aa r X i v : . [ m a t h . R T ] J un Representations of group rings and groups
Ted Hurley ∗ Abstract
An isomorphism between the group ring of a finite group and a ring of certain block diagonalmatrices is established. The group ring RG of a finite group G is isomorphic to the set of groupring matrices over R . It is shown that for any group ring matrix A of C G there exists a matrix P (independent of the entries of A ) such that P − AP = diag( T , T , . . . , T r ) for block matrices T i of fixed size s i × s i where r is the number of conjugacy classes of G and s i are the ranks of thegroup ring matrices of the primitive idempotents. Using the isomorphism of the group ring to thering of group ring matrices followed by the mapping A P − AP (where P is of course fixed) givesan isomorphism from the group ring to the ring of such block matrices. Specialising to the groupelements gives a faithful representation of the group. Other representations of G may be derivedusing the blocks in the images of the group elements.Examples are given demonstrating how interesting and useful representations of groups can bederived using the method.For a finite abelian group Q an explicit matrix P is given which diagonalises any group ringmatrix of C Q . The matrix P is defined directly in terms of roots of unity depending only on anexpression for Q as a product of cyclic groups. The characters and character table of Q may be readoff directly from the rows of the diagonalising matrix P . This has applications to signal processingand generalises the cyclic group case. For background on groups and group rings, including information on conjugacy classes and representationtheory, see [7], and for group ring matrices see [4]. Further information on representation theory andcharacter theory may be found in [3] and/or [6]. Results are given over the complex numbers C butmany of the results hold over other suitably chosen fields.A matrix A is said to be diagonalised by P if P − AP = D where D is a diagonal matrix. A circulantmatrix can be diagonalised by the Fourier matrix of the same size. The diagonalising Fourier matrixis independent of the particular circulant matrix; this is the basis for the finite Fourier transform andthe convolution theorem, see for example [2]. The Fourier n × n matrix satisfies F F ∗ = nI n , (and isthus a complex Hadamard matrix) and when the rows are labelled by { , g, g , . . . , g n − } , it gives thecharacters and character table of the cyclic group C n generated by g . The ring of circulant matrices over R is isomorphic to the ring of group ring matrices over R of the cyclic group, see for example [4].The group ring of a finite group is isomorphic to the ring of group ring matrices as determined in[4]. The group ring matrices are types of matrices determined by their first rows; see section 2 belowfor precise formulation. For example circulant matrices are the group ring matrices of the cyclic groupand matrices of the form (cid:18) A BB T A T (cid:19) , where A, B are circulant matrices, are determined by their firstrows and correspond to the group ring matrices of the dihedral group. See Sections 2,4 and 5 below forfurther examples.Group rings and group ring matrices will be over C unless otherwise stated. Results may hold overother fields but these are not dealt with here.An isomorphism from the ring of group ring matrices of a finite group G into certain block diagonalmatrices is established. More precisely it is shown that for a group ring matrix A of a finite group G there exists a matrix P (independent of the particular A ) such that P − AP = diag( T , T , . . . , T r ) for ∗ National Universiy of Ireland Galway, email: [email protected] T i of fixed size s i × s i where r is the number of conjugacy classes of G and the s i are theranks of the group ring matrices of the primitive idempotents. Thus the group ring C G is isomorphic tomatrices of the type diag( T , T , . . . , T r ). A faithful representation of the group itself may be