On Orbits of the Ring Z m n under the Action of the Group SL(m, Z n )
OOn Orbits of the Ring Z nm under the Action of the Group SL ( m ,Z n ) Petr Novotný, Ji ř í Hrivnák Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, B ř ehová 7, 115 19 Prague 1, Czech Republic Ing. Petr Novotný [email protected] 222311333 Ing. Ji ř í Hrivnák [email protected] 222311333 Keywords : Ring, finite group
Abstract
We consider the action of the finite matrix group SL ( m ,Z n ) on the ring Z nm . We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A.A. Kirillov for m= n prime number.
1. Introduction
The important role of symmetries in classical and quantum physics is well known. We focus on so called discrete quantum physics; it means that the corresponding Hilbert space is finite dimensional [1,2]. Well known are also 2×2 Pauli matrices, besides spanning real Lie algebra su(2), they form a fine grading of sl(2,C). The fine gradings of a given Lie algebra are preferred bases which yield quantum observables with additive quantum numbers. The generalized n × n Pauli matrices were described in [3]. For n= n ,C) associated with these generalized Pauli matrices were studied only recently in [6]. This work pointed out the importance of the finite group SL (2,Z n ) as the group of symmetry of the Pauli gradings. The additive quantum numbers, mentioned above, form in this case the finite associative additive ring Z n ×Z n . The action of SL (2,Z n ) on Z n ×Z n then represents the symmetry transformations of Pauli gradings of sl( n ,C). The orbits of this action form such points in Z n ×Z n which can be reached by symmetries. For the purpose of so called graded contractions [7], it became convenient to study the action of SL (2,Z n ) on various types of Cartesian products of Z n [8]. Note that the orbits of SL (2,Z p ) on Z p2 , where p is prime number were considered in [9] § SL ( m ,Z n ) on Z nm where m,n are arbitrary natural numbers.
2. Action of the group SL ( m ,Z n ) In the whole article we shall use the following notation: N:={1,2,3,…} denotes the set of all natural numbers and P:={2,3,5,…} denotes the set of all prime numbers. Let n be a natural number, then the set {0,1,…, n -1} forms, together with operations + mod n , ⋅ mod n , an associative ommutative ring with unity. We will denote this ring, as usual, by Z n . It is well known that for n prime the ring Z n is a field. Let us consider m , n be arbitrary natural numbers. We denote by
44 344 21 K m nnnmn ZZZZ ×××= the Cartesian product of m rings Z n . It is clear that Z nm with operations + mod n , ⋅ mod n defined elementwise is an associative commutative ring with unity again. It contains divisors of zero and we call its elements row vectors or points . Furthermore we call the zero element (0,…,0) zero vector and denote it simply by 0. We denote by Z nm,m the set of all m × m matrices with elements in the ring Z n. For k ∈ N and A ∈ Z nm,m we will denote by (A) mod k a matrix which arose from matrix A after application of operation modulo k on its elements. In the following we shall frequently use a product on the set Z nm,m defined as matrix multiplication together with operation modulo n , i.e. A,B ∈ Z nm,m → (AB) mod n . (2.1) This product is, due to associativity of matrix multiplication, associative again and the set Z nm,m equipped with this product forms a semigroup. If we take matrices A,B ∈ Z nm,m , such that det(A)=det(B)=1 (mod n ), then det((AB) mod n ) = 1 (mod n ) holds. It follows that the subset of Z nm,m formed by all matrices with determinant equal to unity modulo n is a semigroup. Definition 2.1:
