Coninvolutions on Upper Triangular Matrix Group over the Ring of Gaussian Integers and Quaternions integers modulo p
aa r X i v : . [ m a t h . R A ] A ug CONINVOLUTIONS ON UPPER TRIANGULAR MATRIXGROUP OVER THE RING OF GAUSSIAN INTEGERS ANDQUATERNIONS INTEGERS MODULO p IVAN GARGATE AND MICHAEL GARGATE
Abstract.
In this article we give various formulates for compute the numberof all coninvolutions over the group of upper triangular matrix with entries intothe ring of Gaussian integers module p and the ring of Quaternions integersmodule p , with p an odd prime number. Introduction
Let R be an finite ring endowed with a compatible complex conjugate struc-ture. Denote by T n ( R ) the n × n upper triangular matrix group with entries in R . A matrix A ∈ T n ( R ) is called an Involution iff A = I n with I n the identity in T n ( R ). Involutions on T n ( R ) has been studied by several authors, see for instance[2,4] also over Incidence Algebras see [3]. On the other hand, Coninvolution matri-ces has been studied by [1] and [5]. In the present article we study how computethe number of Coninvolution matrices over two specially finite rings, the ring ofGaussian Integers module p and the ring of Quaternions Integers module p , this ismotivated by the work initiated by Slowik in [2].An matrix A = ( a rs ) ∈ T n ( R ) is called a Coninvolution iff A ¯ A = I n , where¯ A = (¯ a rs ) and ¯ a denote the complex conjugate of the number a . Denote by CI ( n, R )the number of all coninvolution matrices there are in T n ( R ).In the section 2 we show how construct a Coninvolution matrix and with thisalgorithm we compute the number of all Coninvonlutions with R = Z p [ i ], the ringof Gaussian Integers module p , i.e, we calculate the number CI ( n, Z p [ i ]).In the section 3, similarly, we show how construct a Coninvolution matrixbut in the block upper triangular matrix group T n ( M s ( Z p ) that is isomorphicto the group T n ( Z p [ i, j, k ]) whose entries are in the Ring of Quaternion Inte-gers module p . With this isomorphism its possible describe all Coninvolutionsin T n ( Z p [ i, j, k ]). With this result, we finally compute the number of all Coninvo-lutions CI ( n, Z p [ i, j, k ]). In both cases we assume that p is an odd prime number.In the section 4 we present various Tables of the numbers of coninvolutionsin T n ( Z p [ i ]) and T n ( Z p [ i, j, k ]).Our main results are the followings: Theorem 1.1.
Let Z p [ i ] the Gaussian integers module p . Then the number ofConinvoltions in the group T n ( Z p [ i ]) is equal to CI ( n, Z p [ i ]) = | U ( Z p [ i ]) | n × p ( n − n , Key words and phrases.
Coninvolutions, Gaussian integers, Quaternions integers, upper trian-gular matrices . where | U ( Z p [ i ]) | denote the cardinality of U ( Z p [ i ]) = { z ∈ Z p [ i ] , | z | = 1 } . And, for the ring of Quaternion Integers module p we have the followingTheorem. Theorem 1.2.
Consider s = | SL ( Z p ) | = ( p − p . Then the number of coinvolu-tions over T n ( Z p [ i, j, k ]) , denote by CI ( n, Z p [ i, j, k ]) , is equal to min { s,n } X j =1 X n + n + ··· + n j = nn ≤ n ≤···≤ n j s !( s − j )! g ( n , n , · · · , n j ) (cid:18) nn n · · · n s (cid:19) p [ n − j X u =1 n u +3 n u ( n u − , where g ( n , · · · , n j ) = r ! · r ! · · · r t ! if and only if m = m = · · · = m r = m r +1 = m r +2 = · · · = m r + r = m r + r +1 = · · · Coninvolutions over the Gaussian Integers module p Consider p > Z p [ i ] = { a + ib, a, b ∈ Z p and i = − } bethe Gaussian integers module p , then if z = a + ib ∈ Z p [ i ] denote by Re ( z ) = a and Im ( z ) = b the real and imaginary parts of z , respectively. Also, define the naturalcomplex conjugation ¯ z = a + ( p − ib = a − ib and | z | = z · ¯ z = a + b the naturalmodulus. For a matrix A = ( z rs ) ∈ T n ( Z p [ i ]) define their complex conjugation as¯ A = (¯ z rs ). A matrix A ∈ T n ( Z p [ i ]) is called a coninvolution if A ¯ A = I n where I n isthe identity matrix. Simillarly to [2] we have the following theorem: Theorem 2.1.
