AP05_5.vp 1 Introduction The important role of symmetries in classical and quan- tum physics is well known. We focus on so called discrete quantum physics; this 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 obser- vables with additive quantum numbers. The generalized n×n Pauli matrices were described in [3]. For n � 3 these 3×3 Pauli matrices form one of four non-equivalent gradings of sl(3,C). Other fine gradings are Cartan decomposition and the grading which corresponds to Gell-Mann matrices [4, 5]. The symmetries of the fine grad- ing of sl(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, Zn) as the group of sym- metry of the Pauli gradings. The additive quantum numbers, mentioned above, form in this case the finite associative addi- tive ring Zn×Zn. The action of SL(2, Zn) on Zn×Zn then represents the symmetry transformations of Pauli gradings of sl(n, C). The orbits of this action form such points in Zn×Zn 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, Zn) on various types of Cartesian products of Zn [8]. Note that the orbits of SL(2, Zp) on Z p 2 , where p is a prime number were, considered in [9] §16.3. The purpose of this paper is to generalize this result to orbits of SL(m,Zn) on Zn m where m, n are arbitrary natural numbers. 2 Action of the group SL(m, Zn) Throughout the paper 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 commutative ring with unity. We will denote this ring, as usual, by Zn. It is well known that for n prime the ring Zn is a field. Let us consider m, n to be arbitrary natural numbers. We denote by Z Z Z Zn m n n n m � � � �� � ��� ��� the Cartesian product of m rings Zn. It is clear that Zn m with operations �mod n , ×mod n defined elementwise is an associa- tive commutative ring with unity again. It contains divisors of zero and we call its elements row vectors or points. Further- more we call the zero element (0, …, 0) zero vector and denote it simply by 0. We denote by Zn m m, the set of all m×m matrices with ele- ments in the ring Zn. For k �N and A Z� n m 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 Zn m m, defined as matrix multiplication together with oper- ation modulo n, i.e. A, B Z AB)� �n m m n , mod( . (2.1) This product is, due to the associativity of matrix multipli- cation, associative again and the set Zn m m, equipped with this product forms a semigroup. If we take matrices A, B Z� n m 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 Zn m m, formed by all matrices with the determinant equal to unity modulo n is a semigroup. Definition 2.1: For m n, �N, n � 2 we define SL m nn n m m( , ): { |det (mod )},Z A Z A� � � 1 . Now we show that SL(m,Zn) with operation (2.1) forms a group. Because SL(m, Zn) is a semigroup, it is sufficient to show that there exists a unit element and a right inverse ele- ment. Unit matrix is clearly the unit element. In order to find a right inverse element consider the following equation AA A)Iadj � det( . (2.2) The symbol Aadj denotes the adjoint matrix defined by (A A( , )adj) : ( ) det,i j i j j i� � �1 , where A( , )j i is the matrix ob- tained 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, Zn), and evidently holds after application of operation modulo n on both sides. Consequently, for A Z�SL m n( , ), we have AA Iadj � (mod )n , i.e. (AA Iadj)mod n � . Therefore Aadj is the right inverse element corresponding to matrix A, and consequently SL(m, Zn) is a group. © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 39 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 5/2005 On Orbits of the Ring Zn m under Action of the Group SL(m, Zn) P. Novotný, J. Hrivnák We consider the action of the finite matrix group SL(m,Zn ) on the ring Zn m. 