supercharacter table of certain finite groups
aa r X i v : . [ m a t h . G R ] F e b Supercharacter Table of Certain Finite Groups
H. Saydi ,a , M. R. Darafsheh ,b and A. Iranmanesh ,c (1) College of Mathematical Science, Tarbiat Modares University, Tehran, Iran(2) School of mathematics, statistics and computer science, College of science, University ofTehran, Tehran, Iran(a) e-mail: [email protected](b) e-mail: [email protected](c) e-mail: [email protected] Abstract
Supercharacter theory is developed by P. Diaconis and I. M. Isaacs as a natural generaliza-tion of the classical ordinary character theory. Some classical sums of number theory appearas supercharacters which are obtained by the action of certain subgroups of GL d ( Z n ) on Z dn .In this paper we take Z dp , p prime, and by the action of certain subgroups of GL d ( Z p ) we findsupercharacter table of Z dp . Keywords: Supercharacter, Superclass, Ramanujan sum,Kloosterman sum, Char-acter table.
Let
Irr ( G ) denote the set of all the irreducible complex characters of a finite group G , and let Con ( G ) denote the set of all the conjugacy classes of G . The identity element of G is denoted by1 and the trivial character is denoted by 1 G . By definition a supercharacter theory for G is a pair( X , K ) where X and K are partitions of Irr ( G ) and G respectively, |X | = |K| , { } ∈ K , and foreach X ∈ X there is a character σ X such that σ X ( x ) = σ X ( y ) for all x, y ∈ K , K ∈ K . We call σ X supercharacter and each member of K a superclass. We write Sup ( G ) for the set of all thesupercharacter theories of G .Supercharacter theory of a finite group were defined by Diaconis and Isaacs [3] as a general caseof the ordinary character theory. In fact, in a supercharacter theory characters play the role ofirreducible ordinary characters and union of conjugacy classes play the role of conjugacy classes.1n [3] it is shown that { G } ∈ X and if X ∈ X then σ X is a constant multiple of P χ ∈ X χ (1) χ , andthat we may assume that σ X = P χ ∈ X χ (1) χ .For any finite group there are two trivial supercharacter theories as follows. In the first case, X = S { χ } χ ∈ Irr ( G ) and K is the set of all conjugacy classes of G . In the second case, X = { G } ∪ { Irr ( G ) − { G }} and K = { } ∪ { G − { }} . In the first case, supercharacters are just irreducible characters andsuperclasses are conjugacy classes. In the second case, the non-trivial supercharacter is ρ G − G ,where ρ G denotes the regular character of G . These two supercharacter theories of G are denotedby m ( G ) and M ( G ) respectively.It is mentioned in [6] that the set of supercharacter theories of a group form a Lattice in thefollowing natural way. Sup ( G ) can be made to a poset by defining ( X , K ) ≤ ( Y , L ) if X ≤ Y inthe sense that every part of X is a subset of some part of Y . In [6] it is shown that this definitionis equivalent to ( X , K ) ≤ ( Y , L ) if K ≤ L . By this definition m ( G ) is the least and M ( G ) is thelargest element of Sup ( G ).Among construction of supercharacter theories of a finite group G the following is of greatimportance which is a lemma by Brauer on character tables of groups. Let A be a subgroup of A ut ( G ) and Irr ( G ) = { χ = 1 G , . . . , χ h } , Con ( G ) = {C = { } , . . . , C h } . Suppose for each α ∈ A , C iα = C j , 1 ≤ i ≤ h , and χ iα ( g ) = χ i ( g α ) for all g ∈ G , α ∈ A , then the number of conjugacyclasses fixed by α equals the number of irreducible characters fixed by α , and more over the numberof orbits of A on Con ( G ) equals the number of orbits of A on Irr ( G ), [4]. It is easy to see that theorbits of A on Irr ( G ) and Con ( G ) yield a supercharacter theory for G . This supercharacter theoryof G is called automorphic. In [7] it is shown that all the supercharacter theories of the cyclic groupof order p , p prime, are automorphic.Another aspect of the supercharacter theory of finite