aa r X i v : . [ m a t h . R T ] N ov REPRESENTATIONS OF FINITE POLYADIC GROUPS
M. SHAHRYARIAbstract.
We prove that there is a one-one correspondence betweensets of irreducible representations of a polyadic group and its Post’scover. Using this correspondence, we generalize some well-known prop-erties of irreducible characters in finite groups to the case of polyadicgroups. Introduction
A non-empty set G together with an n -ary operation f : G n → G is calledan n -ary group or a polyadic group , if the operation f is associative and forall x , x , . . . , x n ∈ G and fixed i ∈ { , . . . , n } there exists a unique element z ∈ G such that f ( x i − , z, x ni +1 ) = x . In the binary case (i.e., for n = 2), a polyadic group is just usual group.In this paper, a sequence of elements x i , x i +1 , . . . , x j is denoted by x ji . If x i +1 = x i +2 = . . . = x i + t = x , then instead of x i + ti +1 we write ( t ) x . In thisconvention f ( x , . . . , x n ) = f ( x n ) and hence the associativity of f can beformulated as f ( x i − , f ( x n + i − i ) , x n − n + i ) = f ( x j − , f ( x n + j − j ) , x n − n + j ) , where 1 ≤ i, j ≤ n , and x , . . . , x n − ∈ G .The idea of investigations of such polyadic group goes back to E. Kasner’slecture [3] at the fifty-third annual meeting of the American Association forthe Advancement of Science in 1904. But the first paper concerning thetheory of n -ary groups was written (under inspiration of Emmy Noether) byW. D¨ornte in 1928 (see [1]). In this paper D¨ornte observed that any n -arysystem ( G, f ) of the form f ( x n ) = x ◦ x ◦ . . . ◦ x n ◦ b , where ( G, ◦ ) is agroup and b is its fixed element belonging to the center of ( G, ◦ ), is an n -arygroup. Such n -ary group is called b -derived from the group ( G, ◦ ), and wewill denote it by der nb ( G, ◦ ). In the case when b is the identity of ( G, ◦ ), wesay that such n -ary group is reducible to the group ( G, ◦ ) or derived from( G, ◦ ) and we denote it by der n ( G, ◦ ). For every n > n -arygroups which are not derived from any group. An n -ary group ( G, f ) is
Date : October 29, 2018.
MSC(2010): 20N15Keywords: Polyadic Groups; Representations; Character degrees; Post’s cover . M. SHAHRYARI derived from some group if and only if it contains an element e (called an n -ary identity ) such that f ( ( i − e , x, ( n − i ) e ) = x holds for all x ∈ G and for all i = 1 , . . . , n .From the definition of an n -ary group ( G, f ) we can directly see that forevery x ∈ G there exists only one z ∈ G satisfying the equation f ( ( n − x , z ) = x. This element is called skew to x and is denoted by x . In a ternary group( n = 3) derived from a binary group ( G, ◦ ), the skew element coincides withthe inverse element in ( G, ◦ ). Thus, in some sense, the skew element is ageneralization of the inverse element in binary groups.Nevertheless, the concept of skew elements plays a crucial role in thetheory of n -ary groups. Namely, as D¨ornte proved, the following theorem istrue. Theorem 1.1.
In any n -ary group ( G, f ) the following identities f ( ( i − x , x, ( n − i ) x , y ) = f ( y, ( n − j ) x , x, ( j − x ) = y,f ( ( k − x , x, ( n − k ) x ) = x hold for all x, y ∈ G , i, j n and k n . Suppose (
G, f ) is an n -ary group. A map Λ : G → GL m ( C ) with theproperty Λ( f ( x n )) = Λ( x )Λ( x ) . . . Λ( x n )is a representation of G . The function χ ( x ) = T r Λ( x )is called the corresponding character of Λ. The number m is the degree ofrepresentation. Note that, Λ is a representation of ( G, f ), iff it is an n -aryhomomorphism G → der n ( GL m ( C )).In [2], a joint paper with W. Dudek, we studied representation theory ofpolyadic group, but representations we dealt with in that paper were consid-ered to have non-empty kernels. In this paper, we study representations ofpolyadic groups without that assumption, i.e. the representations we dealwith in this paper, may have empty kernels, as well. We will prove thatthere is a one-one correspondence between the sets of irreducible represen-tations of polyadic groups and their Post cover . Using this correspondence,we will generalize some well-known properties of irreducible characters offinite groups to finite polyadic groups.