Let Z p [ i ] the ring of Gaussian integers module p . A matrix A =( z rs ) ∈ T n ( Z p [ i ]) is a convinvolution if and only if A is described by the followingstatements: (i) For all z rr with ≤ r ≤ n we have | z rr | = 1 . (ii) If z rr = ¯ z ss then z rs = iγ , if s = r + 1 − ( z rr ) − · s − X t = r +1 z rt ¯ z ts + iγ , if s > r + 1 , (iii) If z rr = − ¯ z ss then z rs = γ , if s = r + 1 γ − ( z rr ) − · s − X t = r +1 z rt ¯ z ts , if s > r + 1 , (iv) If z rr = z ss then a.) If Re ( z rr ) = 0 we have z rs = − [ Re ( z rr ) − Im ( z rr ) − i ] γ , if s = r + 1 − (2 Re ( z rr )) − · s − X t = r +1 z rt ¯ z ts − [ Re ( z rr ) − Im ( z rr ) − i ] γ , if s > r + 1b.) If Im ( z rr ) = 0 we have z rs = [1 − iIm ( z rr ) − Re ( z rr )] γ , if s = r + 1(1 − iIm ( z rr ) − Re ( z rr )) γ − i (2 Im ( z rr )) − · s − X t = r +1 z rt ¯ z ts , if s > r + 1 , ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p (v) If z rr = − z ss then a.) If Re ( z rr ) = 0 we have z rs = [1 + i Re ( z rr ) − Im ( z rr )] γ , if s = r + 1(2 Re ( z rr )) − · ( s − X t = r +1 z rt ¯ z ts ) + [1 + i Re ( z rr ) − Im ( z rr )] γ, if s > r + 1 , b.) If Im ( z rr ) = 0 we have z rs = [( Im ( z rr ) − Re ( z rr ) + i ] γ , if s = r + 1 i (2 Im ( z rr )) − · ( s − X t = r +1 z rt ¯ z ts ) + [( Im ( z rr ) − Re ( z rr ) + i ] γ, if s > r + 1 , (vi) If z rr = ± z ss and z rr = ± ¯ z ss then z rs = [ − ( Re ( z rr + z ss )) − · Im ( z rr + z ss ) + i ] γ , if s = r + 1 Re ( z rr + z ss ) − Re ( − s − X t = r +1 z rt ¯ z ts )++[ − ( Re ( z rr + z ss )) − · Im ( z rr + z ss ) + i ] γ, if s > r + 1 , where γ ∈ Z p , in all cases, may be arbitrary.Proof. For r = s consider z rs = x + iy with x, y ∈ Z p .(i) If A = ( z rs ) is a coninvolution then z rr ¯ z rr = | z rr | = 1 for all r . Also, wehave if s > r + 1, the following equation z rr ¯ z rs + z r ( r +1) ¯ z ( r +1) s + · · · + z r ( s − ¯ z ( s − s + z rs ¯ z ss = 0 . or z rr · ¯ z rs + z rs · ¯ z ss = − s − X p = r +1 z rp · ¯ z ps . (1)And, if s = r + 1 we have only the equation z rr · ¯ z r,r +1 + z r,r +1 · ¯ z r +1 ,r +1 = 0 . So, we analise only the case if s > r + 1.(ii) In this case, z rr · ¯ z rs + z rs · ¯ z ss = z rr · Re ( z rs ) = − s − X t = r +1 z rt · ¯ z ts , then x = Re ( z rs ) = ( z rr ) − · ( − s − X t = r +1 z rt · ¯ z ts ) , here z rs = x + iy with y ∈ Z p may be arbitrary.(iii) Is similar that (ii).(iv) If z rr = z ss , denote by z rr = a + ib and z ss = α + iβ , with a, b, α, β ∈ Z p ,then the equation (1) is expressed in the form: − s − X t = r +1 z rt · ¯ z ts = [( a + α ) x + ( b + β ) y ] + i [( b − β ) x + ( α − a ) y ] . (2) IVAN GARGATE AND MICHAEL GARGATE
In the case that a = 0 then we have x = (2 a ) − ( − s − X t = r +1 z rt · ¯ z ts − by ) , so z rs = x + iy = (2 a ) − ( − s − X t = r +1 z rt · ¯ z ts − by ) + iy = (2 a ) − ( − s − X t = r +1 z rt · ¯ z ts ) − (cid:8) a − b − i (cid:9) y, with y ∈ Z p may be arbitrary. The case b = 0 is similar.(v) If z rr = − z ss and a = 0 then follows from the equation (2) we have that y = ( − ai ) − ( − s − X t = r +1 z rt · ¯ z ts − bix ) , so z rs = x + iy = x + ( − ai ) − ( − s − X t = r +1 z rt · ¯ z ts − bix ) i = ( − a ) − ( − s − X t = r +1 z rt · ¯ z ts ) + (cid:8) ia − b (cid:9) x, with x ∈ Z p may be arbitrary. The case b = 0 is similar.