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 � 2 and n prime number. Keywords: ring, finite group. The group SL(m, Zn) is finite and its order was computed by You Hong and Gao You in [10] (see also [11], p. 86). If n n� �N, 2 is written in the form n pi k i r i� � 1 , where pi are distinct primes, then according to [10], the order of SL(m, Zn) is SL m n p n m i j j m i r ( , )Z � � � � � � � � � �� 2 1 21 1 1 . (2.3) Let G be a group and X � 0 a set. Recall that a mapping �: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: 1. � � �( , ) ( , ( , ))gh x g h x� for all h g G, � . 2. �( , )e x x� , where e is the unit element of G. Let � be an action of a group G on a set X. A subset of G, { | ( , ) }g G g a a� �� is called a stability subgroup of the ele- ment a �X. A subset of X, { | , ( , )}b X g G b g a� � � � � is called an orbit of the element a �X with respect to the action � of group G. Let us note that if � is an action of a group G on a set X then relation ~ defined by formula a b a b g G g a b, , ~ , ( , )� � � � �X � (2.4) is an equivalence on the set X and the corresponding equiva- lence classes are orbits. Definition 2.2: For m n, �N, n � 2 we define a right action � of the group SL(m, Zn) on the set Zn m as right multiplication of the row vector a n m� Z by the matrix A Z�SL m n( , ) modulo n: �( , ): ( )modA Aa a n� . Henceforth we will omit the symbol mod n and write this action simply as aA. 3 Orbits for n � p prime number The purpose of this section is to describe orbits of the ring Z p m under the action of the group SL(m, Zp), where p is prime. Trivially, for m � 1 is SL(1, Zp) � {(1)} and any orbit has the form {a} for a p� Z . Consequently we will further consider m � 2. It is clear that the zero element can be transformed by the action of SL(m, Zp) to itself only, thus it forms a one-point orbit and its stability subgroup is the whole SL(m, Zp). Let us take a nonzero element, for instance ( , , , )0 0 1� � Z p m, and find its orbit. An arbitrary matrix A from SL(m, Zp) acts on this element as follows ( , , , ) , , , , , , , 0 0 1 1 1 1 2 1 1 1 1 2 1 1 � � � � � � � A A A A A A A A m m m m m m � � � m m m m m m m A A A A p , , , , ,( , , ) (mod ). 2 1 2 � � � � � � �� � � � � �� � � Thus the orbit of element ( , , , )0 0 1� contains the last row of any matrix from SL(m, Zp). 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 p m. Let ( , , ), , ,A A Am m m m p m 1 2 � � Z be a nonzero element, which means � �j m{ , , , }1 2 � such that Amj � 0, then matrix A can be cho- sen with the determinant equal to 1. Without loss of generality consider j � 1: A B � � � � � �� � � � � �� 0 0 1 2 � �A A Am m m m, , , , where � �B � � � �diag 1 1 1 1 1 1, , , ( ) ( ),� m mA . Here (Am,1) �1 denotes the inverse element to Am,1 in the field Zp. 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. ( pm�1)-point orbit Z p m\{0} represented by the element (0, …, 0, 1) 4 Orbits for n natural number We consider an arbitrary natural number n of the form n pi k i r i� � 1 , where pi are distinct primes and ki are natural numbers. The action of the group SL(m, Zn) on the ring Zn m was established in definition 2.2 as a right multiplication of a row vector from Zn m by a matrix from SL(m, Zn) mod- ulo n. We define an equivalence induced by this action on the ring Zn m according to (2.4). Elements a a a am� ( , , , )1 2 � , b b b bm n m� �( , , , )1 2 � Z are equivalent a~b if and only if there exists A Z�SL m n( , ) such that aA � b i.e. a A b n i mj i j i j m , (mod ), { , , , }� � � � � 1 1 2 � . (4.1) Definition 4.1: Let ~ be the equivalence on Zn m defined by (4.1). For any divisor d of n, we will denote by Orm,n(d) the class of equivalence (orbit) containing the point (0, …, 0, (d)mod n), i.e. Or Zm n n m nd a a d, mod( ) { | ~ ( , , ( ) )}� � 0 � . (4.2) Note that the orbit Orm, n(n) contains only the zero vector, because the zero vector can