given bytaking images of the group elements. Other representations of G may be obtained using the blocks inthe images of the of the group elements.See Sections 4 and 5 below for applications and examples; these show how interesting and usefulrepresentations of the groups, and group rings, may be derived by the method.The finite abelian group ring is a special case but is dealt with independently in Section 5 as moredirect information and direct calculations may be made. The diagonalising matrix is obtained directlyfrom Fourier type matrices, the diagonal entries are obtained from the entries of the first row of thegroup ring matrix and the character table may be read off from the diagonalising matrix.More precisely, for a given finite abelian group H it is shown explicitly that exists a matrix P suchthat P − BP is diagonal where B is any group ring matrix of H . The matrix P is independent of theentries of the particular group ring matrix B and the diagonal entries are given precisely in terms of theentries of the first row of B . The matrix P may be chosen so that P P ∗ = nI n and when the rows of P are labelled appropriately to the structure of the group as a product of cyclic groups, then the rows of P give the characters and character table of H .Many results for circulant n × n matrices (= group ring matrices over the cyclic group C n ) hold notjust over C but over any field F which contains a primitive n th root of unity. Similarly some results herehold over fields other than C but this aspect is not dealt with here.The idea of using group ring matrices of complete orthogonal sets of idempotents originated in [5]where these are used in the study and construction of types of multidimensional paraunitary matrices. Certain classes of matrices are determined by their first row or column. A particular type of such matricesare those corresponding to group rings. It is shown in [4] that the group ring RG where | G | = n may beembedded in the ring of n × n matrices over R in a precise manner.Let { g , g , . . . , g n } be a fixed listing of the elements of G . Consider the following matrix: g − g g − g g − g . . . g − g n g − g g − g g − g . . . g − g n ... ... ... ... ... g − n g g − n g g − n g . . . g − n g n Call this the matrix of G (relative to this listing) and denote it by M ( G ). RG -matrix Given a listing of the elements of G , form the matrix M ( G ) of G relative to this listing. An RG -matrix over a ring R is a matrix obtained by substituting elements of R for the elements of G in M ( G ). If w ∈ RG and w = n X i =1 α i g i then σ ( w ) is the n × n RG -matrix obtained by substituting each α i for g i inthe group matrix.Precisely σ ( w ) = α g − g α g − g α g − g . . . α g − g n α g − g α g − g α g − g . . . α g − g n ... ... ... ... ... α g − n g α g − n g α g − n g . . . α g − n g n It is shown in [4] that w σ ( w ) gives an isomorphism of the group ring RG into the ring of n × n matrices over R .Given the entries of the first row of an RG -matrix the entries of the other rows are determined fromthe matrix M ( G ) of G .An RG -matrix is a matrix corresponding to a group ring element in the isomorphism from the2roup ring into the ring of R n × n matrices. The isomorphism depends on the listing of the elementsof G . For example if G is cyclic, an RG -matrix is a circulant matrix relevant to the natural listing G = { , g, g , . . . , g n − } where G is generated by g . An RG -matrix when G is dihedral is one of the form (cid:18) A BB A (cid:19) where A is circulant and B is reverse circulant but also one of the form (cid:18) A BB T A T (cid:19) whereboth A, B are circulant, in a different listing of G . Other examples are given within [4].In general given a group ring element w , and a fixed listing of the elements of the group, the corre-sponding capital letter W is often used to denote the image of w , σ ( w ), in the ring of RG -matrices. Listing