For m , n ∈ N, n ≥ SL ( m ,Z n ) := {A ∈ Z nm,m | det A = 1 (mod n )}. Now we show that SL ( m ,Z n ) with operation (2.1) forms a group. Because SL ( m ,Z n ) is a semigroup, it is sufficient to show that there exists a unit element and a right inverse element. Unit matrix is clearly the unit element. In order to find a right inverse element consider the following equation AA adj = det(A)I. (2.2) The symbol A adj denotes the adjoint matrix defined by (A adj ) i,j := (-1) i+j detA(j,i), where A(j,i) is the matrix obtained from matrix A by omitting the j-th row and the i-th column. The equation (2.2) holds for an arbitrary matrix, hence it holds for matrices from SL ( m ,Z n ), and evidently holds after application of operation modulo n on both sides. Consequently, for A ∈ SL ( m ,Z n ), we have AA adj = I (mod n ), i.e. (AA adj ) mod n = I . Therefore A adj is the right inverse element corresponding to the matrix A, and consequently SL ( m ,Z n ) is a group. The group SL ( m ,Z n ) is finite and its order was computed by You Hong and Gao You in [10] (see also [11], p. 86). If n ∈ N, n ≥ n = ∏ = ri ki i p , where p i are distinct primes, then according to [10], the order of SL ( m ,Z n ) is | SL ( m ,Z n ) | = ∏ ∏ = =− − ri mj jim pn . (2.3) Let G be a group and X ≠ / a set. Recall that a mapping ψ: G ×X → X is called a right action of the group G on the set X if the following conditions hold for all elements x ∈ X: ψ ( gh , x )= ψ ( g , ψ ( h , x )) for all h , g ∈ G. ψ ( e , x ) = x , where e is the unit element of G . Let ψ be an action of a group G on a set X. A subset of G , { g ∈ G | ψ ( g , a )= a } is called a stability subgroup of the element a ∈ X. A subset of X, { b ∈ X | ∃ g ∈ G , b = ψ ( g , a )} is called an orbit of the element a ∈ X with respect to the action ψ of the group G . Let us note that if ψ is an action of a group G on a set X then relation ∼ defined by formula a , b ∈ X, a ∼ b ⇔ ∃ g ∈ G , ψ (g, a )= b (2.4) is an equivalence on the set X and the corresponding equivalence classes are orbits. Definition 2.2:
For m , n ∈ N, n ≥ ψ of the group SL ( m ,Z n ) on the set Z nm as right multiplication of the row vector a ∈ Z nm by the matrix A ∈ SL ( m ,Z n ) modulo n : ψ (A, a ) := ( a A) mod n Henceforth we will omit the symbol mod n and write this action simply as a A.
3. Orbits for n=p prime number
The purpose of this section is to describe orbits of the ring Z pm under the action of the group SL ( m ,Z p ), where p is prime. Trivially, for m =1 is SL (1,Z p )={(1)} and any orbit has the form { a } for a ∈ Z p . Consequently we will further consider m ≥
2. It is clear that the zero element can be transformed by the action of SL ( m ,Z p ) to itself only, thus it forms a one-point orbit and its stability subgroup is the whole SL ( m ,Z p ). Let us take a nonzero element for instance (0,…,0,1) ∈ Z pm , and find its orbit. An arbitrary matrix A from SL ( m ,Z p ) acts on this element as follows ( ) ( ) mmmmmmmm mmmm m AAAAAA AAA AAA ,2,1,,1,1, ,11,11,1 ,12,11,1 ,,,1,0,,0
