Algebra of Matrix Arithmetic

210, 194]215 Ž1998. JA987527

JOURNAL OF ALGEBRA ARTICLE NO.

Algebra of Matrix Arithmetic Gautami BhowmikU and Olivier Ramare ´† Department of Mathematics, Uni¨ ersite´ Lille 1, Unite´ associee ´ au CNRS, URA 751, 59655 Villeneu¨ e d’Ascq Cedex, France Communicated by Walter Feit Received November 3, 1997

We study the algebra of the arithmetic of integer matrices. A link is established between the divisor classes of matrices and lattices. The algebra of arithmetical functions of integral matrices is then shown to be isomorphic to an extension of the Hecke algebra, also called a Hall algebra in combinatorics. The dictionary helps translate results from one setting to another. One important application is the study of subgroups of a finite abelian group. Q 1998 Academic Press

I. INTRODUCTION Although integral matrices have been intensively studied Žsee, for instance, wHx, wNex, wN4x, wT3x. and some arithmetical notions like GCD and divisibility have been introduced, the set of divisor classes of a given matrix still remains unsatisfactorily understood; in this paper we wish to fill this gap. Let M r be the algebra of r = r matrices with coefficients in Z. We shall pay special attention to nonsingular matrices, i.e., to In¨ r , the subset of M r consisting of those matrices M for which detŽ M . / 0. The set of invertible elements Ur Žsome authors use GLr ŽZ. instead., which is the subset of M r consisting of matrices M verifying detŽ M . s "1, will feature prominently in our discussion. Although we do not do so in this paper, we could as well have worked with singular matrices, the arithmetic of which would be adapted on the lines of wBNx. We recall that if M is in In¨ r , a left divisor class of M is an integral matrix A that is a canonical representative of A ? Ur for which there exists an integral matrix B such that AB s M. In particular, we take A in U †

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Hermite Normal Form ŽHNF.. We use the notation A N M to indicate divisibility. The set containing Žleft. divisor classes of M is denoted by LDŽ M . and is known to be of finite cardinality. The classical way to study the arithmetic of commutative structures is through the description of ideals of M r . This approach is not obviously adapted to noncommutative situations, and in this paper we shall derive information from the action of M r over Z r. To do so, we choose a basis B of Z r and identify a matrix M with the endomorphism w M whose matrix in B is M. We consider the image M ŽZ r ., which depends only on the unimodular class of M, i.e., the HNF of M and its cokernel GŽ M . s Z rrM ŽZ r .. ŽIf we wished to study singular matrices as well, we would take only the torsion part of the cokernel.. It is known that GŽ M . is a finite abelian group independent of the chosen basis, and its invariant factors are the same as that of M. Notwithstanding partial efforts like those of Hua wH, Chap. 14x or Thompson wT2, T3x, in the past matrices have been used only as a tool in the study of finite abelian groups Žsee, e.g., wBx, wNe, II.21.ax. without any formal connection having been established. Here we prove THEOREM 1.1.

Let M g In¨ r . The arrow D : LD Ž M . ª  subgroups of G Ž M . 4 , A ¬ A Ž Z r . rM Ž Z r .

is one-to-one and order-preser¨ ing Ž i.e., if A1 N l A 2 then DŽ A1 . > DŽ A 2 ... Furthermore, GŽ Ay1 M . ( DŽ A.. As an immediate corollary we get COROLLARY 1.2. The number of di¨ isor classes of M g In¨ r is equal to the number of subgroups of GŽ M .. The interpretation of divisor classes in terms of lattices and finite groups has other applications. The left GCD D l and right multiple Mr of A and B in In¨ r can be defined simply as D l ŽZ r . s AŽZ r . q B ŽZ r . and Mr ŽZ r . s AŽZ r . l B ŽZ r .. This enables us to give another proof of a recent result of Thompson Žsee wT1x and wN1x for two other proofs. on the classical lines of Grassman’s formula. COROLLARY 1.3. GŽ A. = GŽ B . ( GŽ D l . = GŽ Mr .. As another important application, we determine indŽ S ., the index of a matrix S, which is the number of Hermite Normal Forms H having a given Smith Normal Form S. We recall that S is a canonical representative of Ur ? H ? Ur .

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COROLLARY 1.4. Let p be a prime number and l s Ž l1 , . . . , l r . be a partition. The number of Hermite Normal Forms whose Smith Normal Form is diagŽ p lr , . . . , p l1 . is gi¨ en by r lX1 y lX2 , lX2 y lX3 , . . . , lXl1

l1

X

X

p Ý is 1 l iq 1Ž ry l i . , p

where lX is the conjugate partition of l and w . . . x p is the gaussian multinomial. We shall see that this index function is the classical homomorphism from an abstract Hecke algebra to C. It is well known that if G is a finite abelian p-group, there exist positive integers l1 G l 2 G ??? G l s such that G is isomorphic to Ł i Zrp li Z. The partition l s Ž l1 , . . . , l s . is called the type of G Žalso sometimes called its Segre characteristic .. A subgroup H of G has a type m and a cotype n , which, by definition, is the type of GrH. Klein ŽwKx. has shown that there exists a polynomial gml, n with integer coefficients such that the number of subgroups of G of type m and cotype n is gml, n Ž p .. These polynomials are called Hall polynomials. The corresponding notion for matrices is the notion of invariant factors. If M g In¨ r has a determinant that is a power of a prime p, then Ur ? M ? Ur contains a unique diagonal matrix diagŽ p lr , . . . , p l1 . with l1 G ??? G l r . This representative is called the Smith Normal Form of M and is denoted by SNFŽ M .. As a corollary to Theorem 1.1, we get COROLLARY 1.5. Let p be a prime number, let M g In¨ r be of determinant a power of p, and let m and n be partitions of length r. We put SNFŽ M . s diagŽ p lr , . . . , p l1 .. Then the number of di¨ isors A of M such that SNFŽ A. s diagŽ p m r , . . . , p m1 . and SNFŽ Ay1 M . s diagŽ p n r , . . . , p n 1 . is equal to gml, n Ž p .. In wT3x Thompson established that the two quantities under consideration are simultaneously nonzero. In the same paper Thompson addresses the following important problem: given three finite abelian groups G, H, and K, what would be the necessary and sufficient conditions in terms of divisibility relations between the invariant factors of these three groups for K to be a subgroup of G with GrK ( H. In the same paper Thompson proves some intricate inequalities necessarily verified by these invariant factors. A necessary and sufficient condition is of course given by the nonvanishing of the corresponding Hall polynomial and in turn by the existence of a rather intricate combinatorial object called a Littlewood] Richardson sequence Žhereafter abbreviated as LR-sequence.. We shall describe LR-sequences in a way that helps us give simpler proofs for some