be transformed by the action of SL(m, Zn) only to itself. We shall see later that any orbit in Zn m has the form (4.2). Definition 4.2: A greatest common divisor of the element a a a am n m� �( , , , )1 2 � Z 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( , ): gcd( , , , , )a n a a a nm� 1 2 � . (4.3) Lemma 4.3: The action of the group SL(m, Zn) on the ring Zn m preserves the greatest common divisor of an arbitrary ele- ment a n m� Z and the number n, i.e. 40 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 5/2005 Czech Technical University in Prague gcd( , ) gcd( , ) , ( , )a n a n a SL mn m nA Z A Z� � � � � . Proof: It follows from a a A a Ai ii m i i mi m A � � � � � � �� �, ,, ,11 1� and gcd( , )| ,a n a Ai i ji m �� 1 , � �j m{ , , , }1 2 � that gcd( , )|gcd( , )a n a nA , i.e. the greatest common divisor cannot decrease during this action. If we take an element aA and a matrix A�1 we obtain gcd( , )|gcd( , ) gcd( , )a n a n a nA AA 1� � and together with the first condition we have gcd( , ) gcd( , )a n a nA � . QED Corollary 4.4: For any divisor d of n the orbit Orm,n(d) is a subset of { |gcd( , ) }a a n dn m� �Z . We will show that the orbit Orm,n(1) is equal to the set { |gcd( , ) }a a nn m� �Z 1 . From corollary 4.4 we know that Orm,n(1) is the subset of { |gcd( , ) }a a nn m� �Z 1 and we prove that they have the same number of elements. At first we determine the number of points in Orm,n(1). For this pur- pose we determine the stability subgroup of the element (0, …, 0, 1). It is obviously formed by matrices of the form A � � � � � � � � � � A A A A A A m m m m m 1 1 1 2 1 1 1 1 2 1 0 0 1 , , , , , , � � � � � � � � � � � �, det( ) (mod )A 1 n . Expansion of this determinant gives 1 1� � � ��det( ) ( ) det ( , ) det ( , ) (mod )A A Am m m m m m n . Therefore the stability subgroup of the point (0, …, 0, 1) is: S A A SL m SL m m m n: ( , )| ( , , , � � � � � � � � � � � � � � �A B Z B 1 2 0 0 1 1 � � Zn ) � � � � � � � � � � � � � � , and its order is S n pm m i j j m i r � �� � � � � � 2 1 2 1 1 1( ). (4.4) According to the Lagrange theorem, the 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, Zn) a left equivalence induced by the stability subgroup S by formula A,B Z A B AB� � � ��SL m Sn S( , ) 1 , then we obtain equivalence classes of the form S SB AB A� �{ | }, B Z�SL m n( , ), i.e. right cosets from SL(m, Zn)/S. The number of these cosets is, by definition, the index of subgroup S. These cosets correspond one-to- -one with the 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. A similar calcula- tion 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, Zn) 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 Orm,n(1) is equal to Or =m n m i m i r n p, ( ) ( )1 1 1 � � � . (4.5) Now we will determine the number of all elements in Zn m that 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 �m:N N� defined by �m n mn a a n( ) { |gcd( , ) }� � �Z 1 (4.6) is called the Jordan function of the order m. We present, without proof, some basic properties of the Jordan function which can be found in [12]. Proposition 4.7: For the Jordan function �m of the order m �N and for any n �N holds: 1. �m m m p n p n n p( ) ) | , � � � � ( P 1 (4.7) 2. �m d n d md n( ) | , � � � N (4.8) 3. �m n d m n m n d a a n d a a n d � � � � � � � � � � � { |gcd( , ) } { |gcd( , ) Z Z 1 } . (4.9) The number of all elements in Zn m, which are co-prime with n, given by the first property of the Jordan function �m(n) (4.7), is equal to the number of points in the orbit Orm,n(1). Therefore the orbit Orm,n(1) is formed by all ele- ments in Zn m which are co-prime with n. Proposition 4.8: For m n, �N, m � 2 holds Or Zm n n ma a n, ( ) { |gcd( , ) }1 1� � � . 