EPRESENTATIONS OF FINITE POLYADIC GROUPS 3 Generalities
Suppose (
G, f ) is an n -ary group and a ∈ G is any fixed element. Let G ∗ a = { ( x, i ) : x ∈ G, i ∈ Z n − } . We define a binary operation of G ∗ a by( x, i ) · ( y, j ) = ( f ∗ ( x, ( i ) a , y, ( j ) a , ¯ a, ( n − − i ∗ j ) a ) , i ∗ j ) , where i ∗ j ≡ i + j +1 ( mod n − f ∗ indicates that f is used one or twice,depending on the value of i ∗ j . The set G ∗ a together with this operation is anordinary group (see [4]), and we call it Post’s cover of (
G, f ). The element( a, n −
2) is the identity of the group G ∗ a . The inverse element has the form( x, i ) − = ( f ∗ ( a, ( n − − i ) a , x, ( n − x , a, ( i +1) a ) , k ) , where k = ( n − − i )(mod ( n − Post’s coset theorem , and it is proved in [4] and also in[5].
Theorem 2.1.
Suppose H = { ( x, n −
2) : x ∈ G } . Then H E G ∗ a and G ∗ a /H ∼ = Z n − . Further, we can identify G with the subset { ( x,
0) : x ∈ G } , and in under this identification, G is a coset of H and we have f ( x n ) = x x · · · x n . It is proved that (see [4]), Post’s cover G ∗ a does not depend on a , i.e. forany a, b ∈ G , we have G ∗ a ∼ = G ∗ b . Proposition 2.2.
Suppose A is an ordinary group and a ∈ A . Then forany n ≥ , we have ( der n ( A )) ∗ a ∼ = A × Z n − . Proof . It is enough to suppose a = e , the identity element of A . We have( der n ( A )) ∗ e = { ( x, i ) : x ∈ A, i ∈ Z n − } , and also ( x, i ) · ( y, j ) = ( xy, i ∗ j )= ( xy, i + j + 1)= ( x, i )( y, j )( e, . This shows that ( der n ( A )) ∗ e = der e, ( A × Z n − ) . Now, define a map ϕ : der e, ( A × Z n − ) → A × Z n − , by ϕ ( x, i ) = ( x, i + 1).It is easy to check that ϕ is an isomorphism. M. SHAHRYARI
As a result, we see that if a ∈ GL m ( C ), then( der n ( GL m ( C ))) ∗ a ∼ = GL m ( C ) × Z n − . Now, let (
G, f ) be an n -ary group and suppose Λ : G → der n ( GL m ( C ))is any representation. Let a ∈ G be fixed. We define a new mapΛ ∗ a : G ∗ a → ( der n ( GL m ( C ))) ∗ Λ( a ) by the rule Λ ∗ a ( x, i ) = (Λ( x ) , i ) . Lemma 2.3. Λ ∗ a is a group homomorphism. Proof . Let B = Λ( a ). Note that we have Λ(¯ a ) = Λ( a ) − n = ¯ B . Now, forany x, y ∈ G and i, j ∈ Z n − , we haveΛ ∗ a (( x, i ) · ( y, j )) = Λ ∗ ( f ∗ ( x, ( i ) a , y, ( j ) a , ¯ a, ( n − − i ∗ j ) a ) , i ∗ j )= (Λ( x )Λ( a ) i Λ( y )Λ( a ) j Λ(¯ a )Λ( a ) n − − i ∗ j , i ∗ j )= (Λ( x )Λ( a ) i Λ( y )Λ( a ) j Λ( a ) − n Λ( a ) n − − i ∗ j , i ∗ j )= (Λ( x )Λ( a ) i Λ( y )Λ( a ) j − i ∗ j , i ∗ j ) . On the other hand,Λ ∗ a ( x, i ) · Λ ∗ a ( y, j ) = (Λ( x ) , i ) · (Λ( y ) , j )= (Λ( x ) B i Λ( y ) B j ¯ BB n − − i ∗ j , i ∗ j )= (Λ( x ) B i Λ( y ) B j B − n B n − − i ∗ j , i ∗ j )= (Λ( x ) B i Λ( y ) B j − i ∗ j , i ∗ j )= (Λ( x )Λ( a ) i Λ( y )Λ( a ) j − i ∗ j , i ∗ j ) . This shows that Λ ∗ a is a group homomorphism.Note that we have an isomorphism q : ( der n ( GL m ( C ))) ∗ Λ( a ) → ( der n ( GL m ( C ))) ∗ I , where I is the identity matrix. It is easy to see that q ( X, i ) = ( X Λ( a ) i , i ) , for any X ∈ GL m ( C ). As we saw in the previous section, we have also anisomorphism ϕ : ( der n ( GL m ( C ))) ∗ I → GL m ( C ) × Z n − , such that ϕ ( X, i ) = (
X, i + 1). Now, let π : GL m ( C ) × Z n − → GL m ( C ) bethe projection. Combining all of these maps, we obtainΛ ∗ = πϕq Λ ∗ a : G ∗ a → GL m ( C ) , EPRESENTATIONS OF FINITE POLYADIC GROUPS 5 which is an ordinary representation of G ∗ a . Note thatΛ ∗ ( x, i ) = πϕq Λ ∗ a ( x, i )= πϕq (Λ( x ) , i )= πϕ (Λ( x )Λ( a ) i , i )= π (Λ( x )Λ( a ) i , i + 1)= Λ( x )Λ( a ) i . Conversely, suppose Γ : G ∗ a → GL m ( C ) is an ordinary representation of G ∗ a . Since G ⊆ G ∗ a , so by restriction we obtain a representation Γ G for( G, f ). Lemma 2.4.