(vi) The equation (2) can be write in the following linear system( a + α ) x + ( b + β ) y = Re ( − s − X t = r +1 z rt · ¯ z ts ) , (3)( b − β ) x + ( α − a ) y = Im ( − s − X t = r +1 z rt · ¯ z ts ) . (4)Here, of the equation (3) we have that x = ( a + α ) − · [ Re ( − s − X t = r +1 z rt · ¯ z ts ) − ( b + β ) y ]and substituting into the equation (4) we have( b − β ) · ( a + α ) − · [ Re ( − s − X t = r +1 z rt · ¯ z ts ) − ( b + β ) y ] + ( α − a ) y = Im ( − s − X t = r +1 z rt · ¯ z ts ) , then y (cid:8) ( α − a ) − ( b − β )( a + α ) − (cid:9) == Im ( − s − X t = r +1 z rt · ¯ z ts ) − ( b − β )( a + α ) − · Re ( − s − X t = r +1 z rt · ¯ z ts ) . Notice that, by the item (i) we have that a + b = α + β = 1 and here b − β = α − a = ( α − a )( α + a ) then ( α − a ) = ( b − β )( a + α ) − . ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p Follow that the second side of the last identity is null and also, ( α − a ) − ( b − β )( a + α ) − = 0, so y ∈ Z p can be arbitrary. Finally, we have z rs = x + iy = ( a + α ) − · [ Re ( − s − X t = r +1 z rt · ¯ z ts ) − ( b + β ) y ] + iy = ( a + α ) − · Re ( − s − X t = r +1 z rt · ¯ z ts ) + (cid:8) − ( a + α ) − · ( b + β ) + i (cid:9) y. (cid:3) Then, with this, we show the Theorem 1.1:
Proof of Theorem 1.1.
Follow immediately from Theorem 2.1 that, if A is a Con-involution in T n ( Z p [ i ]) then over the main diagonal we can choose any element in U ( Z p [ i ]) and independent from these choices all entries over the main diagonal canbe choose that depending from one variable. How we can ( n − n entries over themain diagonal then we conclude that we are | U ( Z p [ i ]) | n × p ( n − n and this conclude the proof. (cid:3) Coninvolutions over the Quaternion Integers module p Consider the set { i, j, k } such that satisfies the relations i = j = k = − ij = − ji = k , and define the set Z p [ i, j, k ] = { z = x + x i + x j + x k, x , x , x , x ∈ Z p } , with natural operations of sum and product. The set Z p [ i, j, k ] is called the ring ofQuaternions Integers module p . We define the conjugation of the number z as¯ z = x + ( p − x i + ( p − x j + ( p − x k = x − x i − x j − x k. If A = ( A rs ) ∈ T n ( Z p [ i, j, k ]) is an upper triangular matrix, then define thecomplex conjugation of A as ¯ A = ( ¯ A rs ) . An matrix A ∈ T n ( Z p [ i, j, k ]) is called anconinvolution if A ¯ A = I n where I n is the identity in T n ( Z p [ i, j, k ]). In order to cal-culate the number of coninvolutions on T n ( Z p [ i, j, k ]) we consider the isomorphimsmultiplicative ϕ : Z p [ i, j, k ] → M ( Z p ) define by ϕ ( z ) = ϕ ( x + x i + x j + x k ) == x (cid:20) (cid:21) + x (cid:20) p − (cid:21) + x (cid:20) a bb p − a (cid:21) + x (cid:20) b p − ap − a p − b (cid:21) , where a, b ∈ Z p such that a + b = p −
1. These numbers exists if p is an oddprime number and in this case ϕ is an isomorphism multiplicative (see [8]). For A = (cid:20) M M M M (cid:21) ∈ M ( Z p ) define the ϕ -conjugation of A and denote by e A tothe matrix (cid:20) M − M − M M (cid:21) . If ϕ ( z ) = A then is not difficult show that ϕ (¯ z ) = e A .Using this isomorphism we can enunciated the following result. Lemma 3.1.
Let z ∈ Z p [ i, j, k ] and consider ϕ ( z ) = A ∈ M ( Z p ) their respectiverepresentation. Then z · ¯ z = 1 if and only if det( A ) = 1 . So, there are ( p − p elements in Z p [ i, j, k ] that satisfies the equation z ¯ z = 1 . IVAN GARGATE AND MICHAEL GARGATE
Proof.