KKK MMMM KK = −−− (mod p ). Thus the orbit of element (0,…,0,1) contains last row of any matrix from SL ( m ,Z p ). It follows from det(A) = 1 that these rows cannot be zero and we show that they can be equal to an arbitrary nonzero element from Z pm . Let ( A m,1 ,A m,2 ,…,A m,m ) ∈ Z pm be a nonzero element, that means ∃ j ∈ {1,2,…, m } such that A m,j ≠ j = 1: = mmmm AAA BA ,2,1, KM , where B = diag(1,…,1,(-1) ( A m ,1 ) -1 ). Here ( A m, ) -1 denotes the inverse element to A m ,1 in the field Z p . We conclude that in the case of n=p prime there are only two orbits: 1. one-point orbit represented by the zero element (0,…,0,0) 2.( p m -1)-point orbit Z pm \{0} represented by the element (0,…,0,1) . Orbits for n natural number We consider an arbitrary natural number n of the form ∏ = = ri ki i pn , where p i are distinct primes and k i are natural numbers. The action of the group SL ( m ,Z n ) on the ring Z nm was established in definition 2.2 as a right multiplication of a row vector from Z nm by a matrix from SL ( m ,Z n ) modulo n . We define an equivalence induced by this action on the ring Z nm according to (2.4). Elements a =( a , a ,…, a m ), b =( b , b ,…, b m ) ∈ Z nm are equivalent a ∼ b if and only if there exists A ∈ SL ( m ,Z n ) such that a A= b i.e. imj jij bAa = ∑ = (mod n ), ∀ i ∈ {1,2,…, m }. (4.1) Definition 4.1:
Let ~ be the equivalence on Z nm defined by (4.1). For any divisor d of n , we will denote by Or m,n ( d ) the class of equivalence (orbit) containing the point (0,…,0,( d ) mod n ), i.e. Or m,n ( d ) = { a ∈ Z nm | a ~ (0,…,0, ( d ) mod n )}. (4.2) Note that the orbit Or m,n ( n ) contains only the zero vector, because the zero vector can be transformed by the action of SL ( m ,Z n ) only to itself. We shall see later that any orbit in Z nm has the form (4.2). Definition 4.2: A greatest common divisor of the element a = ( a , a ,…, a m ) ∈ Z nm and the number n ∈ N is the greatest common divisor of all components of the element a and the number n in the ring of integers Z. We denote it by gcd( a , n ):= gcd( a , a ,…, a m , n ). (4.3) Lemma 4.3:
The action of the group SL ( m ,Z n ) on the ring Z nm preserves the greatest common divisor of an arbitrary element a ∈ Z nm and the number n , i.e. gcd( a A, n )=gcd( a , n ) ∀ a ∈ Z nm , ∀ A ∈ SL ( m ,Z n ). Proof:
It follows from a A=( imi i Aa ∑ = ,…, mimi i Aa ,1 ∑ = ) and gcd( a , n )| jimi i Aa ,1 ∑ = , ∀ j ∈ {1,2,…, m } that gcd( a , n )|gcd( a A, n ), i.e. the greatest common divisor cannot decrease during this action. If we take an element a A and a matrix A -1 we obtain gcd( a A, n )|gcd( a AA -1 , n ) = gcd( a , n ) and together with first condition we have gcd( a A, n )=gcd( a , n ). QED Corollary 4.4:
For any divisor d of n the orbit Or m,n ( d ) is a subset of { a ∈ Z nm | gcd( a,n )= d }. We will show that the orbit Or m,n (1) is equal to the set { a ∈ Z nm | gcd( a,n )=1}. From corollary 4.4 we know that Or m,n (1) is the subset of { a ∈ Z nm | gcd( a,n )=1} and we prove that they have the same number of elements. At first we determine the number of points in Or m,n (1). For this purpose we determine the stability subgroup of the element (0,…,0,1). It is obviously formed by matrices of the form = −−− ,11,11,1 ,12,11,1 KK MMMM K mmmm m
AAA AAAA , det(A) = 1 (mod n ). Expansion of this determinant gives 1 = det(A) = (-1) m+m detA( m , m ) = detA( m , m ) (mod n ). Therefore the stability subgroup of the point (0,…,0,1) is: S := −∈∈ = )Z,1(B | )Z,(100A nn,2,1 mSLmSLAB A mm K M , and its order is | S |= ∏ ∏ = −= −−− − ri mj jimm pn )1( . (4.4) According to the Lagrange theorem, product of the order and the index of an arbitrary subgroup of a given finite group is equal to the order of this group. If we define on the group SL ( m ,Z n ) a left equivalence induced by the stability subgroup S by formula A,B ∈ SL ( m ,Z n ) A ≈ S B ⇔ AB -1 ∈ S , then we obtain equivalence classes of the form S B = {AB | A ∈ S }, B ∈ SL ( m ,Z n ), i.e. right cosets from SL ( m ,Z n )/ S . The number of these cosets is, by definition, the index of the subgroup S . These cosets correspond one-to-one with points of the orbit which includes the point (0,…,0,1). Therefore the index of the stability subgroup S is equal to the number of points in this orbit. Similar calculation can be done for an arbitrary point in an arbitrary orbit. Thus we have the following proposition. Proposition 4.5:
The number of elements in an orbit is equal to the order of the group SL ( m ,Z n ) divided by the order of the stability subgroup of an arbitrary element in this orbit. Using (2.3) and (4.4) we obtain that the number of points in the orbit Or m,n (1) is equal to | Or m,n (1)| = ∏ = − − ri mim pn )1( . (4.5) Now we will determine the number of all elements in Z nm , which have the greatest common divisor with the number n equal to unity. This number is equal to the Jordan function. Definition 4.6:
For m ∈ N a mapping ϕ m :N → N defined by ϕ m ( n ) = | { a ∈ Z nm | gcd( a,n )=1} | (4.6) is called the Jordan function of the order m . e present, without proof, some basic properties of the Jordan function which can be found in [12]. Proposition 4.7:
For the Jordan function ϕ m of the order m ∈ N and for any n ∈ N holds: 1. ϕ m ( n )= ( ) ∏ ∈ − − P,| pnp mm pn (4.7) 2. mdnd m nd = ∑ ∈ )( N,| ϕ (4.8) 3. ϕ m dn = | { a ∈ dn Z m | gcd( a , dn ) = | = | { a ∈ Z nm | gcd( a,n ) =d } | . (4.9) The number of all elements in Z nm , which are co-prime with n , given by the first property of the Jordan function ϕ m ( n ) (4.7), is equal to the number of points in the orbit Or m,n (1). Therefore the orbit Or m,n (1) is formed by all elements in Z nm which are co-prime with n . Proposition 4.8:
For m,n ∈ N, m ≥ m,n (1) = { a ∈ Z nm | gcd( a , n ) = 1}. n=p k power of a prime Let us now consider n of the form n=p k , where p is a prime number and k ∈ N, and determine orbits in this case.
Definition 4.1.1:
For j ∈ N, j ≤ k , we define a mapping F j : mp k Z → mp k Z by the formula F j ( a ) = ( p j .a k p mod ) for any a ∈ mp k Z . Lemma 4.1.2:
Let a and b be two equivalent elements from mp k Z and j ≤ k . Then the elements F j ( a ) and F j ( b ) are equivalent as well. Proof:
Let a , b ∈ mp k Z , a ∼ b . It follows from definition of equivalence ∼ that there exists a matrix A ∈ SL ( m , k p Z ) such that a A= b . Consequently F j ( a A) = F j ( b ), where F j ( a A) = ( p j a A k p mod ) = ( p j a k p mod ) (A k p mod ) = F j ( a )A. Since we have F j ( a )A= F j ( b ) and therefore F j ( a ) ∼ F j ( b ). QED Proposition 4.1.3:
Any orbit in the ring mp k Z has the form )(Or k pm, j p = { a ∈ mp k Z | gcd( a , p k ) = p j }, 0 ≤ j ≤ k, and consists of | )(Or k pm, j p | = ϕ m ( p k-j ) points. Proof:
From Lemma 4.1.2 it is clear that F j maps the orbit )1(Or k pm, into the orbit )(Or k pm, j p and from Corollary 4.4 we have F j ( )1(Or k pm, ) ⊂ )(Or k pm, j p ⊂ { a ∈ mp k Z | gcd( a , p k ) = p j }. onversely, { a ∈ mp k Z | gcd( a , p k ) = p j }={ p j a | a ∈ mp jk − Z , gcd( a , p k-j )=1} ⊆ ⊆ { k p mod )( ap j | a ∈ mp k Z , gcd( a , p k ) = 1} = F j ( )1(Or k pm, ). Thus we have F j ( )1(Or k pm, )= )(Or k pm, j p ={ a ∈ mp k Z | gcd( a , p k ) = p j }. QED n=pq , gcd( p,q )=1 Let us now consider n of the form n=pq , where p,q ∈ N are co-prime numbers. In this case it will be very useful to apply the Chinese remainder theorem [13].