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of Thompson’s results. Note that links between the arithmetic of matrices and that of partitions have been shown earlier in some special situations Žsee wN2x, wB0x, wB3x, wNex, and wMx.. In the final part of this paper we interpret arithmetical functions of matrices in the context of divisibility in terms of lattices. A function f : In¨ r ª C is said to be arithmetical whenever f Ž A. depends only on the Smith Normal Form of A. This formalization helps us determine the pointwise value of arithmetical functions that are given as a convolution of two simpler functions Žfor instance, the number of divisors of a given matrix.. Pointwise evaluations have been treated in wCx Žfor the symmetric Euler-f function., in wN2x Žfor the norm, the t-norms, Euler-f t functions, and Mobius function., in wRSx and wN3x Žfor Ramanujan sums., and more ¨ w recently in B3x Žfor the divisor function.. A unified account of these results will be found in wBNx. Since these quantities are often difficult to evaluate, it is of interest to have a description that enables us to compare them or to have an idea of their forms. With this aim we shall give formulas for the Dirichlet series of a convolution product of two functions in the context of the Hall algebra Žgeneralizing the result of wBRx.. We shall finally use the divisor class]sublattice correspondence to deal with the algebra of arithmetical functions as defined in wN4x. Note, however, that unlike in wN4x, we restrict our attention to r = r integral matrices with nonzero determinant to get a more complete description. We shall identify in a natural way the ‘‘ p-component’’ of this algebra as the completion of the abstract Hecke algebra built over In¨ r, p and the unimodular group Žsee wKrx., where In¨ r, p is the set of r = r integer matrices whose determinant is a power of p. From this we shall deduce that this p-component is isomorphic to the algebra of formal power series with r indeterminates over C. As a further consequence we shall get THEOREM 1.6. The ring of arithmetical functions is isomorphic as a C-algebra to the ring of formal power series in countably many unknowns o¨ er C. From the above we infer that this ring does not have any zero divisors and that it is factorial Ža property shown by Cashwell and Everett Žcf. wCEx. while studying the case r s 1.. II. NOTATIONS Let r G 1 be the dimension. M r s  integer r = r matrices with coefficients in Z 4 , In¨ r s  M g M rrdet Ž M . / 0 4 ,

Ur s  M g M rrdet Ž M . s "1 4 .

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We need similar notations for the p-primary components when p is a prime number. We will use M r, p to denote the set of matrices of determinant "p n , n g N, and In¨ r , p s  M g M r , prdet Ž M . / 0 4 . We finally put < M < s
g U : Z rrV1 ª Z rrV2 x ¬ g Ž x. .

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We have LEMMA 3.1. E¨ ery morphism f : Z rrV1 ª Z rrV2 can be written as f s g U with g g In¨ r and g Ž V1 . ; V2 . Moreo¨ er, f is injecti¨ e if and only if g Ž V1 . s V2 l g ŽZ r .. Proof. Let Ž e1 , e2 , . . . , e r . be a basis of Z r such that Ž a1 e1 , . . . , a r e r . is a basis of V1 , and let Ž e 1 , e 2 , . . . , e r . be a basis of Z r such that Ž b1 e 1 , . . . , br e r . is a basis of V2 . We can write r

f Ž ei . s

Ý Ž m i j q bj Z . e j

Ž j s 1, . . . r .

js1

for some choice of m i j . Let N be a large integer parameter. We modify the m i j into mXi j by mXii s m ii q bi N. If N is taken large enough, the determinant of the map g Ž e i . s Ý rjs1 mXi j e j for i s 1, . . . r is asymptotic to N r b1 ??? br and thus is nonzero if N G N0 , say. We take N s N0 , and verify further that g Ž x . s f Ž x .. Moreover, f Ž0. s 0, so that g Ž V1 . ; V2 . To describe HomŽZ r , V1 . we define EpiŽ V1 , V2 . as the subset of HomŽ V1 , V2 . consisting of elements g such that g Ž V1 . s V2 . We also let IsoŽ V . s EpiŽ V, V .. Every element of IsoŽ V . induces an isomorphism on V Žsince its determinant is "1.. We have LEMMA 3.2. If g g EpiŽZ r , V . and c g HomŽZ r , V . then there exists k g HomŽZ r , Z r . such that c s gk . Proof. For any x in Z r , c Ž x . is in V. Thus there exists a unique y in Z r such that g Ž y . s c Ž x .. LEMMA 3.3.

Let g 0 g EpiŽZ r , V .. Then HomŽZ r , V . s g 0 (HomŽZ r , Z r ..

Proof. One inclusion is clear. For the other let B be a basis of Z r and g g HomŽZ r , V .. Then g Ž B . can be expressed in terms of g 0 Ž B .. Hence the result. Let us fix a basis B s Ž e1 , e2 , . . . , e r . of Z r. With any M g M r we associate the linear mapping w M of Z r , whose matrix is M in B and M ŽZ r . s w M ŽZ r .. Whenever convenient, we shall write M instead of w M . We also define GŽ M . s Z rrM ŽZ r .. It is well known that GŽ M . is in one-to-one correspondence with the SNF of M. We now define divisors: Left Di¨ isors: V1 is a left divisor of V2 iff V2 ; V1. The set of left divisors of V will be denoted by LDŽ V .. Note that LDŽ V . is naturally ordered by inclusion.