By the isomorpshim ϕ this is equivalent to proof that A e A = I . If A = (cid:20) M M M M (cid:21) then the above equation is true if det( A ) = M M − M M = 1. Weconclude the proof by the observation that | SL ( Z p ) | = ( p − p. (cid:3) We consider the block upper triangular matrix group T n ( M ( Z p )) where theentries are 2 × Z p . In this group, M = ( M rs ) ∈ T n ( M ( Z p )) is called an coninvolution if M f M = I where f M = ( f M rs ) . The iso-morphism multiplicative ϕ can be extended naturally to an isomorphism multi-plicative into the groups T n ( Z p [ i, j, k ]) and T n ( M ( Z p )), so, by the isomorphism ϕ ,we can conclude that A = ( a rs ) is a coninvolution in T n ( Z p [ i, j, k ]) if and only if M = ( ϕ ( a rs )) is a coninvolution in T n ( M ( Z p )). The following Theorem study howwe can construct a coninvolution in T n ( M ( Z p )). Theorem 3.2.
A block upper triangular matrix M = ( Z rs ) ∈ T n ( M ( Z p )) is aconvinvolution if and only if M described by the following statements: For all Z rr we have Z rr e Z rr = I and we can conclude that det( Z rr ) = 1 . Denote by Z rr = (cid:20) a bc d (cid:21) , Z ss = (cid:20) x yw z (cid:21) and in the case that s > r + 1 denote by − s − X t = t +1 Z rt e Z ts = (cid:20) A BC D (cid:21) and θ = det ( Z ss − Z rr ) . Then, forall Z r,s we have a.) If Z rr = Z ss , then (i) If = a = 0 then Z rs is equals to: β (cid:20) − a − d (cid:21) + β (cid:20) a − c (cid:21) + β (cid:20) a − b (cid:21) , if s = r +1 (cid:20) a − A (cid:21) + β (cid:20) − a − d (cid:21) + β (cid:20) a − c (cid:21) + β (cid:20) a − b (cid:21) , if s > r +1(ii) If b = 0 then Z rs is equal to: β (cid:20) b − d (cid:21) + β (cid:20) − b − c (cid:21) + β (cid:20) b − a (cid:21) , if s = r +1 (cid:20) − b − A (cid:21) + β (cid:20) b − d (cid:21) + β (cid:20) − b − c (cid:21) + β (cid:20) b − a (cid:21) , if s > r +1(iii) If c = 0 then Z rs is equal to: β (cid:20) c − d (cid:21) + β (cid:20) − c − b (cid:21) + β (cid:20) c − a (cid:21) , if s = r +1 (cid:20) − c − A (cid:21) + β (cid:20) c − d (cid:21) + β (cid:20) − c − b (cid:21) + β (cid:20) c − a (cid:21) , if s > r +1 ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p (iv) If d = 0 then Z rs is equal to: β (cid:20) d − c
10 0 (cid:21) + β (cid:20) d − b
01 0 (cid:21) + β (cid:20) − d − a
00 1 (cid:21) , if s = r +1 (cid:20) d − A
00 0 (cid:21) + β (cid:20) d − c
10 0 (cid:21) + β (cid:20) d − b
01 0 (cid:21) + β (cid:20) − d − a
00 1 (cid:21) , if s > r +1 , where β , β , β ∈ Z p are arbitrary elements, so, depending from threevariables. b.) If Z rr = Z ss then we consider the following cases (i) If a = x, b = y, c = w and d = z then Z rs is equal to γ (cid:20) z − d ) − ( y − b ) − ( z − d ) − ( c − w ) 1 (cid:21) , if s = r + 1 (cid:20) ( b − y ) − B ( c − w ) − W − ( z − d ) − ( c − w ) − ( y − b ) C ( z − d ) − C (cid:21) , if s > r + 1 , + γ (cid:20) z − d ) − ( y − b ) − ( z − d ) − ( c − w ) 1 (cid:21) where W = A − D − ( z − d )( b − y ) − B . (ii) If b = y, a = x, c = w and d = z then Z rs is equal to: γ (cid:20) ( c − w ) − ( a − x ) 01 − ( c − w ) − ( z − d ) (cid:21) , if s = r + 1 (cid:20) ( z − d ) − W − ( z − d ) − ( c − w ) − ( a − x ) C ( x − a ) − B c − w ) − C (cid:21) , if s > r + 1+ γ (cid:20) ( c − w ) − ( a − x ) 01 − ( c − w ) − ( z − d ) (cid:21) where W = A − D − ( c − w )( x − a ) − B . (iii) If c = w, a = x, b = y and d = z then Z rs is equal to: γ (cid:20) − ( b − y ) − ( x − a ) 10 − ( b − y ) − ( z − d ) (cid:21) , if s = r + 1 (cid:20) ( b − y ) − B z − d ) − C ( a − x ) − W − ( b − y ) − ( a − x ) − ( z − d ) B (cid:21) , if s > r + 1+ γ (cid:20) − ( b − y ) − ( x − a ) 10 − ( b − y ) − ( z − d ) (cid:21) where W = A − D − ( y − b )( z − d ) − C . (iv) If d = z, a = x, b = y and c = w then Z rs is equal to γ (cid:20) − ( b − y )( x − a ) − − ( x − a ) − ( c − w ) 0 (cid:21) , if s = r + 1 (cid:20) x − a ) − B ( y − b ) − W − ( x − a ) − ( y − b ) − ( c − w ) B ( c − w ) − C (cid:21) , if s > r + 1 , + γ (cid:20) − ( b − y )( x − a ) − − ( x − a ) − ( c − w ) 0 (cid:21) where W = A − D − ( a − x )( c − w ) − C . IVAN GARGATE AND MICHAEL GARGATE (v) If a = x, b = y, c = w and d = z then Z rs is equal to γ (cid:20) − ( z − d ) − ( c − w ) 1 − ( z − d ) − ( c − w ) β − [ z ( z − d ) − ( c − w )+ w ] β − [ z ( z − d ) − ( c − w ) + w ] (cid:21) ,,if s = r +1 (cid:20) ( z − d ) − A z − d ) − C − ( z − d ) − ( c − w ) β − W β − W (cid:21) + γ (cid:20) − ( z − d ) − ( c − w ) 1 − ( z − d ) − ( c − w ) β − [ z ( z − d ) − ( c − w )+ w ] β − [ z ( z − d ) − ( c − w )+ w ] (cid:21) ,,if s > r +1 where W = A + b ( z − d ) − C − z ( z − d ) − A and β = b ( z − d ) − ( c − w )+ a . (vi) If a = x, c = w, b = y and d = z then Z rs is equal to: · If w = 0 γ (cid:20) w − a (cid:21) , if s = r + 1 (cid:20) ( b − y ) − B − w − ( A − z ( b − y ) − B + b ( z − d ) − C )( z − d ) − C (cid:21) , if s > r + 1+ γ (cid:20) w − a (cid:21) · If a = 0 then γ (cid:20) − a − w (cid:21) , if s = r + 1 (cid:20) ( b − y ) − B z − d ) − C − a − ( A − z ( b − y ) − B + b ( z − d ) − C ) (cid:21) , if s > r + 1+ γ (cid:20) − a − w (cid:21) (vii) If a = x, d = z, b = y and c = w then Z rs is equal to γ (cid:20) − ( c − w ) − ( y − b )1 0 (cid:21) , if s = r +1 (cid:20) ( b − y ) − B ( c − w ) − ( A − D )0 ( c − w ) − C (cid:21) + γ (cid:20) − ( c − w ) − ( y − b )1 0 (cid:21) , if s > r +1(viii) If b = y, c = w, d = z and a = x then Z rs is equal to γ (cid:20) − ( z − d ) − ( a − x ) 00 1 (cid:21) , if s = r +1 (cid:20) ( z − d ) − ( A − D ) ( x − a ) − B ( z − d ) − C (cid:21) + γ (cid:20) − ( z − d ) − ( a − x ) 00 1 (cid:21) , if s > r +1(ix) If b = y, d = z, c = w and a = x then Z rs is equal to ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p · If z = 0 : γ (cid:20) z − b
01 0 (cid:21) , if s = r + 1 (cid:20) z − ( A + w ( x − a ) − B − a ( c − w ) − D ) ( x − a ) − B c − w ) − D (cid:21) , if s > r + 1+ γ (cid:20) z − b
01 0 (cid:21) · If b = 0 : γ (cid:20) b − z (cid:21) , if s = r + 1 (cid:20) x − a ) − B − b − ( A + w ( x − a ) − B − a ( c − w ) − D ) ( c − w ) − D (cid:21) , if s > r + 1+ γ (cid:20) b − z (cid:21) (x) If c = w, d = z, b = y and a = x then Z rs is equal to γ (cid:20) ( b − y ) − β − ( x − a )( b ( y − b ) − ( a − x )+ w ) − β − ( b ( y − b ) − ( a − x )+ w ) − ( y − b ) − ( a − x ) 1 (cid:21) , if s = r +1 (cid:20) ( b − y ) − ( B − ( x − a ) − β − W ) β − W ( y − b ) − ( A − D ) 0 (cid:21) + γ (cid:20) ( b − y ) − β − ( x − a )( b ( y − b ) − ( a − x )+ w ) − β − ( b ( y − b ) − ( a − x )+ w ) − ( y − b ) − ( a − x ) 1 (cid:21) , if s > r +1 where W = A − z ( b − y ) − B + b ( y − b ) − ( A − D ) and β = − z ( b − y ) − ( x − a ) − w . (xi) If all entries of Z rr and Z ss are different, then Z rs is equal to γ (cid:20) ( x − a )( z − d ) − − ( z − d ) − ( b − y ) − ( z − d ) − ( c − w ) 1 (cid:21) , if s = r +1 (cid:20) ( b − y ) − [ B + ( x − a )( z − d ) − θ − W ] − ( z − d ) − θ − W ( z − d ) − C (cid:21) + , if s > r +1+ γ (cid:20) ( x − a )( z − d ) − − ( z − d ) − ( b − y ) − ( z − d ) − ( c − w ) 1 (cid:21) where W = ( z − d )[( b − y )( A − D ) − ( z − d ) B ] − ( y − c ) C and, in allof the above cases, γ ∈ Z p can be an arbitrary element.Proof. Whitout lost of generality, denote by Z rr = (cid:20) a bc d (cid:21) , Z rs = (cid:20) z z z z (cid:21) and Z ss = (cid:20) x