Changing the listing of the elements of the group gives an equivalent RG -matrix and one is obtainedfrom the other by a sequence of processes consisting of interchanging two rows and then interchangingthe corresponding two columns. Matrices, when diagonalisable, may be simultaneously diagonalised if and only if they commute. Howevera set of matrices may be simultaneously block diagonalisable in the sense that there exist a matrix U such that U − AU has the form diag( T , T , . . . , T r ), where each T i is of fixed r i × r i size, for every matrix A in the set – and for U independent of A . This is the case for group ring matrices.Idempotents will naturally play an important part. (See [5] where these are used for paraunitarymatrices.) Say e is an idempotent in a ring R if e = e and say { e, f } are orthogonal if ef = 0 = f e .Say { e , e , . . . , e k } is a complete orthogonal set of idempotents in a ring R if e i = e i , e i e j = 0 for i = j and e + e + . . . + e k = 1 where 1 is the identity of R . Now tr A denotes the trace of a matrix A . Proposition 3.1
Suppose { e , e , . . . , e k } is a complete orthogonal set of idempotents. Consider w = α e + α e + . . . + α k e k with α i ∈ F , a field. Then w is invertible if and only if each α i = 0 and in thiscase w − = α − e + α − e + . . . + α k − e k . Proof:
Suppose each α i = 0. Then w ∗ ( α − e + α − e + . . . + α k − e k ) = e + e + . . . + e k = e + e + . . . + e k = 1.Suppose w is invertible and that some α i = 0. Then we i = 0 and so w is a (non-zero) zero-divisorand is not invertible. (cid:3) Lemma 3.1
Let { E , E , . . . , E s } be a set of orthogonal idempotent matrices. Then rank( E + E + . . . + E s ) = tr ( E + E + . . . + E s ) = tr E + tr E + . . . + tr E s = rank E + rank E + . . . + rank E s . Proof:
It is known that rank A = tr A for an idempotent matrix, see for example [1], and so rank E i =tr E i for each i . If { E, F, G } is a set of orthogonal idempotent matrices so is { E + F, G } . From this itfollows that rank( E + E + . . . + E s ) = tr( E + E + . . . E s ) = tr E + tr E + . . . + tr E s = rank E +rank E + . . . + rank E s . (cid:3) Corollary 3.1 rank( E i + E i + . . . + E i k ) = rank E i + rank E i + . . . + rank E i k for i j ∈ { , , . . . , s } , i j = i l for j = l . Let A = a E + a E + . . . + a k E k for a complete set of idempotent orthogonal matrices E i . Then A is invertible if and only if each a i = 0 and in this case A − = a − E + a − E + . . . + a k − E k . Thisis a special case of the following. Proposition 3.2
Suppose { E , E , . . . , E k } is a complete symmetric orthogonal set of idempotents in F n × n . Let Q = a E + a E + . . . + a k E k . Then the determinant of Q is | Q | = a rank E a rank E . . . a rank E k k . Let RG be the group ring of a finite group G over the ring R . Let { e , e , . . . , e k } be a completeorthogonal set of idempotents in RG and { E , E , . . . , E k } the corresponding RG -matrices (relevant to3ome listing of the elements of G ). Such a set of idempotents is known to exist when R = C , the complexnumbers, and also over other fields, see for example [3] or [7]. We will confine ourselves here to C butmany of the results hold over these other fields. The idempotent elements from the group ring satisfy e ∗ = e and so the idempotent matrices are symmetric, E ∗ = E , and satisfy E = EE ∗ = E .We now specialise the E i to be n × n matrices corresponding to the group ring idempotents e i , thatis σe i = E i . Define the rank of e i to be that of E i .Consider now the group ring F G where F = C the complex numbers and G is a finite group. Asalready mentioned, F G contains a complete orthogonal set of idempotents { e , e , . . . , e k } which may betaken to be primitive, [7]. Theorem 3.1
Let A be a F G -matrix with F = C . Then there exists a non-singular matrix P inde-pendent of A such that P − AP = T where T is a block diagonal matrix with blocks of size r i × r i for i = 1 , , . . . , k and r i are the ranks of the e i . Proof:
Let { e , e , . . . , e k } be the orthogonal idempotents and S = { E , E , . . . , E k } the group ringmatrices corresponding to these, that is, σ ( e i ) = E i in the embedding of the group ring into the C G -matrices. Any column of E i is orthogonal to any column of E j for i = j as E i E ∗ j = 0. Now letrank E i = r i . Then P ki =1 r i = n . Let S i = { v i, , v i, , . . . v i,r i } be a basis for the column space of E i consisting of a subset of the columns of E i ; do this for each i . Then each element of S i is orthogonal toeach element of S j for i = j . Since P ki =1 r i = n it follows that S = { S , S , . . . , S k } is a basis for F n .Let V i,j denote the F G -matrix determined by the column vector v i,j , let S i ( G ) denote the set of F G -matrices obtained by substituting V i,j for v i,j in S i and let S ( G ) denote the set of F G -matricesobtained by substituting S i ( G ) for S i in S .As S is a basis for F n the first column of AE i is a linear combination of elements from S . Thefirst column of AE i determines AE i , as AE i is an F G -matrix, and hence AE i is a linear combinationof elements of S ( G ). By multiplying AE i through on the right by E i , and orthogonality, it follows that AE i is a linear combination of S i ( G ) = { V i, , V i, , . . . , V i,r i } . Now each V i,j consists of columns whichare a permutation of the columns of E i . Also E i contains the columns S i . Thus equating AE i to thelinear combination of S i ( G ) implies that each AV i,j is a linear combination of S i .Let P then be the matrix with columns consisting of the first columns S i for i = 1 , , . . . , k . Then AP = P A where T is a matrix of blocks of size r i × r i arranged diagonally for i = 1 , , . . . k . Since P isinvertible it follows that P − AP = T . (cid:3) The proof is constructive in the sense that the matrix P is constructed from the complete orthogonalset of idempotents. Method:1. Find complete orthogonal set of idempotents { e , e , . . . , e k } for F G .2. Construct the corresponding
F G -matrices { E , E , . . . , E k } .3. Find a basis S i for the column space of E i for 1 ≤ ≤ k .4. Let P be the matrix made up of columns of the union of the S i .5. Then P − AP is a block diagonal matrix consisting of blocks of size r i × r i where r i is the rank of E i .However this algorithm requires being able to construct a complete orthogonal set of idempotents. Ifthe matrix P could be obtained directly then indeed this would be a way for the construction of theidempotents. Corollary 3.2
The group ring
F G is isomorphic to a subring of such block diagonal matrices. Theisomorphism is given by w σ ( w ) = W P − W P . The isomorphism includes an isomorphic embedding of the group G itself into the set of such blockdiagonal matrices. Other linear representations of G may be obtained by using the block images of thegroup elements. 4 heorem 3.2 Suppose A is an F G -matrix where F = C . Then there exists a unitary matrix P suchthat P T AP = T where T is a block diagonal matrix with blocks of size r i × r i for i = 1 , , . . . , k alongthe diagonal. Proof:
The diagonalising matrix in the proof of Theorem 3.1 may be made unitary by constructingan orthonormal basis for space generated by { V i, , V i, , . . . , V i,r i } for each i = 1 , , . . . , k . Let S i = { W i, , W i, , . . . , W i,r i } be an orthonormal basis for the space spanned by { V i, , V i, , . . . , V i,r i } . Thenˆ S = { S , S , . . . , S k } is an orthonormal basis for F n . Set P to be the matrix with elements of ˆ S ascolumns. Then P is unitary and P T AP = T as required. (cid:3) The group ring is isomorphic to the ring of RG -matrices, [4], and the ring of RG matrices is isomorphicto the ring of such block diagonal matrices under the mapping w σ ( w ) = W P − W P for this fixed P . See for example [3, 6, 7] for information on representation theory including characters and charactertables. See for example [4] for information on group ring matrices and in particular on the method forobtaining the corresponding group ring matrix from a group ring element.