4.1 Orbits for n � pk power of a prime Let us now consider n of the form n � pk, where p is a prime number and k �N, and determine orbits in this case. Definition 4.1.1: For j j k� �N, , we define a mapping F Z Zj p m p m k k: � by the formula F j j p a p a k( ) ( )mod � for any a p m k� Z . Lemma 4.1.2: Let a and b be two equivalent elements from Z p m k and j k� . Then the elements F j a( ) and F j b( ) are equiva- lent as well. Proof: Let a b p m k, Z� , a ~ b. It follows from the definition of equivalence ~ that there exists a matrix A Z�SL m p k ( , ) such that aA � b. Consequently F A Fj ja b( ) ( )� , where F A A) ) A) F A mod mod mod j j p j p p ja p a p a ak k k( ) ( ( ( ( )� � � . © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 41 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 5/2005 Since we have F A Fj ja b( ) ( )� and therefore F Fj ja b( ) ~ ( ). QED Proposition 4.1.3: Any orbit in the ring Z p m k has the form Or Z m p j p m k j k kp a a p p, ( ) { |gcd( , ) }� � � , 0 � �j k, and consists of Or m p j m k j k p p, ( ) ( )� �� points. Proof: From Lemma 4.1.2 it is clear that F j maps the orbit Or m p k, ( )1 into the orbit Or m p j k p, ( ) and from Corollary 4.4 we have F p p a a p pj m p j m p j p m k j k k k( ( )) ( ) { |gcd( , ) }, , Or Or Z! ! � � . Conversely, { |gcd( , ) } { | ,gcd( , ) }a a p p p a a a p p m k j j p m k j k k j� � � � �� �Z Z 1 ! � � �{( ) | , gcd( , ) } ( ( )). mod , p a a a pj p p m k j m pk k k Z F Or1 1 Thus we have F Or Or Zj m p m p j p m k j k k kp a a p p( ( )) ( ) { |gcd( , ) }, , 1 � � � � . QED 4.2 Orbits for n � pq, gcd(p,q) � 1 Let us now consider n of the form n � pq, where pq �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 a1 Z, 2 � . Let p p1 N, 2 � be co-prime numbers. Then there exists x � Z, such that x a p ii i� � �(mod ), ,1 2. If x is a solution, then y is a solution if and only if x y p p� (mod )1 2 . Definition 4.2.2: For p q, �N, gcd(p, q) � 1 we define a map- ping G Z Z Z: pq m p m q m� � by the formula � �G( ): ( ) ,( )mod moda a ap q� for any a pqm� Z , and a mapping g Z Z Z: ( , ) ( , ) ( , )SL m SL m SL mpq p q� � by the formula � �g A A A( ): ( ) ,( )mod mod� p q for any A Z�SL m 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 Zp m q m� . For this purpose we define the action of the Cartesian product of groups SL m SL mp q( , ) ( , )Z Z� on ring Z Zp m q m� by the formula � �a a a a ap qA A A A A� �( , )( , ) ( ) , ( )mod mod1 2 1 2 1 1 2 2 for any a a a p m q m� � �( , )1 2 Z Z and any A A A Z Z� � �( , ) ( , ) ( , )1 2 SL m SL mp q . It follows from the definition of this action that orbits in Z Zp m q m� are Cartesian products of orbits in Z p m and Zq m. Proposition 4.2.4: Let p q, �N be co-prime numbers. Then the mapping G provides one-to-one correspondence between the orbits in Z pq m and the Cartesian products of the orbits in Z p m and Zq m. Moreover, if p p1| , q q1| and the orbits Orm,p(p1), Orm,q(q1) are of the form Or Z Or Z m p p m m q q m p a a p p q a , , ( ) { |gcd( , ) }, ( ) { |gcd 1 1 1 � � � � � ( , ) },a q q� 1 then � �Or G Or Or Z m pq m p m q pq m p q p q a a , , ,( ) ( ) ( ) { |gcd( , 1 1 1 1 1� � � � � pq p q) }.� 1 1 Proof: First, we prove that G and G�1 preserve equivalence, i.e. a b a b~ ( ) ~ ( )� G G for all a b pq m, � Z . From the definition of equivalence we have a b SL m a b a bpq~ ( , ), ( ( )� � � � � �A Z A G A) G , where � � � � G A) A) A) ) ) A) A) mod mod mod mod mod ( ( , ( ( , ( ( , ( a a a a a p q p q p � � � �mod G )g A). q a � � ( ( Because G and g are one-to-one correspondences we obtain a b a b a b a b~ ( ( ( ) ( ( )� � � � �A G )g A) G G ) ~ G . Since the mapping G is an isomorphism and G, G�1 pre- serve equivalence, the orbits in the ring Z pq m correspond one-to-one with the orbits in the ring Z Zp m q m� , and these are Cartesian products of orbits on Z p m and Zq m. Now remain to prove that the orbit Orm pq p q, ( )1 1 cor- responds to the orbit Or Orm p m qp q, ,( ) ( )1 1� . It follows from the Chinese remainder theorem that G maps the set { |gcd( , ) }a a pq p qpq m� �Z 1 1 on the set {( , ) |gcd( , ) , gcd( , ) }a a a p p a q qp m q m 1 2 1 1 2 1� � � �Z Z , which is equal to the orbit Or Orm p m qp q, ,( ) ( )1 1� . Therefore the set { |gcd( , ) }a a pq p qpq m� �Z 1 1 forms an orbit and from Corol- lary 4.4 it follows that Or Zm pq pq mp q a a pq p q, ( ) { |gcd( , ) }1 1 1 1� � � . 