yw z (cid:21) . So we have that e Z rr = (cid:20) d − b − c a (cid:21) , e Z rs = (cid:20) z − z − z z (cid:21) and e Z ss = (cid:20) z − y − w x (cid:21) . s = r + 1, in this case we obtain Z rr e Z r,r +1 + Z r,r +1 e Z r +1 ,r +1 = 0 . So, we obtain the system z − w − b ab − y x − a z − d c − wd − c − y x z z z z = . (5)In the case that s > r + 1 we obtain Z rr e Z rs + Z r,r +1 e Z r +1 ,s + · · · + Z r,s − e Z s − ,s + Z rs e Z ss = 0or Z rr e Z rs + Z rs e Z ss = − s − X t = r +1 Z rt e Z ts . (6)By hipothese, denote by − s − X t = r +1 Z rt e Z ts = (cid:20) A BC D (cid:21) , then from equation(6) we obtain the following system z − w − b ab − y x − a z − d c − wd − c − y x z z z z = ABCD . (7)2a.) If Z rr = Z r +1 ,r +1 then from the system (5) we have the equation dz − cz − bz + az = 0 . How det( Z rr ) = 1 then some entry a, b, c or d is not zero, from this, forexample, if d = 0 then we can write z = d − ( cz + bz − az ) . From this Z r,r +1 = z (cid:20) d − c
10 0 (cid:21) + z (cid:20) d − b
01 0 (cid:21) + z (cid:20) − d − a
00 1 (cid:21) , with arbitrary z , z , z ∈ Z p . If Z rr = Z ss with s > r + 1 then, fromthe system (7), necessarily B = C = 0 and A = D . Here we obtain theequation dz − cz − bz + az = A. Similarly to the above item, we can consider d = 0, then z = d − ( A + cz + bz − az ) . Thus Z rs = z (cid:20) d − c
10 0 (cid:21) + z (cid:20) d − b
01 0 (cid:21) + z (cid:20) − d − a
00 1 (cid:21) + (cid:20) d − A
00 0 (cid:21) , where z , z , z ∈ Z p are arbitrary elements.2b.) In the case that Z rr = Z ss we study the following cases: ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p (i) If a = x, b = y, c = w and d = z . Then from the equation (6) we obtainthat ( b − y ) z = B and from here z = ( b − y ) − B. From the first and thefourth equation of the system (7) we obtain( z − d ) z + ( c − w ) z + ( y − b ) z = A − D or ( c − w ) z + ( y − b ) z = A (8)with A = A − D − ( z − d )( b − y ) − B. From equation (8) and the thirdequation of the system (7) we obtain the new following system (cid:26) ( z − d )( c − w ) z + ( z − d )( y − b ) z = ( z − d ) A ( y − b )( z − d ) z + ( y − b )( c − w ) z = ( y − b ) C. , and substraying both equation, we obtain( z − d )( c − w ) z − ( y − b )( c − w ) z = ( z − d ) A − ( y − b ) C. then z = ( z − d ) − ( c − w ) − { ( z − d ) A − ( y − b ) C + ( y − b )( c − w ) z } . Also, from the third equation of the system (7) we obtain z = ( z − d ) − { C − ( c − w ) z } . So Z rs = z (cid:20) z − d ) − ( y − b ) − ( z − d ) − ( c − w ) 1 (cid:21) ++ (cid:20) ( b − y ) − B ( c − w ) − A − ( z − d ) − ( c − w ) − ( y − b ) C ( z − d ) − C (cid:21) for arbitrary z ∈ Z p . The cases (ii),(iii) and (iv) solves similarly.(v) In this case we have that a = x, b = y, c = w and d = z , then from system(7), we have B = 0. From first and fourth equations into the same systemwe obtain z = ( z − d ) − [ A − ( c − w ) z ] , (9)and from second equation of the same system we obtain z = ( z − d ) − [ C − ( c − w ) z ] . (10)Substituting the above relations into first equation of the system (7) weobtain − [ z ( z − d ) − ( c − w ) + w ] z + [ b ( z − d ) − ( c − w ) + a ] z = E, (11)with E = A + b ( z − d ) − C − z ( z − d ) − A. Now, analyzing the term β = b ( z − d ) − ( c − w ) + a . Suppose that β = 0 then b ( z − d ) − ( c − w ) + a = 0or az − ad + bc − bw = 0 . (12)By item (1) of our proof, we obtain that det( Z rr ) = ad − bc = 1. Thensubstituting in the equation (12) we have az − bw = 1 , (13) but, by hipotesse Z rr = Z ss then Z rr e Z ss = (cid:20) az − bw ∗∗ ∗ (cid:21) = I , in par-ticular az − bw = 1 and this contraries the equation (13). So β = 0 and bythe equation (11) we obtain that z = β − [ E + [ z ( z − d ) − ( c − w ) + w ] z ] . (14)From the relations (9), (10) and (14) we finally obtains that Z rs is equal to (cid:20) ( z − d ) − A z − d ) − C − ( z − d ) − ( c − w ) β − E β − E (cid:21) ++ z (cid:20) − ( z − d ) − ( c − w ) 1 − ( z − d ) − ( c − w ) β − [ z ( z − d ) − ( c − w )+ w ] β − [ z ( z − d ) − ( c − w )+ w ] (cid:21) , where E = A + b ( z − d ) − C − z ( z − d ) − A and β = b ( z − d ) − ( c − w ) + a .So, Z rs depending only one variable.