When G = C n , the cyclic group of order n , the matrix P of Theorem 3.1 is the Fourier matrix and T isa diagonal matrix. The case when G is any abelian group is dealt with fully in section 5. The dihedral group D n is generated by elements a and b with presentation: h a, b | a n = 1 , b = 1 , bab = a − i It has order 2 n , and a natural listing of the elements is { , a, a , . . . , a n − , b, ab, a b, . . . , a n − b } .As every element in D n is conjugate to its inverse, the complex characters of D n are real. Thecharacters D n are contained in an extension of Q of degree φ ( n ) / Q only for 2 n ≤
6. Here φ is the Euler phi function. Let S n denote the symmetric group of order n . The characters of S n arerational. S = D Consider D . Note that D = S . The conjugacy classes are { } , { a, a } , { b, ab, ab } . The central(primitive, symmetric) idempotents are e = 1 / a + a + b + ba + ba ) , e = 1 / a + a − b − ba − ba ) , e = 1 / − a − a ).This gives the corresponding group ring matrices: E = 16 , E = 16 − − −
11 1 1 − − −
11 1 1 − − − − − − − − − − − − , E = 13 − − − − − − − −
10 0 0 − −
10 0 0 − − . Now E , E have rank 1 and E has rank 4 from general theory.Thus we need a set consisting of one column from each of E , E and 4 linearly independent columnsfrom E to form a set of 6 linearly independent vectors. It is easy to see that v = (1 , , , , , T , v = (1 , , , − , − , − T , v = (2 , − , − , , , T ,v = ( − , , − , , , T , v = (0 , , , , − , − T , v = (0 , , , − , , − T is such a set.Now let P = ( v , v , v , v , v , v ). Then for any C D matrix A , P − AP = diag( a, b, D ) where D isa 4 × RG -matrices, and the ring of RG -matrices isisomorphic to the ring of such block diagonal matrices under the mapping A P − AP for this fixed P (Theorem 3.1).Now consider the image of D itself under this isomorphism. The matrix A of a ∈ S = D in thisisomorphism is mapped to P − AP . Here A = and P − AP = − − −
10 0 0 0 1 − .(In some cases it is easier to work out AP and then solve for D in P D where D is of the correct blockdiagonal type.)Similarly the image of b is obtained; B is the RG -matrix of b and P − BP = − .Representations of S = D may be obtained using the block matrices of the images of the groupelements. For example a − − −
10 0 1 − , b gives a representation of D = S .It may be shown directly from the structure of P and of A corresponding to a group element a thatthe 4 × P − AP has the form T = (cid:18) X Y (cid:19) or of the form S = (cid:18) XY (cid:19) where X, Y are 2 × T if it has the form T and say a matrix is in S if it has the form S . Interestinglythen generally T S ∈ S , ST ∈ S , T T ∈ T , S S ∈ T , for S, S , S ∈ S , T, T , T ∈ T . Now P may be made orthogonal by finding an orthogonal basis for the 4 linearly independent columnsof E and then dividing each of the resulting set of 6 vectors by their lengths.An orthogonal basis for the columns of E is { (2 , − , − , , , T , (0 , , − , , , T , (0 , , , , − , − T , (0 , , , , , − T } .Construct an orthonormal basis: v = q (1 , , , , , T , v = q (1 , , , − , − , − T , v = q (2 , − , − , , , T ,v = q (0 , , − , , , T , v = q (0 , , , , − , − T , v = q (0 , , , , , − T .Now construct the unitary (orthogonal in this case) matrix P = ( v , v , v , v , v , v ). Then for any C D matrix A , P ∗ AP = diag( a, b, D ) where D is a 4 × P is unitary, = orthogonal in this case, then P T AP and P T BP are unitary as A, B areorthogonal. The diagonal 4 × T AP = P ∗ AP = − / √ / −√ / − / − / −√ /
20 0 0 0 √ / − / The 4 × The character tables for D n may be derived from [3, 6] and are also available at various on-line resourcessuch as that of Jim Belk. We outline how the results may be applied in the case of D .The character table of D is the following: b a a − π/
5) 2 cos(4 π/ π/