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Weak Complementary Di¨ isors: Let V1 and V2 be two submodules of Z r. We say that Ur ? V1 is a weak complementary divisor of V2 if there exists g g EpiŽ V1 , V2 .. Such a definition would make sense only if V1 s uŽ V1X . for u g Ur , and if for a morphism g of EpiŽ V1 , V2 . the map belongs to EpiŽ V1X , V2 .. The set of weak complementary divisors of V will be denoted by WCDŽ V .. Furthermore, for M g In¨ r , we let LDŽ M . be the set of left divisor classes of M. This set is naturally ordered by divisibility. We now describe the links between these three sets of ‘‘divisors.’’ To this end we define the three following arrows: Q: Ž LD Ž M . , N . ª Ž LD Ž M Ž Z r . . , > . , M1 ? Ur ¬ M1 Z r ,

˜ Ž LD Ž M Ž Z r . . , > . ª Ž LD Ž M . , N . , Q: V1 ¬ M1 ? Ur , and C : LD Ž M . ª WCD Ž M Ž Z r . . , r M1 ? Ur ¬ Ur ? My1 1 MŽZ . ,

where M1 is the matrix in B of any u g EpiŽZ r , V1 .. We show that these arrows are well defined. This is clearly so for Q and C. We note also that ˜ is well defined by LemC depends only on the HNF of M and that Q ma 3.3. ˜ s Id LD Ž M . and Q(Q ˜ ˜s We check that Q(Q s Id LDŽ M ŽZ r .. , so that Q Qy1 . We show that C is a surjection by taking an element of WCDŽ M ŽZ r .., Ur ? V2 , and one of EpiŽ V2 , M ŽZ r .. and then letting V1 s g ŽZ r . > M ŽZ r .. We then have gy1w M Z r s V2 , as required. ˜ are order-preserving. Let M1 < M2 < M. Finally, we prove that Q and Q Then M2 s M1 N for some N g In¨ r , which implies M2 ŽZ r . ; M1ŽZ r ., as required. Conversely, if M1ŽZ r . > M2 ŽZ r ., then M1 N M2 by the sheer fact ˜ associated with M2 is well defined. that the mapping Q ˜ are We have thus proved a sharper form of Theorem 1.1, i.e., Q and Q one-to-one and order-preser¨ ing and are in¨ erses of one another. Furthermore, C is a surjection. ˜ In fact, a bit more can be We will henceforth use Qy1 instead of Q. said, since C actually identifies the SNF of the complementary divisor. THEOREM 3.4. Let V1 be a left di¨ isor of V and choose V2 so that CQy1 Ž V1 . s Ur ? V2 . Then there exists g g EpiŽ V2 , V . l EpiŽZ r , V1 .. For any such choice Ž V2 , g ., we ha¨ e an exact sequence, gU

0 ª Z rrV2 ª Z rrV ª Z rrV1 ª 0.

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r Proof. With obvious notations we have Ur ? V2 s Ur ? My1 1 MZ . Thus y1 r y1 V2 s u( M1 MZ and g s M1 ( u is a point of EpiŽ V2 , V . such that g ŽZ r . s V1. Thus g U is injective and g U ŽZ rrV2 . s V1rV, which proves the exactness of the above sequence.

Theorem 3.4 tells us that V1rV is isomorphic to the SNF of the complementary divisor. COROLLARY 3.5. The number of di¨ isor classes M1 of M such that . SNFŽ M1 . s S1 and SNFŽ My1 1 M s S2 is equal to the number of subgroups G of GŽ M . such that G ( GŽ S2 . and GŽ M .rG ( GŽ S1 .. Furthermore, we have Corollary 1.2. We now consider lattices V verifying w Ž V . ; V for every w g HomŽZ r , Z r .. It is easily seen that the lattices mZ r verify this property and that they are the only ones to do so. Furthermore, given any lattice V, we have V > detŽ V .Z r Žcorresponding to the fact that any algebraic extension of Q is contained in a Galois extension.. This property extends to several lattices, so that the intersection of two lattices is yet another lattice. We shall call such lattices diagonal lattices whose mere existence is enough to prove Corollary 1.4. Proof of Corollary 1.4. For this proof we need the material presented in Section IV. Let m s g 1 q ??? qgr . The HNF’s we are looking for are the lattices L contained in p m Z r whose cotypes are n s Žgr , . . . , g 1 .. It is thus . . . , m. Ž . Ž the sum over all possible types a of g aŽ m, p cf. Section IV.. The first ,n part of the theorem follows readily. To get the precise expression, we first use the fact that gml, n s gnl, m , and then use Birkhoff’s result ŽProposition 4.5.. We note, finally, that this index is an important quantity in Hecke algebras. Following Krieg wKrx, it would already have been possible to show that the index is a polynomial in p. One obtains that the index is a homomorphism from Hr to Z, and that Hr is a polynomial algebra generated by some Tr, 0 , . . . , Tr, r for which we know that indŽTr, j . s rj p . ŽSee Remark 7.3.b, Proposition 7.2 and Theorem 8.1 of Chapter 5 as well as Corollary 4.5 of Chapter 1 of wKrx.. Obtaining the degree of the polynomial by this approach does not seem to be easy. We shall give definitions of left GCD and right LCM in terms of lattices. Given A and B in In¨ r , there exists a D l , unique up to multiplication on the right by a unimodular matrix, such that AŽZ r . q B ŽZ r . s D l ŽZ r . Žwhich is, by the way, the analogue of the definition given in wH, Theorem 9.7, Chap. 14x.. v

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Similarly, AŽZ r . l B ŽZ r . s Mr ŽZ r ., where Mr is a right common multiple Žthis intersection is yet another lattice by virtue of the remark following Corollary 3.5.. Note that we have the exact sequence v

0ªG Ž Mr . ªG Ž A . = G Ž B . ªG Ž D l . ª 0 x

¬

Ž x, yx . Ž x, y .