(vi) In this case, from the second equation of the system (7) we obtain z = ( b − y ) − B, and from the thrid equation z = ( z − d ) − C, and substuting in the first equation zz − wz − bz + az = A. If w = 0, we obtain z = − w − ( A − zz + bz − az ) , and from here, Z rs is equal to: (cid:20) ( b − y ) − B − w − ( A − z ( b − y ) − B + b ( z − d ) − C )( z − d ) − C (cid:21) + z (cid:20) w − a (cid:21) , and if a = 0, we obtain z = − a − ( A − zz wz + bz ) , and from here, Z rs is equal to (cid:20) ( b − y ) − B z − d ) − C − a − ( A − z ( b − y ) − B + b ( z − d ) − C ) (cid:21) + z (cid:20) − a − w (cid:21) . (vii) If a = x, d = z, b = y and c = w then, from the system we have immediatelythat z = ( b − y ) − B, (15) z = ( c − w ) − C, (16) z = ( c − w ) − [ A − D − ( y − b ) z ] . (17)So Z rs is equal to: (cid:20) ( b − y ) − B ( c − w ) − ( A − D )0 ( c − w ) − C (cid:21) + z (cid:20) − ( c − w ) − ( y − b )1 0 (cid:21) where z ∈ Z p and here, Z rs depending only one variable.(viii) Solves similarly that (vii).(ix) Solves similarly that (vi).(x) Solves similarly that item (v). ONINVOLUTIONS OVER GAUSSIAN INTEGERS AND QUATERNIONS INTEGERS MODULO p (xi) In this case Z rr = Z ss , a = x, b = y, c = w and d = z .From first and fourth equation of the system (7) we obtain( z − d ) z + ( c − w ) z + ( y − b ) z + ( a − x ) z = A − D (18)( b − y ) z + ( x − a ) z = B (19)( z − d ) z + ( c − w ) z = C (20)From the equations (18) and (19) we have[( c − w )( b − y ) − ( z − d )( x − a )] z +( b − y )( y − b ) z +( b − y )( a − x ) z = ( b − y )( A − D ) − ( z − d ) B (21)and from equations (21) and (20) we have( z − d )[( c − w )( b − y ) − ( z − d )( x − a )] z +[( a − x )( b − y )( z − d ) − ( y − b )( b − y )( c − w )] z = E (22)where E = ( z − d )[( b − y )( A − D ) − ( z − d ) B ] − ( y − b ) C , or, from the lastequation we simplify and obtain[( x − a )( z − d ) − ( c − w )( b − y )] · [( z − d ) z + ( b − y ) z ] = − E (23) Claim 3.3. θ = ( x − a )( z − d ) − ( c − w )( b − y ) = 0Notice that θ = det (cid:18)(cid:20) x yw z (cid:21) + (cid:20) − a − b − c − d (cid:21)(cid:19) = det( Z ss + ( − Z rr ))and use the following formula for 2 × A + B ) = det A + det B + det A · T race ( A − B ) . and observing that T race ( Z − ss ( − Z rr )) = T race ( (cid:20) z − y − w x (cid:21) (cid:20) − a − b − c − d (cid:21) )= T race (cid:18)(cid:20) − za + yc ∗∗ bw − xd (cid:21)(cid:19) = ( yc − az ) + ( bw − xd ) . Substitying in the above formula θ = 2 + ( yc − az ) + ( bw − xd )How Z rr = Z ss then Z rr Z − ss = Z rr e Z ss = I and for this we obtain that xd − cy = 1 and az − bw = 1 and for this we conclude that θ = 0 . So,follows from equation (23) that z = ( z − d ) − [ − θ − E − ( b − y ) z ] . So z = ( b − y ) − (cid:2) B − ( x − a )( z − d ) − [ − θ − E − ( b − y ) z ] (cid:3) , and z = ( z − d ) − [ C − ( c − w ) z ] . So, Z rs is equal to: (cid:20) ( b − y ) − [ B + ( x − a )( z − d ) − θ − E ] − ( z − d ) − θ − E ( z − d ) − C (cid:21) ++ z (cid:20) ( x − a )( z − d ) − − ( z − d ) − ( b − y ) − ( z − d ) − ( c − w ) 1 (cid:21) with z ∈ Z p an arbitrary element. (cid:3) Remark 3.4.
