpful discussions. They are also grateful to the referee for a number of valuable suggestions and comments, in particular for the second proof of Theorem 2.3 which is due to him/her. References [1] I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976. [2] I.M. Isaacs, M.R. Pournaki, Generalizations of Fermat’s Little Theorem via group theory, Amer. Math. Monthly 112 (8) (2005) 734–739. [3] G.D. James, The Representation Theory of the Symmetric Groups, Lecture Notes in Math., vol. 682, Springer-Verlag, Berlin, 1978. [4] A.R. Madadi, M. Shahryari, Symmetry classes of tensors as sln (C)-modules, Linear Multilinear Algebra 56 (5) (2005) 517–541. [5] M. Marcus, Finite Dimensional Multilinear Algebra, part I, Marcel Dekker, 1973. [6] R. Merris, Multilinear Algebra, Gordon and Breach Science Publishers, 1997. [7] R. Merris, Pattern inventories associated with symmetry classes of tensors, Linear Algebra Appl. 29 (1998) 225–230. [8] B.Y. Wang, M.P. Gong, The subspaces and orthonormal bases of symmetry classes of tensors, Linear Multilinear Algebra 30 (1991) 195–204. [9] D.E. White, A Polya interpretation of the Schur function, J. Combin. Theory 28 (1980) 272–281. [10] S.G. Williamson, Symmetry operators of Kranz products in enumerative combinatorial theory, J. Combin. Theory 11 (1971) 122–138.