¬

x q y,

which finally yields Corollary 1.3; although first we need two auxiliary results on the ranks of finite abelian groups. LEMMA 3.6. If H is a subgroup of a finite abelian group G, there exists H0 , a direct factor of G, of the same rank as H, such that H ; H0 ; G. Proof. We lift the situation in Z r where the rank of G is r. We take V ; Z r such that Z rrV ( G and denote this surjection by s. Now there exists a submodule W of Z r such that V ; W ; Z r and sŽW . s H with rankŽW . s rankŽ H .. We can find a basis Ž e1 , . . . , e r . of Z r for which there exist integers a1 , . . . , a r such that W s Ý ris1 Z ? a i e i . With a suitable reordering of the e i-s we could assume that for i F t, t being the rank of H, a i / 0 and that a tq1 s ??? s a r s 0. We set W0 s Ýtis1 Z ? e i and W1 s Ý ristq1 Z ? e i , so that Z r s W1 [ W0 and W ; W0 . From this we infer that G s sŽW1 . q sŽW0 .. To prove that this sum is direct, we simply show that ŽW1 q V . l ŽW0 q V . s V. Let w 1 s w 0 q ¨ . Then ¨ s ¨ 1 q ¨ 0 . This gives w 1 y ¨ 1 s w 0 q ¨ 0 s 0. Hence w 1 s ¨ 1 g V. Now H0 s sŽW0 . satisfies our requirement. LEMMA 3.7. Let G, H, K be finite abelian groups with rankŽ G . G rankŽ H . q rankŽ K .. If there exists an exact sequence f

g

0 ª H ª G ª K ª 0, then G ( H = K. Proof. By Lemma 3.6 we can write G s H0 [ H, where H0 > f Ž H . and is of the same rank as H. With obvious notations, let Ž g Ž k 11 q k 01 ., . . . , g Ž k 1s q k 0s .. be a basis of K. But Ž k 11 , . . . , k 1s . generates H1 , for the cardinality of the subgroup of H1 generated by the above elements is the cardinality of K and hence of H1. Let H2 be generated by ² k 11 q k 01 , . . . ,k 1s q k 0s :. We check that H2 q H0 s G. To prove that the above sum is direct, we use the fact that dim Ž H2rpG . q dim Ž H0rpG . F dim Ž GrpG . , which gives H2 l H0 s  04 . Thus H2 [ H0 s G. By definition H2 ( K. Hence g Ž H0 . s 0, i.e., H0 s Ker g s f Ž H ..

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Proof of Corollary 1.3. We notice that D l is a GCD of A and B if and only if pDl is a GCD of pA and pB. The same is true for LCM. We can thus assume the ranks of both GŽ A. and GŽ B . to be equal to r. Since the ranks of GŽ D l . and GŽ Mr . are at most r, the conclusion follows. We note that other proofs of Corollary 1.3 can be found in wT1x and wN1x, and that by duality we also obtain GŽ A. = GŽ B . ( GŽ Dr . = GŽ Ml .. IV. PARTITIONS AND MATRICES Throughout this section p is a fixed prime number. We study some links between partitions and matrices and start with a description of the LRsequence of a subgroup H of an abelian p-group G. By the structure theorem of finite abelian groups Žwhich can easily be derived from the study above., G is isomorphic to a group. Zrp l1 Z = ??? = Zrp lr Z

Ž l1 G l2 G ??? G l r . .

Following wKx and wMx, we define the type of G as being the partition lŽ G . s Ž l1 , . . . , l r .. Given a subgroup H of G, we have access to its type lŽ H . and to its cotype lŽ GrH .. Further invariants are obtained by taking lŽ Grp i H . for i G 0. We now need to recall the definition of a LR-sequence. A LR-sequence of type Ž m , n ; l. is an increasing sequence Ž lŽ1., . . . , lŽ s. . of partitions with lŽ1. s m and lŽ s. s l. A third batch of properties involving n is also imposed. To explain it we build the diagram associated with lŽ1. Žcf. Section II. and write 0 over all of the squares. On top of this diagram, put the one for lŽ2., with the upper left corners of the two diagrams coinciding. Write 1 over the new squares created. Continue in this way with lŽ3., . . . , lŽ s.. We end up with a diagram representing l s lŽ s., where the squares are numbered 0 to s y 1. We assume the following properties to hold true: Ži. When a horizontal line Ži.e., a row. is read from left to right, the symbols are weakly increasing. Žii. When a vertical line Ži.e., a column. is read from top to bottom, starting after the last 0, the symbols Žif any. are strictly increasing. Žiii. For any i G 1 and any k G 1, the number of symbols in the last k columns Žstarting from the left. is not less than the number of symbols i q 1. Živ. For any i G 1, the number of symbols i in the whole diagram is n iX .

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As a useful additional property, one checks that Žv. The kth row Žstarting from the top. contains some or all of the symbols 0, 1, . . . , k. Given such a diagram, its north is the top of the page, its east is the right-hand side of the page, and so on. Here is an example of an LR-sequence of type ŽŽ3, 2, 1., Ž6, 4, 3, 1.; Ž7, 6, 6, 2..: 0 0 0 1 0 0 1 2 0 0 2 1 1 3 2 3 4 3 4 5 6

The LR-sequence itself is the sequence of partitions

Ž Ž 3, 3, 1 . , Ž 4, 4, 2, 1 . , Ž 5, 4, 3, 2 . , Ž 6, 5, 4, 2 . , Ž 6, 6, 5, 2 . , Ž 6, 6, 6, 2 . , Ž 7, 6, 6, 2 . . . We define a string of length k as being a line linking symbols k, k y 1, . . . ,1, with exactly one of these symbols, each segment of this line being oriented between north and northeast. We now have LEMMA 4.1. Gi¨ en a LR-sequence of type Ž m , n ; l., it is possible to build n 1X strings such that each symbol G 1 belongs to a string and that two distinct strings do not intersect. Proof. We prove this lemma by recursion on the highest symbol k. For k s 1, it is trivial. Let us suppose our strings to be built until the symbol k, and let us add the symbol k q 1. Take the symbol k q 1, which is the most east, say square S. Then there exists a symbol k north and east of S by property Žiii.. Such a symbol is forcibly north of S by Ži. and Žii.. We take the easternmost of such symbols. We continue in a similar fashion. Green Žcf. wGx. has shown that the sequence S Ž H . s Ž l Ž GrH . , lŽ GrpH . , . . . , lŽ Grp s H . s lŽ G . .