Using the Theorem 3.2 and appliying the isomorphims inverse ϕ − we can construct coninvolution matrices over T n ( Z [ i, j, k ]) considering each case ofthe theorem. Describe this is too long and we leave it to the readerAnd, with this we can compute the number of Coninvolutions in T n ( Z p [ i, j, k ]): Proof of Theorem 1.2.
By the Lemma 3.1 we know that there are ( p − p ofsolutions of z ¯ z = 1, then we enumerate all solutions and denote q , q , · · · , q s thesesolutions. Denote by n , n , · · · , n s the number of times that q , q , · · · , q s appearsin the diagonal respectively, such that n + n + · · · + n s = n and n i ≥ i = 1 , , · · · , s . Observe that the value n j = 0 means that the solution q j does notappears in the diagonal of matrix. Then, by the Theorem 3.2 we conclude that,if q r and q s appears into the diagonal n r and n s times, respectively, then we havethat count: • p nr ( nr − × p ns ( ns − , this because there are n r diagonals that are equalsand similarly there are n s equal entries in the diagonal. • p n r · n s because there are n r and n s different entries in the diagonal.And the combinatory number means the choose such that we can put the so-lutions q , q , · · · , q s into the diagonal of a matrix. With this we conclude that CI ( n, Z p [ i, j, k ]) is equal to n X n + n + ··· + n s = nn ≥ ,n ≥ , ··· ,n s ≥ (cid:18) nn n · · · n s (cid:19) p X ≤ i
12 ( n i ( n i − . Notice that P ≤ i Number of coninvolutions in the group T n ( Z p [ i ]) p | U ( Z p [ i ]) | n = 2 n = 3 n = 4 n = 52 2 8 64 1024 327683 8 192 13824 2985984 32768 × × × × × × × × × × × × 10 32 19240 32768000 1048576 × × 11 120 158400 120 × × × p Table 2. Number of coninvolutions in the group T n ( Z p [ i, j, k ]) p n = 3 n = 4 n = 53 730944 1935197568 965056268079365 55921200 242942845440 55403265921715207 1097237568 10865680662144 34893873222411302411 63229077120 2297513874543360 25234230121430757504013 284363059344 16960852265255808 304726284071367566553617 3184303946688 423042785745341184 16889274223145926405862419 8670723975360 1607003347848750720 89458185230583343270848023 48451556497344 15918297951624613248 1569990376505187740247513629 390682469205840 257213522072770327680 508193153533449590679553920 References [1] K.D. Ikramov, On condiagonalizable matrices, Linear Algebra Appl. 424 (2007)456465.[2] R. Sowik, Involutions in triangular groups, Linear Multilinear Algebra 61 (7),909916, (2013).[3] I. Gargate., M. Gargate, Involutions on Incidence Algebras of Finite POsets.(2019). arXiv:1907.06805.[4] Hou, X., Li, S., Zheng, Q., Expressing infinite matrices over rings as productsof involutions. Linear Algebra and Its Applications, 532, 257265, (2017).[5] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cam-bridge, Second Edition. (2012).[6] Roy, J., Patra, K., Some Aspects of Addition Cayley Graph of Gaussian Inte-gers Modulo n, MATEMATIKA, Vol.32 (2016), p. 43-52.[7] Roksana Sowik. How to construct a triangular matrix of a given order, Linearand Multilinear Algebra, 62:1, 28-38,(2014).[8] C. J. Miguel, R. Serdio, On the Structure of Quaternion Rings over Z p . Inter-national Journal of Algebra, Vol. 5, 2011, no. 27, 1313 - 1325. UTFPR, Campus Pato Branco, Rua Via do Conhecimento km 01, 85503-390 PatoBranco, PR, Brazil E-mail address : [email protected] UTFPR, Campus Pato Branco, Rua Via do Conhecimento km 01, 85503-390 PatoBranco, PR, Brazil E-mail address ::