groups is to employ the theory to the group U n ( F ), the group of n × n unimodular upper triangular matrices over the Galois field GF ( p m ), p prime. Computation of the conjugacy classes and irreducible characters of U n ( F ) is still open,but in [1] the author has developed an applicable supercharacter theory for U n ( F ). This result isreviewed in [3]. Let G be a finite group and ( X , K ) be a supercharacter theory for G . Suppose X = { X , X , . . . , X h } be a partition for Irr ( G ) with the corresponding supercharacter σ i = P χ ∈ X i χ (1) χ .Let K = { K , K , . . . , K h } be the partion of G into superclasses. In fact X = { G } , K = { } and K i ’s are conjugacy classes of G . The supercharacter table of G corresponding to ( X , K ) is thefollowing h × h array: 2 K · · · K j · · · K h σ σ ( K ) σ ( K ) · · · σ ( K j ) · · · σ ( K h ) σ σ ( K ) σ ( K ) · · · σ ( K j ) · · · σ ( K h )... ... ... ... σ i σ i ( K ) σ i ( K ) · · · σ i ( K j ) · · · σ i ( K h )... ... ... ... σ h σ h ( K ) σ h ( K ) · · · σ h ( K j ) · · · σ h ( K h )Let us set S = ( σ i ( K j )) hi,j =1 , and call it the supercharacter table of G .Recall that a class function on G is a function f : G −→ C which is constant on conjugacyclasses of G . The set of all the class functions on G , Cf ( G ) has the structure of a vector spaceover C with an orthonormal basis Irr ( G ) with respect the inner product h f, g i = 1 | G | P x ∈ G f ( x ) g ( x ).Since supercharacters are constant on superclasses, it is natural to call them superclass functions.We have: h σ i , σ j i = 1 | G | h X k =1 | K k | σ i ( K k ) σ j ( K k )But using the orthogonality of Irr ( G ) we also can write: h σ i , σ j i = h X χ ∈ X i χ (1) χ, X ϕ ∈ X j ϕ (1) ϕ i = δ ij X χ ∈ X i χ (1) Therefore 1 | G | h X k =1 | K k | σ i ( K k ) σ j ( K k ) = δ ij X χ ∈ X i χ (1) . If we set the matrix U = 1 p | G | σ i ( K j ) p | K j | r P χ ∈ X i χ (1) hi,j =1 We see that U is a unitary matrix with the following properties, which are proved in [2]. We have U = U t , U = P where P is a permutation matrix and U = I .In the course of studying the supercharacter theory of a group G finding the supercharactertable of G and the matrix U is of great importance. In this paper, we will do this task for certaingroups acting on certain sets. In this section we follow the method used in [2] considering the group G = Z dn which abelian oforder n d . The automophism group of G is GL d ( Z n ), the group of d × d invertible matrices withentries in Z n . We write elements of G as row vectors y = ( y , . . . , y d ) and let the action of GL d ( Z n )on G be as follows: 3 A = yA for A ∈ GL d ( Z n ).Irreducible characters of G are of degree 1 and the number of them is equal to | G | . For x ∈ G , letus define ψ x : G −→ C × , by ψ x ( ζ ) = e ( x · ζn ), where e ( t ) stands for e ( t ) = e πit and x · ζ is the innerproduct of two elements x and ζ of G as row vectors in G = Z dn . Therefore Irr ( G ) = { ψ x | x ∈ G } and the action of GL d ( Z n ) on Irr ( G ) is as follows: ψ Ax = ψ xA − t where A ∈ GL d ( Z n ), x ∈ G .Now let Γ be a symmetric subgroup of GL d ( Z n ), i. e. Γ t = Γ. Then Γ acts on G and Irr ( G ) asabove Let X be the set of orbits of Γ on Irr ( G ) and K be the set of orbits of GL d ( Z n ) on Irr ( G ).It is shown in [2] that ( X , K ) is a supercharacter theory of G . Following the notations used in[2] we identity ψ x with x and ψ Ax = ψ xA − t = xA − t . Therefore X is identified with the set of orbitsof GL d ( Z n ) on G , by x xA − t , and K is identified with the orbits of the action of GL d ( Z n ) on G by y yA .In [2] using different subgroups of GL d ( Z n ) the authors provide supercharacter tables for G . Forexample the discrete Fourier transform in the case of Γ = { } , or Γ = {± } a group of order 2. TheGauss sums is obtained in the case of G = Z p , p an odd prime, Γ = h g i where g is a primitive rootmodulo p . Kloosterman