The maps Λ Λ ∗ and Γ Γ G are inverse to each other. Proof . We have (Λ ∗ ) G ( x,
0) = Λ ∗ ( x, x )Λ( a ) = Λ( x ) . On the other hand(Γ G ) ∗ ( x, i ) = Γ( x, a, i = Γ( x, a, i − x, · ( a, i − f ∗ ( x, (0) a , a, ( i − a , ¯ a, ( n − − ∗ ( i − a ) , ∗ ( i − f ∗ ( x, ( i ) a , ¯ a, ( n − i − a ) , i )= Γ( x, i ) . So, the maps are inverse to each other.Note that G is a generating set for G ∗ a and hence, Λ is irreducible, iff Λ ∗ is irreducible. Hence, we proved; Theorem 2.5.
Let ( G, f ) be an n -ary group. Then there is a one-onecorrespondence between the set of all irreducible representations of G andthose of G ∗ a , for any a ∈ G . This correspondence is the map Λ Λ ∗ .Especially, the number of irreducible representations of any finite polyadicgroup is finite. Applications
In this section, we apply the correspondence just we obtained, to gener-alize some results of representations theory of finite groups to the case offinite polyadic groups. In this section (
G, f ) is a finite n -ary group. Let a ∈ G be any fixed element. M. SHAHRYARI
Suppose Λ , ..., Λ k are all non-equivalent irreducible representations of G with degrees d , d , . . . , d k . Then the set of irreducible representations of G ∗ a isΛ ∗ , . . . , Λ ∗ k with the same set of degrees. Since for any i , the order of G ∗ a is divisible by d i , and since we have k X i =1 d i = | G ∗ a | , so we have; Theorem 3.1.
The number ( n − | G | is divisible by all d i , and also wehave k X i =1 d i = ( n − | G | . Denote by Irr(
G, f ), the set of all irreducible characters of G . We cangeneralize the orthogonality property of ordinary irreducible characters, forthose elements of Irr( G, f ), which have non-empty kernels.
Theorem 3.2.
Let χ, ψ ∈ Irr ( G, f ) and p ∈ ker χ , q ∈ ker ψ , and a ∈ G .Then we have n − | G | n − X i =0 X x ∈ G χ ( f ( x, ( i ) a , ( n − i − p )) ψ ( f ( x, ( i ) a , ( n − i − q )) ∗ = δ χ,ψ , where ∗ denotes complex conjugation. Proof . Let ˆ χ and ˆ ψ be the corresponding characters of G ∗ a . Suppose Λis a representation of G , whose character is χ . Thenˆ χ ( x, i ) = T r Λ ∗ ( x, i )= T r Λ( x )Λ( a ) i = T r Λ( x )Λ( a ) i Λ( p ) n − i − = T r Λ( f ( x, ( i ) a , ( n − i − p ))= χ ( f ( x, ( i ) a , ( n − i − p )) . Similarly, for ψ , we haveˆ ψ ( x, i ) = ψ ( f ( x, ( i ) a , ( n − i − q )) . EPRESENTATIONS OF FINITE POLYADIC GROUPS 7
Hence, we have δ χ,ψ = δ ˆ χ, ˆ ψ = 1 | G ∗ a | X ( x,i ) ∈ G ∗ a ˆ χ ( x, i ) ˆ ψ ( x, i ) ∗ = 1( n − | G | n − X i =0 X x ∈ G χ ( f ( x, ( i ) a , ( n − i − p )) ψ ( f ( x, ( i ) a , ( n − i − q )) ∗ . References [1] W. D¨ornte,
Unterschungen ¨uber einen verallgemeinerten Gruppenbegriff , Math. Z. (1929), 1 − . [2] W. Dudek, M. Shahryari, Representation theory of polyadic groups , Algebras andRepresentation Theory, DOI 10.1007/S10468-010-9231-9 (2010).[3] E. Kasner,
An extension of the group concept , Bull. Amer. Math. Soc. (1904),290 − . [4] J. Michalski, Covering k -groups of n -groups , Archivum Math. (Brno), (1981),207 − . [5] E. L. Post, Polyadic groups , Trans. Amer. Math. Soc. (1940), 208 − . Department of Pure Mathematics, Faculty of Mathematical Sciences, Uni-versity of Tabriz, Tabriz, Iran
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