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for a large enough s is a LR-sequence of type Ž lŽ H ., lŽ GrH .; lŽ G ... We shall count subgroups according to their LR-sequence. It would be interesting to describe such classes. In this direction we have CONJECTURE 4.2. Let p be a prime number. Let G be a finite abelian p-group and H and K be two of its subgroups. Then there exists an automorphism s of G such that s Ž H . s K if and only if H and K have the same LR-sequence. ŽNote that in the Appendix we prove this conjecture for some special cases.. We recall the following theorem announced by P. Hall in the 1950s Žit was probably already stated by Frobenius in the beginning of the century. and proved by T. Klein in 1969 wKx. THEOREM 4.3 ŽT. Klein.. Let p be a prime number. Let G be a finite abelian p-group of type l. The number of subgroups of G ha¨ ing a gi¨ en LR-sequence and being of type m and cotype n is a monic polynomial in p with integer coefficients and of degree nŽ l. y nŽ m . y nŽ n .. Summing over all possible LR-sequences having a fixed type m and a fixed cotype n tells us that the number of subgroups of G having type m and cotype n is a polynomial in p, which we denote by gml, n Ž p . of degree nŽ l. y nŽ m . y nŽ n ., and the leading coefficient of which is cml, n , the number of LR-sequences of type m and cotype n . Summing over all possible types and cotypes and recalling Corollary 3.5, we get that the number of divisors of a matrix g In¨ r, p is a polynomial in p, a fact that is proved in a very short way in wB3x. Note: In wT3x Thompson also considers LR-sequences. However, the reader should be aware of the fact that his definition is not compatible with McDonald’s or with ours; what Thompson calls an LR-sequence is what we call the type of an LR-sequence, i.e., the triple Ž l, m , n . up to some reordering. There might be several LR-sequences having the same type. Having described LR-sequences and their relations with subgroups, we turn toward their use. The concept of strings will turn out to be helpful. We present a very simple proof of an inequality relating the invariant factors of A, B, and C s AB wT2x. We denote these factors by sr Ž M . N sry1Ž M . N ??? N s1Ž M .. PROPOSITION 4.4. lar matrices. Then

Let C s AB, where A and B are r = r integer nonsingu-

sl 1 Ž C . sl 2Ž C . ??? sl tŽ C . si1Ž A . si 2Ž A . ??? si tŽ A . s j1Ž B . s j 2Ž B . ??? s j tŽ B .

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206 whene¨ er i1 - ??? - i t ,

j1 - ??? - jt ,

l m s i m q jm y m, 1 F m F t.

Proof. It is enough to assume that the determinant of C is a power of a prime p. Let l s lŽ GŽ C .., m s lŽ GŽ B .., and n s lŽ GŽ C .. be the associated partitions. Let um s l i m qj m ym y m m . We will prove t

Ý ms1

t

n jm G

Ý

um .

ms1

The case t s 1 is obvious. For t s 2, we consider the u 1th nonzero element in column l i1 and the Ž j1 y 1. elements to the right of it in the same row. Let these j1 elements comprise what we call sr 1 Žsubrow 1.. We similarly construct sr 2 with Ž j2 y 1. elements. Let k 1 elements of sr 2 be included in the strings passing through the elements of sr 1. By definition, the elements of sr 1 are at least equal to u 1. We let the minimum value be u 1 q u1. Thus n j1 s u 1 q u1 , u1 G 0. Now there are Ž j2 y 1 y k 1 . elements of sr 2 that have not been counted in these strings and which, therefore, give at least Ž j2 y 1 y k 1 . strings of length at least u 2 . For 0 F k 1 - j1 , we already have j2 y j1 strings of length at least u 2 , and hence n j 2 G u 2 . For k 1 s j1 , we consider the column representing l i1. Since a string passing through the l i1th column into sr 2 can utilize at most Ž u 1 q u1 . y u 2 elements of the column, there is an element at least u 2 y u1 on this column, which makes up for the missing string, and we have

n j 2 G u 2 y u1 . We now use induction on t. Let n j n s un y u ny1 q u n for 1 F n F t. For n s t q 1, we use the same reasoning as above to find, in the worst case Ži.e., when k t s jt y t ., one additional string of length u tq1 y u t , given by an element of at least this value, which can be found on the column representing l i t. Thus

n j tq 1 G u tq1 y u t , which concludes the recursion. In fact, the condition satisfied by the indices in Proposition 4.4 can be generalized, as was done by Thompson wT3x. We need the concept of a row LR-sequence, to be differentiated from the Žcolumn. LR that we have used up to now. To the best of our understanding, the definition of a row LR-sequence can be drastically simplified from that of Thompson’s wT3x which we do here.

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207

DEFINITION. The sequence of increasing partitions Ž aŽ0., aŽ1., . . . , aŽ s. . is a row LR-sequence of X typeX Ž a, ˜ ˜b, ˜cX. if and only if the sequence of Ž0. Ž1. conjugate partitions Ž a , a , . . . , aŽ s. . is a column LR-sequence of type Ž aX , bX , cX ., where ˜ n denotes the conjugate partition of n written in increasing order. Ž aŽ0., aŽ1., . . . , aŽ s. . and a Note that the diagram of a row LR-sequence X X X Ž0. Ž1. Ž s. column LR-sequence Ž a , a , . . . , a . are the same; their types are mutual conjugates. EXAMPLE. The column LR-sequence of the last example is a row LR-sequence of type Ž a, b, c ., with a s Ž2, 2, 3., b s Ž1, 1, 2, 3, 3, 4., and c s Ž1, 3, 3, 3, 3, 4, 4.. Thompson’s proof for the condition of divisibility in terms of row and column LR-sequences is very long, and we believe that it can be simplified along the lines of the proof of Proposition 4.4. In 1933 G. Birkhoff had already established a partial result in the direction of Theorem 4.3, and his result Žcf. wBx. has the advantage that the involved polynomials are more explicit. His formulation is rather complicated, and we state his result in the form given in wBux. PROPOSITION 4.5 ŽG. Birkhoff.. Let G be a finite abelian p-group of type l, and let n be the type of a subgroup of G. The number of subgroups of G ha¨ ing type n is gi¨ en by X

X

X

Ł p n iq 1Ž l iy n i .

iG1

X lXi y n iq1 X X n i y n iq1

, p

where w . . . x p is the gaussian polynomial. Using Corollary 1.2 together with the above proposition, we get an explicit expression for the number of divisors of an integer matrix. In wB3x, the first named author has given another way of evaluating this function. The proof is short and thus gives a simple method of obtaining the total number of subgroups of a finite abelian group Žin fact, the number of subgroups having a given cardinality is also obtained..