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 Zn m, m � 2 into orbits with respect to the action of the group SL(m, Zn). Then i) any orbit is equal to the orbit Orm,n(d) for some divisor d of n, i.e. Z Orn m m n d n d� , | ( )� ; 42 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 45 No. 5/2005 Czech Technical University in Prague ii) Or Zm n n md a a n d, ( ) { |gcd( , ) }� � � ; iii)the number of points Orm n d, ( ) in d-orbit is given by the Jordan function Or P m n m m m pd n p d n d n d p, | , ( ) ( )� � � � � � � � � � � � � � 1 . 5 Conclusion We have stepwise determined the orbits on the ring Zn m with respect to the action of the group SL(m, Zn). First, we proceeded in the same way as Kirillov in [9] and we 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 the zero element and the second is formed by all nonzero elements. The next step was the case of n � pk 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. 6 Acknowledgments We would like to thank Prof. Jiří Tolar, Prof. Miloslav Havlíček and Doc. Edita Pelantová for numerous stimulating and inquisitive discussions. References [1] Št’ovíček, P., Tolar, J.: “Quantum Mechanics in a Dis- crete Space-Time.” Rep. Math. Phys. Vol. 20 (1984), p. 157–170. [2] Vourdas, A.: “Quantum Systems with Finite Hilbert Space.” Rep. Progr. Phys. Vol. 67 (2004), p. 267–320. [3] Patera, J., Zassenhaus, H.: The Pauli Matrices in n Di- mensions and Finest Gradings of Simple Lie Algebras of Type An�1.” J. Math. Phys. Vol. 29 (1988), p. 665–673. [4] Gell-Mann, M.: “Symmetries of Baryons and Mesons.” Phys. Rev. Vol. 125 (1962), p. 1067. [5] Havlíček, M., Patera, J., Pelantová, E., Tolar, J.: “The Fine Gradings of sl (3, C) and Their Symmetries,” in Proc. of XXIII. International Colloquium on Group Theoretical Methods in Physics, eds. A. N. Sissakian, G. S. Pogosyan and L. G. Mardoyan, JINR, Dubna, Vol. 1 (2002), p. 57–61. [6] Havlíček, M., Patera, J., Pelantová, E., Tolar, J.: “Auto- morphisms of the Fine Grading of sl (n, C) Associated with the Generalized Pauli Matrices.” J. Math. Phys. Vol. 43, 2002, p. 1083–1094. [7] de Montigny, M., Patera, J.: “Discrete and Continuous Graded Contractions of Lie Algebras and Superalge- bras.” J. Phys. A: Math. Gen., Vol. 24 (1991), p. 525–547 . [8] Hrivnák, J.: “Solution of Contraction Equations for the Pauli Grading of sl (3, C).” Diploma Thesis, Czech Technical University, Prague 2003. [9] Kirillov, A. A.: Elements of the Theory of Representations, Springer, New York 1976. [10] You Hong, Gao You: “Computation of Orders of Classi- cal Groups over Finite Commutative Rings.” Chinese Science Bulletin, Vol. 39 (1994), No. 14, p. 1150–1154. [11] Drápal, A.: Group Theory – Fundamental Aspects (in Czech), Karolinum, Praha 2000. [12] Schulte, J.: “Über die Jordanische Verallgemeinerung der Eulerschen Funktion.” Resultate der Mathematik, Vol. 36 (1999), p. 354–364. [13] Graham, R. L., Kmoth, D. E., Patashnik, O.: Concrete Mathematics, Addison-Wesley, Reading, MA, 1994. Ing. Petr Novotný phone: +420 222 311 333 fujtajflik@seznam.cz Ing. Jiří Hrivnák phone: +420 222 311 333 hrivnak@post.cz Department of Physics Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Břehová 7 115 19 Prague 1, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43 Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 5/2005