Theorem 4.2.1: (Chinese remainder theorem) Let a ,a ∈ Z. Let p ,p ∈ N be co-prime numbers. Then there exists x ∈ Z, such that x=a i (mod p i ), ∀ i =1,2. If x is a solution, then y is a solution if and only if x=y (mod p p ). Definition 4.2.2:
For p,q ∈ N, gcd( p,q )=1 we define a mapping G: Z pqm → Z pm ×Z qm by the formula G( a ) := (( a ) mod p , ( a ) mod q ) for any a ∈ Z pqm , and a mapping g: SL ( m, Z pq ) → SL ( m, Z p ) × SL ( m, Z q ) by the formula g(A) := ((A) mod p , (A) mod q ) for any A ∈ SL ( m, Z pq ). It is clear from definition that G,g are homomorphisms and the Chinese remainder theorem implies that G,g are one-to-one correspondences. Thus we have the following proposition. Proposition 4.2.3:
The mapping G is an isomorphism of rings and the mapping g is an isomorphism of groups. Further we determine orbits on the Cartesian product of rings Z pm ×Z qm . For this purpose we define action of the Cartesian product of groups SL ( m, Z p ) × SL ( m, Z q ) on ring Z pm ×Z qm by the formula a A = ( a , a ) (A ,A )=(( a A ) mod p , ( a A ) mod q ) for any a= ( a ,a ) ∈ Z pm ×Z qm and any A=(A ,A ) ∈ SL ( m, Z p ) × SL ( m, Z q ). It follows from definition of this action that orbits in Z pm ×Z qm are Cartesian products of orbits in Z pm and Z qm . Proposition 4.2.4:
Let p,q ∈ N be co-prime numbers. Then the mapping G provides one-to-one correspondence between orbits in Z pqm and Cartesian products of orbits in Z pm and Z qm . Moreover, if p | p, q | q and the orbits Or m,p ( p ), Or m,q ( q ) are of the form Or m,p ( p )={ a ∈ Z pm | gcd( a,p )= p }, Or m,q ( q )={ a ∈ Z qm | gcd( a,q )= q }, then Or m,pq ( p q ) =G -1 (Or m,p ( p )×Or m,q ( q )) = { a ∈ Z pqm | gcd( a,pq ) = p q }. roof: At first we prove that G and G -1 preserve equivalence, i.e. a ∼ b ⇔ G( a ) ∼ G( b ) for all a,b ∈ Z pqm . From definition of equivalence we have a ∼ b ⇔ ∃ A ∈ SL ( m, Z pq ), a A = b ⇔ G( a A) = G( b ), where G( a A) = (( a A) mod p , ( a A) mod q ) = (( a ) mod p , ( a ) mod q ) ((A) mod p , (A) mod q ) = G( a )g(A). Because G and g are one-to-one correspondences we obtain a ∼ b ⇔ a A = b ⇔ G( a )g(A) = G( b ) ⇔ G( a ) ∼ G( b ). Since the mapping G is an isomorphism and G, G -1 preserve equivalence, orbits in the ring Z pqm one-to-one corresponds to orbits in the ring Z pm ×Z qm , and these are Cartesian products of orbits on Z pm and Z qm . Now remain to prove that the orbit Or m,pq ( p q ) corresponds to the orbit Or m,p ( p )×Or m,q ( q ). It follows from the Chinese remainder theorem that G maps the set { a ∈ Z pqm | gcd( a,pq ) = p q } on the set {( a ,a ) ) ∈ Z pm ×Z qm | gcd( a , p ) = p , gcd( a ,q ) = q }, which is equal to the orbit Or m,p ( p )×Or m,q ( q ). Therefore the set { a ∈ Z pqm | gcd( a,pq ) = p q } forms an orbit and from Corollary 4.4 follows that Or m,pq ( p q ) = { a ∈ Z pqm | gcd( a,pq ) = p q }. QED As a corollary of Propositions 4.1.3 and 4.2.4 we obtain the following theorem. Theorem 4.9:
Consider the decomposition of the ring Z nm , m ≥ SL ( m ,Z n ). Then i) any orbit is equal to the orbit Or m,n ( d ) for some divisor d of n , i.e. )(OrZ | nm,mn d nd U = ; ii) Or m,n ( d ) = { a ∈ Z nm | gcd( a,n ) = d }; iii) the number of points |Or m,n ( d )| in d -orbit is given by the Jordan function |Or m,n ( d )| = ϕ m dn = ∏ ∈ − − Ppnpd mm pdn ,| )1( . Conclusion:
We have stepwise determined the orbits on the ring Z nm with respect to the action of the group SL ( m ,Z n ). At first we have proceeded in the same way as Kirillov in [9] and we have obtained the orbits in the case of n prime number. In this case there are only two orbits, the first is one-point orbit formed by zero element and the second one is formed by all nonzero elements. The next step was the case of n = p k power of prime, there we found k+1 orbits characterized by the greatest common divisor of their elements and number n . Finally the orbits for an arbitrary natural number n were found. Our results are summarized in Theorem 4.9. Acknowledgements
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