V. THE STRUCTURE OF THE ALGEBRA OF ARITHMETICAL FUNCTIONS The value of an arithmetical function is invariant on matrices A,

ž

0 0

0 A

/

BHOWMIK AND RAMARE ´

208 and

ž

1 0

0 . A

/

Furthermore, this value is the same on all matrices equivalent to A. Thus we can confine our attention to nonsingular matrices. We consider the set of functions that depend only on the double cosets, i.e., Ur ª C 4 . Hˆr s  f : Ur _ In¨ rrU Alternatively, Hˆr can be seen as the set of functions In¨ r ª C that depend only on the SNF or as the set of formal infinite linear combinations of double cosets. Ž Hˆr , q, ? . is, of course, a vector space over C. The convolution product wN4x over all divisor classes H of S is given by

Ž fw g . Ž S . s

Ý f Ž H . g Ž SHy1 . , HNS

which we write as

Ž fw g . Ž S. s

Ý

f Ž S1 . g Ž S2 . a Ž S1 , S2 ; S . ,

S1 , S 2 in SNF

where the ‘‘weight’’ a Ž S1 , S2 ; S . is the number of H in HNF that divide S and whose SNF is S1 , and such that the SNF of Hy1 S is S2 . With this product Ž Hˆr , q, w, ? . is a commutative C-algebra. We retrace the definition of this product in the context of an abstract Hecke algebra Hr . Since Ž GLr ŽQ., Ur . is a Hecke pair Žsee wKrx., we simply have a product over the set Hr s  f g Hˆrrf Ž S . s 0 except in finitely many points 4 . The product is defined as follows. Let d lŽ S .

Ur s Ur SU

D Ur Hm

ms1

Ur _ Ur SU Ur disjoint with d l Ž S . s aU

and d r ŽT .

Ur T Ur s

D Ln Ur

ns1

Ur T UrrU Ur . disjoint with d r Ž T . s aU

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ALGEBRA OF MATRIX ARITHMETIC

Note that d r Ž S . s d l Ž S . because of the transposition. It is the number of HNF of SNF S and is denoted by indŽ S . ŽwKrx, p. 11.. We have Ur . ? Ž Ur T Ur . s Ž Ur SU

Ý

Ur , a Ž S, T ; R . Ur RU

Rg Ur_Ur S Ur T UrrUr

where

a Ž S, T ; R . s a  Ž m , n . rHm Ln g Ur R 4

˜ Ur L ˜ ; Ur T Ur , RL ˜y1 g Ur SU Ur , s a Ur LrU

½

5

and this last expression is exactly how we have defined a earlier, so that Hr is a subalgebra of Hˆr . The convolution product of arithmetical functions then gives a natural definition for the Hecke product on Hr . We could use this definition to prove, for example, the associativity of the Hecke product, an otherwise difficult exercise. On the other hand, this correspondence helps us to prove below that Hˆr is factorial. When p is a prime number, we define the primary component Hr, p by Ur ª C, rf Ž S . s 0 except in finitely many points 4 . Hr , p s  f : Ur _ In¨ r , prU In wKrx, it is shown that Hr, p is isomorphic as a C-algebra to Cw X 1 , . . . , X r x and that Hr is isomorphic to the tensor product of the Hr, p’s. Defining Hˆr, p in a similar way, we thus see that Hˆr, p is isomorphic as a C-algebra to the algebra of formal power series in r variables, which we denote by Cww X 1 , . . . , X r xx, from which we deduce that Hˆr, p is a local noetherian factorial ring with no zero divisors. As for Hˆr , we see that it is isomorphic to the ring of formal power series in countably many unknowns Cww X i, p , 1 F i F r, p primexx. The factoriality of this latter ring has been shown by Cashwell and Everett in 1959 Žcf. wCEx.. To prove Theorem 1.6 it is useful to introduce a norm over Hˆr . For any nonzero f in Hˆr , put 5 f 5 s sup  < M
5 f w g 5 F 5 f 5 5 g 5,

5 f 5 - 1 iff f Ž Id . s 0.

It is then a routine matter to identify the invertible elements: denoting by h the function defined by h Ž Id . s 1 and h Ž M . s 0 whenever < M < ) 1, we check that h y g is invertible if and only if 5 g 5 - 1, which is obtained in another way in wN4, Theorem 3.12x.

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VI. APPLICATION: ZETA FUNCTION OF THE CONVOLUTION PRODUCT The interpretation of divisor classes in terms of lattices has another important corollary, which is that the primary component Hr, p of Hr is isomorphic to the Hall algebra Žsee wMx.. It is interesting to realize that when studied from an algebraical point of view this algebra is called a Hecke algebra, and when studied from a combinatorial viewpoint it is called a Hall algebra, although often there is not much interaction between researchers in these two areas. Here we indicate an application of the isomorphism of algebras. If we wish to associate a zeta function with the convolution product of arithmetical functions mentioned in Section V, we realize that the zeta function is not a simple product of those of the components of the convolution. Because of the sublattices involved, there is a weight attached that is the sum of gml, n Ž p ., the Hall polynomials mentioned in Section IV. From the foregoing discussions we find that this weight is exactly the sum of coefficients of the Hecke product of Section V, a Ž S1 , S2 ; S ., where S1 is of type m , S2 is of type n , and S is of type l. More precisely, we have Z Ž r . Ž x 1w x 2 , s . s

Ý SSNF

s

x 1w x 2 Ž S .
Ý S1 , S 2 SNF

s

Ý S1 , S 2 SNF

x 1 Ž S1 . x 2 Ž S2 . < S1 < s < S2 < s x 1 Ž S1 . x 2 Ž S2 . < S1 < s < S2 < s

Ý

a Ž S1 , S2 ; S .

S SNF

Ý gml, n Ž p . .