sums in the case G = Z p , p an odd prime and Γ = { " a a − | = a ∈ Z p } .Heilbronn sums, in the case of G = Z p and Γ = { x p | = x ∈ Z p } . The Ramanujan sums in the caseof G = Z n and Γ = Z × n . It is worth mentioning that all the above sums appear as supercharacters.As a generalization of the group Γ in Kloosterman sum we letΓ = { " a b | a, b ∈ Z × p } a group of order ( p − . Here G = Z p × Z p and orbits of Γ on G are: Y = { (0 , } of size 1 Y = (1 , { ( a, | a ∈ Z × p } of size p − Y = (0 , { (0 , b ) | b ∈ Z × p } of size p − Y = (1 , { ( a, b ) | a, b ∈ Z × p } of size ( p − Orbits of Γ on
Irr ( G ) are as follows: X = { (0 , } of size 1 X = (1 , { ( a, | a ∈ Z × p } of size p − X = (0 , { (0 , b ) | b ∈ Z × p } of size p − X = (1 , { ( a, b ) | a, b ∈ Z × p } of size ( p − Now we form the supercharacter table of G ∼ = Z p × Z p . Let σ i be the supercharacter associatedwith X i , with σ = 1. 4e know σ i = P ψ xi ∈ X i ψ x i , and for y ∈ Y j , σ i ( y ) = P ψ xi ∈ X i ψ x i ( y ) = P x i ∈ X i e ( x i · yp ), 1 ≤ i ≤ Z p Y Y Y Y superclass size 1 p − p − p − σ σ p − − p − − σ p − p − − − σ ( p − − ( p − − ( p − − ( p − U we use the formula written down in section 2 to obtain the 4 × U as follows: U = 1 p √ p − √ p − p − √ p − − p − −√ p − √ p − p − − −√ p − p − −√ p − −√ p − At this point it is convenient to consider the general case of G = Z dp ,Γ = { a a . . . . . . a d | a i ∈ Z × p } the diagonal subgroup of order ( p − d of GL d ( Z p ).Orbits of Γ on G are as follows: Y = { (0 , , . . . , } is one orbit. Let y ( k ) = (1 k , d − k ) be a vector of G with k one’s in differentpositions. Then y ( k ) Γ consists of vectors with non-zero entries in exactly k different positions.Therefor the orbit y ( k ) has size ( p − k . Since this k positions is taken out of d positions, thereforewe have dk ! orbits of this shapes each of size ( p − k . Hence we have d P k =0 dk ! = 2 d orbits of Γon G . Each orbit has size ( p − k . Since d P k =0 dk ! ( p − k = p d = | G | , all the orbits are counted.Orbits of Γ on Irr ( G ) have the same setting as above. In this case if ψ x is a representative ofthe orbit X of Γ on Irr ( G ), then we may assume x = x ( l ) = (1 l , d − l ) is a vector with l ones indifferent positions, hence. σ X ( y ) = X x ∈ X ψ x ( y ) = X x ∈ X e ( x · yp )And it is computable if the inner product x · y is known.5 J-Symmetric groups
Let G = Z dn and Γ be a subgroup of GL d ( Z n ). By [2] we have to assume that Γ is symmetric, i.e.Γ = Γ t , in order to conclude that the action of Γ on G and on Irr ( G ) generate the same orbits.Most of the results on supercharacter theory of G holds if we assume Γ is J-Symmetric. Supposethere is a fixed symmetric invertible matrix J ∈ GL d ( Z n ) such that J Γ = Γ t J . As before the actionof Γ on G is by y yA and by identifying ψ x ∈ Irr ( G ) with x , the action of Γ on Irr ( G ) is by x xA − t for A ∈ Γ.If ( X , Y ) is the supercharacter theory obtained in this way, then we set X = { X , X , . . . , X h } and Y = { Y , Y , . . . , Y h } and σ i = σ X i , 1 ≤ i ≤ r , then the unitary matrix U is replaced by U = 1 √ n d " σ i ( Y j ) p | Y j | p | X i | hi,j =1 .In this section we consider G = Z p , p a prime, and J = ,Γ = { a b c d b a | a, b ∈ Z × p , b, c ∈ Z p } is a subgroup of GL ( Z p ) of order p ( p − . It is obvious that Γ is a J-Symmetric group.Orbits of Γ on G are as follows: Y = { (0 , , } Y = (0 , , { (0 , , a ) | a ∈ Z × p } Y = (0 , , { (0 , d, b ) | d ∈ Z × p , b ∈ Z p } Y = (1 , , { ( a, b, c ) | a ∈ Z × p , b, c ∈ Z p } . We have | Y | = 1 | Y | = p − | Y | = p ( p − | Y | = p ( p − . Since | Y | + | Y | + | Y | + | Y | = p , we deduce that Y , Y , Y and Y are orbits of Γ on G . It is easyto see that the orbits of Γ on Irr ( G ) are as follows: X = { (0 , , } X = (1 , , { ( a, , | a ∈ Z × p } X = (0 , , { ( a, b, | a ∈ Z p , b ∈ Z × p } X = (1 , , { ( a, b, c ) | a, b ∈ Z p , c ∈ Z × p } .