5) 2 cos(8 π/ .This gives the following complete (symmetric) orthogonal set of idempotents in the group ring: e = (1+ a + a + a + a + b + ba + ba + ba + ba ) , e = (1+ a + a + a + a − b − ba − ba − ba − ba ) , e = (1 + cos(2 π/ a + cos(4 π/ a + cos(4 π/ a + cos(2 π/ a ) , e = (1 + cos(4 π/ a + cos(8 π/ a +cos(8 π/ a + cos(4 π/ a ).Let σ ( e i ) = E i – this is the image of the group ring element e i in the group ring matrix. Each of E , E has rank 1 and each of E , E has rank 4. Four linearly independent columns in each of E , E are easy to obtain and indeed four orthogonal such may be derived if required. The matrix P is formedusing the first columns of E , E and 4 linearly independent columns of each of E and E . Then P − AP = diag( α , α , T , T ) for any group ring matrix A of D where T , T are 4 × w σ ( w ) = W P − W P is an isomorphism. Representations ofthe group may be obtained by specialising to blocks of the images of the group elements.The form of P is (cid:18) A C D B C D (cid:19) for suitable 5 × A, , C, D, C , D . Then it may beshown that in P − AP the two 4 × (cid:18) X Y (cid:19) or else the form (cid:18) XY (cid:19) for 2 × X, Y when A corresponds to a group element a . The five primitive central idempotents { e , e , e , e , e } of C K where K is the quaternion groupof order 8 is given in [7] page 186. K = h a, b | a = 1 , a = b , bab − = a − i and is listed as { , a, a , a , b, ab, a b, a b } . e = 1 / a + a + a + b + ab + a b + a b ) e = 1 / a + a + a − b − ab − a b − a b ) e = 1 / − a + a − a + b − ab + a b − a b ) e = 1 / − a + a − a − b + ab − a b + a b ) e = 1 / − a )([7] has − ab in e which should be + ab as above.)The group ring matrices { E , E , E , E } corresponding to { e , e , e , e } respectively have rank 1and the group ring matrix E corresponding to e has rank 4, which can be seen from theory. Thus takethe first columns of E , E , E , E and 4 linearly independent columns of E to form a matrix P . Then P − AP = diag( T , T , T , T , T ) where T , T , T , T are scalars and T is a 4 × A of K .Precisely we may take: 7 = − − − − − − − − − − − − − − − − and then P − AP = diag( α , α , α , α , T ) for any group ring matrix A of K where T is a 4 × K to these block matrices given by w σ ( w ) = W P − W P . Representations of K may be obtained by specialising to the group elements.The following then gives an embedding of K : a − − − −
10 0 0 0 0 0 1 0 , b − − − − Using the blocks gives other representations. For example a − −
10 0 1 0 , b − − gives a representation of K .It may be shown directly from the block form of P that the image of a group element has the 4 × (cid:18) X Y (cid:19) or else the form (cid:18) XY (cid:19) for 2 × X, Y . The abelian group case follows from the general case, Section 3, but may be tackled directly and moreilluminatingly as follows.Let { A , A , . . . , A k } be an ordered set of matrices of the same size. Then the block circulant matrix formed from the set is A = circ( A , A , . . . , A k ) = A A . . . A k A k A . . . A k − ... ... ... ... A A . . . A If the A i have size m × t then A has size km × kt . The block circulant formed depends on the orderof the elements in { A , A , . . . , A k } .Let P be an n × n matrix. Then the block Fourier matrix P f corresponding to P is P ⊗ F , the tensorproduct of P and F where F is the Fourier n × n matrix.Thus P f = P ⊗ F = P P P . . . PP ωP ω P . . . ω n − P ... ... ... ... P ω n − P ω n − . . . ω ( n − n − P It is clear then that:
Proposition 5.1 P f is invertible if and only if P is invertible and the inverse when it exists is P − ⊗ F ∗ . Here F ∗ denotes the inverse of the Fourier matrix. If the Fourier matrix is normalised in C , then F ∗
8s the complex conjugate transposed of F .The following theorem may be proved in a manner similar to the proof that the Fourier matrixdiagonalises a circulant matrix. Theorem 5.1
Suppose { A , A , . . . , A k } are matrices of the same size and can be simultaneously diag-onalised by P with P − A i P = D i where each D i is diagonal. Then the block circulant matrix A formedfrom these matrices can be diagonalised by P f = P ⊗ F = P P P . . . PP ωP ω P . . . ω k − PP ω P ω P . . . ω k − P ... ... ... ... ... P ω k − P ω k − P . . . ω ( k − k − P where ω is a primitive k th root of unity.Moreover P − f AP f = D where D is diagonal and D = diag( D + D + . . . + D k , D + ωD + . . . + ω k − D k , D + ω D + ω D + . . . + ω k − D k , . . . , D + ω k − D + ω k − D + . . . + ω ( k − k − D k ) Proof:
The proof of this is direct, involving working out AP f and showing it is P f D , with D as given. Since P f is invertible by Proposition 5.1 the result will follow. This is similar to a proof that the Fourier matrixdiagonalises a circulant matrix. (cid:3) The simultaneous diagonalisation process of the Theorem may then be repeated.Suppose now G = K × H , the direct product of K, H , and H is cyclic. Then a group ring matrixof G is of the form M = circ( K , K , . . . , K h ) where K i are group ring matrices of K and | H | = h . Ifthe K i can be diagonalised by P then M can be diagonalised by the Fourier block matrix formed from P by Theorem 5.1. A finite abelian group is the direct product of cyclic groups and thus repeating theprocess enables the simultaneous diagonalisation of the group ring matrices of a finite abelian group andit gives an explicit diagonalising matrix. The characters and character table of the finite abelian groupmay be read off from the diagonalising matrix.Since the Fourier n × n matrix diagonalises a circulant n × n matrix, and the Fourier matrix is aHadamard complex matrix, the diagonalising matrix P of size q × q , constructed by iteration of Theorem5.1, of a group ring matrix of a finite abelian group is then seen to satisfy P P ∗ = qI and to have rootsof unity as entries. It is thus a special type of Hadmard complex matrix.The examples below illustrate the method. • Consider G = C × C . Now P = ω ω ω ω where ω is a primitive 3 rd root of unitydiagonalises any circulant 3 × C . Then P f = P P PP ωP ω PP ω P ωP diagonalises any group ring matrix of C × C .Written out in full: P f = ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω
11 1 1 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω C × C may be read off from the rows of P f by labellingthe rows of P f appropriate to the listing of the elements of C × C when forming the group ringmatrices. The listing here is { , g, g , h, hg, hg , h , h g, h g } where the C are generated by { g, h } respectively. Thus the character table of C × C is g g h hg hg h hg h g ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω
11 1 1 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω Note that √ P f is unitary and that P f is a Hadamard complex matrix. • For C × C consider P = (cid:18) − (cid:19) and note that a primitive 4 th root of 1 is i = √−
1. Then i = − , i = − i, i = 1. Now form Q = P P P PP iP − P − iPP − P P − PP − iP − P iP . The characters of C × C canbe read off from Q , Q is a Hadamard complex matrix and √ Q is unitary. • For C × C consider that C × C ∼ = C . Then the diagonalising matrix obtained using the naturalordering in C × C is equivalent to the diagonalising matrix using the natural ordering in C . • Consider C n . Let P = (cid:18) − (cid:19) and inductively define for n ≥ P n = (cid:18) P n − P n − P n − − P n − (cid:19) .Then P n diagonalises any C C n -matrix and the characters of C n may be read off from the rows of P n . Note that P n is a Hadamard (real) matrix. References [1] Oskar M. Baksalary, Dennis S. Bernstein, Gtz Trenkler, “On the equality between rank and traceof an idempotent matrix”, Applied Mathematics and Computation, 217, 4076-4080, 2010.[2] Richard E. Blahut,
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Representation Theory of Finite Groups and Associative Algebras ,AmerMathSoc., Chelsea, 1966.[4] Ted Hurley, “Group rings and rings of matrices”, Inter. J. Pure & Appl. Math., 31, no.3, 2006,319-335.[5] Barry Hurley and Ted Hurley, “Paraunitary matrices and group rings”, Int. J. of Group Theory,Vol. 3, no.1, 31-56, 2014. (See also arXiv:1205.0703v1.)[6] I. Martin Isaacs,
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