To determine Z Ž r . Ž x 1w x 2 , s . in the two-dimensional case, we have computed the constant a Ž S1 , S2 ; S .. Thus gml, n Ž p . for m s Ž m q k, k ., n s Ž n q l, l ., and l s Ž t q m q k q l, k q l q n y t . is known, and we have

¡p Ž 1 q 1rp . pn

if t s 0, m s n, if t s 0, m / n,

p nyt Ž 1 y 1rp . pm

if 0 - t - n y m, if t s n y m / 0,

n

gml, n Ž p . s

~

¢

wBR, Proposition 1x. However, a general result of this kind is not yet accessible.

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211

The understanding of this zeta function should throw light on the Hall algebra. In particular, if we consider both x 1 and x 2 to be the constant function | Ž|Ž M . s 1 for all M ., then the zeta function associated has the easier formulation, Z

Žr.

Ž | |, s . s w

Ý

gml, n Ž p . ,
about which some information is already available Žfor example, in wBWx..

APPENDIX: ENDOMORPHISMS OF GŽ M . This appendix provides some evidence in support of our belief that for two subgroups H and K of a finite abelian p-group, there exists an automorphism s such that s Ž H . s K if and only if H and K have the same LR-sequence ŽConjecture 4.2.. A step toward understanding this situation is the study of the automorphisms of G. We define an invariant r Ž y . of an element y of G such that r Ž y . s r Ž z . if and only if there exists an automorphism f of G such that f Ž y . s z. This characterization, in turn, reveals part of the structure of G as a module over HGŽ M ., its ring of endomorphisms, and enables us to prove the above conjecture when H is an HGŽ M .-submodule of G Žin this case the proof of the conjecture reduces to showing that no two distinct HGŽ M .-submodules of G have the same LR-sequence.. Consider the set of classes of matrices T such that TMZ r s MHZ r for some H g M r , i.e., such that My1 TM g M r . If T1 and T2 are congruent modulo M and if T1 is a homomorphism, then so is T2 . Note that we have a multiplication over HGŽ M . induced by composition, so that HGŽ M . is a ring with unity. We first describe it through coordinates. Let us take M in the form diagŽ p li ., where l1 F l 2 ??? F l r . Then T s Ž t i, j . is a homomorphism if and only if t i , j ' 0 p l jy l i

Ž j - i. .

Note also that t i, j is to be taken modulo p l j. In this way we easily get the type of the abelian group HGŽ M . to be ŽminŽ l i , l j .. i, j . Note further that for any subgroup H ; G there exists a homomorphism T such that T Ž G . s H Žobtained by taking a divisor class.. Let us see a counterexample. The condition ImŽ f . s ImŽ g . does not ensure that there exists hrf s gh. For example, assume p 2 e1 s pe2 s 0 and define f by f Ž e1 . s e2 and f Ž e2 . s pe1 q e2 and g Ž e1 . s pe1 and g Ž e2 . s e2 . We verify that h does not exist.

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On applying standard results we see the following. THEOREM A.1. Let y g GŽ M .. Then HGŽ M . ? y s  z s Ý z i e ir p r i Ž y . N z i 4 . Furthermore, the type of this subgroup is Ž l k y r k Ž y .. k and its cotype is Ž r k Ž y .. k . We present another interesting counterexample: it is false to say that if y and z have the same order and the depth of y is equal to that of z Žthe depth being defined as the largest h such that ' x with y s p h x ., then there exists f g HGŽ M . such that f Ž y . s z. To see this, take p 3 e1 s p 2 e2 s pe1 s 0, y s pe1 q pe2 q e3 , and z s pe1 q e2 q pe3 . We verify that this ‘‘property’’ can be violated only if r G 3. Before going any further, we study this sequence r Ž y . s Ž r k Ž y .. k associated with y. THEOREM A.2. The sequence r Ž y . ¨ erifies l k y l kq1 G r k Ž y . y r kq 1Ž y . G 0 for 1 F k F r y 1, or equi¨ alently, the sequences r Ž y . and l y r Ž y . are decreasing. Reciprocally, for any sequence u ¨ erifying these properties there exists a y with u s r Ž y .. Note that as a corollary of this theorem, we have that r k Ž y . s r kq1Ž y . as soon as l k s l kq1. Proof. All of that follows from the definition. It is worth mentioning that r Ž y . is a fixed point for R: Ž m k . ¬

ž min ŽmaxŽ 0, l 1FiFr

k

y li . q mi . .

/

We now give another characterization of the r k Ž y .’s. LEMMA A.3. HomŽ GŽ M ., Zrp lk Z. ? y s p r k Ž y . Zrp l k Z. Proof. Since ' f g HGŽ M . with f Ž y . s p r k Ž y . e k , we get one inclusion. For the reverse assume that there exists F g HomŽ GŽ M ., Zrp lk Z. such that F Ž y . s p r k Ž y .y1. Then f Ž z . s F Ž z . e k is in HGŽ M . }hence the contradiction. Using the invariant r Ž y ., we can be even more precise. THEOREM A.4. There exists an automorphism f such that f Ž y . s z if and only if r Ž y . s r Ž z .. Proof. It is, of course, a necessary condition. To prove that it is sufficient, we shall send y by an automorphism to z s Ý p r k Ž y . e k . Let y s Ý ris1 x i e i . By using mappings of the type e k ¬ e k q t k p maxŽ0 , l hyl k . e h ,

ei ¬ ei Ž i / k .

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213

for k / h, we can change x h modulo gcdŽ x k p maxŽ0, l hyl k ., k / h.. Putting x i s p m i a i with 0 F m i F l i and Ž a i , p . s 1, we see that x i can be taken modulo p t with t s minŽ m k q maxŽ0, l h y l k ., k / h.. Now, if m h s r hŽ y ., we do not move it, else t s r hŽ y ., and we get m h F r hŽ y . Ži.e., equality. without changing the other m k ’s. Using this process repeatedly, we reach a point yX s

Ý a k p r Ž y .ek , k

Ž a k , p . s 1.

Note that we have only used transforms of determinant 1. It is then easy to conclude the proof. We say that a subgroup H ; GŽ M . is characteristic if f Ž H . ; H for any f g HGŽ M .. Such subgroups are the submodules of GŽ M . for its structure of the HGŽ M .-module. The smallest of these subgroups are the HGŽ M . ? y, and any characteristic subgroup is a sum of such subgroups. Since Ž p minŽ r 1Ž y., r 1Ž z .. e1 , . . . , p minŽ r r Ž y ., r r Ž z .. e r . is a basis of HGŽ M . ? y q HGŽ M . ? z, we see that such a sum is yet another HGŽ M . ? x, so that any characteristic subgroup H is in fact a HGŽ M . ? x, and that we can define r Ž H . to be r Ž x .. We easily prove that a characteristic subgroup is characterized by its function r Ž H .. From the above we see that in case f Ž H . s K, H is a characteristic subgroup if and only if K is one. Furthermore, f Ž H . s K implies that r Ž H . s r Ž K ., which in turn gives that the cotypes of H and K are equal ŽTheorem A.4.. Thus a necessary and sufficient condition for H and K to be equal is that r Ž H . s r Ž K ., and we have proved the conjecture for our restricted H. We are now in a position to characterize ideals of HGŽ M .. THEOREM A.5. The correspondences between ideals of HGŽ M . and characteristic subgroups of GŽ M . gi¨ en by I s  frf Ž H . s 0 4 ,

H s  xrI ? x s 0 4

are one to one and are reciprocals of one another. Proof. The set I Ž H . s  frf Ž H . s 04 is, of course, an ideal Ži.e., a left and right ideal.. Reciprocally, let I be an ideal of HGŽ M ., and let H be the intersection of the kernels of points of I. It is a characteristic subgroup since if y g H, g g HGŽ M ., and f g I, then fg Ž y . s 0, since fg g I. Thus HGŽ M . ? H ; H. Then I ; I Ž H .. Moreover, for any y f H, there exists f g I with f Ž y . / 0. Take y s p maxŽ0, r k Ž H .y1. e k . Then composing with a projector, we can find f k g I such that f k ŽÝ x i e i . s f k Ž x k e k . and f k Ž x k e k . s 0 if and only if p r k Ž H . N x k Žsince there exists an f with f Ž p maxŽ0, r k Ž H .y1. e k . / 0.. Furthermore, we can take f k, hŽÝ x i e i . s

214

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x k p maxŽ0, l hyr k Ž H .. e h . Then any f in I Ž H . is a linear combination of these f k, h ; we get the inclusion I Ž H . ; I as required. COROLLARY A.6.

Ideals of HGŽ M . are principal.

Proof. Put f Ž e k . s p lkyr k Ž H . e k and look at the ideal J s HGŽ M . ? f ? HGŽ M .. We have f Ž H . s 0 and thus hfg Ž H . s 0, which means that J ; I Ž H .. Since KerŽ f . s H, we get the reverse inclusion and hence the result.

REFERENCES wB0x G. Bhowmik, Completely multiplicative arithmetical functions of matrices and certain partition identities, J. Ind. Math. Soc. 56 Ž1991., 73]83. wB1x G. Bhowmik, Divisor functions of integer matrices: evaluations, average orders and applications, Asterisque 209 Ž1992., 169]177. ´ wB2x G. Bhowmik, Average orders of certain functions connected with arithmetic of matrices, J. Ind. Math. Soc. 59 Ž1993., 97]106. wB3x G. Bhowmik, Evaluation of the divisor function of matrices, Acta Arith. 74 Ž1996., 155]159. wBNx G. Bhowmik and V. C. Nanda, Arithmetic of matrices, manuscript. wBRx G. Bhowmik and O. Ramare, ´ Average orders of multiplicative arithmetical functions of integer matrices, Acta Arith. 66 Ž1994., 45]62. wBWx G. Bhowmik and J. Wu, Zeta functions of subgroups of abelian groups, preprint, 1997. wBx G. Birkhoff, Subgroups of abelian groups, Proc. London Math. Soc. 38Ž2. Ž1933., 385]401. wBux L. M. Butler, A unimodality result in the enumeration of subgroups of a finite abelian group, Proc. Amer. Math. Soc. 101 Ž1987., 771]775. wCEx E. D. Cashwell and C. J. Everett, The ring of number-theoretic functions, Pacific J. Math. 9 Ž1959., 975]985. ¨ wCx U. Christian, Uber teilerfremde symmetrische Matrizenpaare, J. Reine Angew. Math. 229 Ž1968., 43]49. wGx J. A. Green, ‘‘Symmetric Functions and p-Modules,’’ Lecture Notes, Manchester Univ. Press, Manchester, 1961. wHx L. K. Hua, ‘‘Introduction to Number Theory,’’ Springer-Verlag, BerlinrNew York, 1982. wKx T. Klein, The Hall polynomial, J. Algebra 12 Ž1969., 61]78. wKrx A. Krieg, ‘‘Hecke Algebras,’’ Mem. Amer. Math. Soc. 87 Ž1990.. wMx I. G. Macdonald, ‘‘Symmetric Functions and Hall Polynomials,’’ Oxford Univ. Press, Oxford, 1979. wN1x V. C. Nanda, On GCD and LCM of matrices, J. Ind. Math. Soc., to appear. wN2x V. C. Nanda, Arithmetic functions of matrices and polynomial identities, Colloq. Math. Soc. Janos ´ Bolyai 34 Ž1982., 1107]1126. wN3x V. C. Nanda, Generalizations of Ramanujan’s sum to matrices, J. Ind. Math. Soc. 48 Ž1984., 177]187. wN4x V. C. Nanda, On arithmetical functions of integral matrices, J. Ind. Math. Soc. 55 Ž1990., 175]188. wNex M. Newman, ‘‘Integral Matrices,’’ Academic Press, New YorkrLondon, 1972.

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wRSx K. G. Ramanathan and M. V. Subbarao, Some generalizations of Ramanujan’s sum, Canad. J. Math. 32 Ž1980., 1250]1260. wT1x R. C. Thompson, Left multiples and right divisors of integral matrices, Linear and Multilinear Algebra 19 Ž1986., 287]295. wT2x R. C. Thompson, Smith invariants of a product of integral matrices, in ‘‘Linear Algebra and Its Role in System Theory,’’ Brunswick, Maine, 1984, Contemp. Math 47 Ž1985., 401]435. wT3x R. C. Thompson, An inequality for invariant factors, Proc. Amer. Math. Soc. 86 Ž1982., 9]11.