Algebra of Matrix Arithmetic

Transcription

1 JOURNAL OF ALGEBRA 210, ARTICLE NO JA Algeba of Matix Aithmetic Gautami Bhowmik and Olivie Ramae Depatment of Mathematics, Uniesite Lille 1, Unite associee au CNRS, URA 751, Villeneue d Ascq Cedex, Fance Communicated by Walte Feit Received Novembe 3, 1997 We study the algeba of the aithmetic of intege matices A link is established between the diviso classes of matices and lattices The algeba of aithmetical functions of integal matices is then shown to be isomophic to an extension of the Hecke algeba, also called a Hall algeba in combinatoics The dictionay helps tanslate esults fom one setting to anothe One impotant application is the study of subgoups of a finite abelian goup 1998 Academic Pess I INTRODUCTION Although integal matices have been intensively studied see, fo instance, H, Ne, N4, T3 and some aithmetical notions like GCD and divisibility have been intoduced, the set of diviso classes of a given matix still emains unsatisfactoily undestood; in this pape we wish to fill this gap Let be the algeba of matices with coefficients in We shall pay special attention to nonsingula matices, ie, to In, the subset of consisting of those matices M fo which det M 0 The set of invetible elements U some authos use GL instead, which is the subset of consisting of matices M veifying det M 1, will featue pominently in ou discussion Although we do not do so in this pape, we could as well have woked with singula matices, the aithmetic of which would be adapted on the lines of BN We ecall that if M is in In, a left diviso class of M is an integal matix A that is a canonical epesentative of A U fo which thee exists an integal matix B such that AB M In paticula, we take A in bhowmik@gatuniv-lille1f amae@gatuniv-lille1f $2500 Copyight 1998 by Academic Pess All ights of epoduction in any fom eseved 194

2 ALGEBRA OF MATRIX ARITHMETIC 195 Hemite Nomal Fom HNF We use the notation A M to indicate divisibility The set containing left diviso classes of M is denoted by LD M and is known to be of finite cadinality The classical way to study the aithmetic of commutative stuctues is though the desciption of ideals of This appoach is not obviously adapted to noncommutative situations, and in this pape we shall deive infomation fom the action of ove To do so, we choose a basis B of and identify a matix M with the endomophism M whose matix in B is M We conside the image M, which depends only on the unimodula class of M, ie, the HNF of M and its cokenel GM M If we wished to study singula matices as well, we would take only the tosion pat of the cokenel It is known that GM is a finite abelian goup independent of the chosen basis, and its invaiant factos ae the same as that of M Notwithstanding patial effots like those of Hua H, Chap 14 o Thompson T2, T3, in the past matices have been used only as a tool in the study of finite abelian goups see, eg, B, Ne, II21a without any fomal connection having been established Hee we pove THEOREM 11 Let M In The aow : LD M subgoups of G M 4, A A M is one-to-one and ode-peseing ie, if A A then A A 1 l Futhemoe, G A M A As an immediate coollay we get COROLLARY 12 The numbe of diiso classes of M In is equal to the numbe of subgoups of G M The intepetation of diviso classes in tems of lattices and finite goups has othe applications The left GCD Dl and ight multiple M of A and B in In can be defined simply as D A B and M l A B This enables us to give anothe poof of a ecent esult of Thompson see T1 and N1 fo two othe poofs on the classical lines of Gassman s fomula COROLLARY 13 G A GB GD GM As anothe impotant application, we detemine ind S, the index of a matix S, which is the numbe of Hemite Nomal Foms H having a given Smith Nomal Fom S We ecall that S is a canonical epesentative of U H U l

3 196 BHOWMIK AND RAMARÉ COROLLARY 14 Let p be a pime numbe and,, 1 be a patition The numbe of Hemite Nomal Foms whose Smith Nomal Fom is diag p,, p 1 is gien by 1 p Ý i1 i1 i,,,, p whee is the conjugate patition of and p is the gaussian multino- mial We shall see that this index function is the classical homomophism fom an abstact Hecke algeba to It is well known that if G is a finite abelian p-goup, thee exist positive integes such that G is isomophic to Ł p i 1 2 s i The patition,, is called the type of G 1 s also sometimes called its Sege chaacteistic A subgoup H of G has a type and a cotype, which, by definition, is the type of GH Klein K has shown that thee exists a polynomial g, with intege coefficients such that the numbe of subgoups of G of type and cotype is g p, These polynomials ae called Hall polynomials The coesponding notion fo matices is the notion of invaiant factos If M In has a deteminant that is a powe of a pime p, then U M U contains a unique diagonal matix diag p,, p 1 with 1 This epesentative is called the Smith Nomal Fom of M and is denoted by SNF M As a coollay to Theoem 11, we get COROLLARY 15 Let p be a pime numbe, let M In be of detemi- nant a powe of p, and let and be patitions of length We put SNF M diag p,, p 1 Then the numbe of diisos A of M such that 1 1 SNF A diag p,, p and SNF A M diag p,, p 1 is equal to g p, In T3 Thompson established that the two quantities unde consideation ae simultaneously nonzeo In the same pape Thompson addesses the following impotant poblem: given thee finite abelian goups G, H, and K, what would be the necessay and sufficient conditions in tems of divisibility elations between the invaiant factos of these thee goups fo K to be a subgoup of G with GK H In the same pape Thompson poves some inticate inequalities necessaily veified by these invaiant factos A necessay and sufficient condition is of couse given by the nonvanishing of the coesponding Hall polynomial and in tun by the existence of a athe inticate combinatoial object called a Littlewood Richadson sequence heeafte abbeviated as LR-sequence We shall descibe LR-sequences in a way that helps us give simple poofs fo some

4 ALGEBRA OF MATRIX ARITHMETIC 197 of Thompson s esults Note that links between the aithmetic of matices and that of patitions have been shown ealie in some special situations see N2, B0, B3, Ne, and M In the final pat of this pape we intepet aithmetical functions of matices in the context of divisibility in tems of lattices A function f: In is said to be aithmetical wheneve f A depends only on the Smith Nomal Fom of A This fomalization helps us detemine the pointwise value of aithmetical functions that ae given as a convolution of two simple functions fo instance, the numbe of divisos of a given matix Pointwise evaluations have been teated in C fo the symmetic Eule- function, in N2 fo the nom, the t-noms, Eule-t functions, and Mobius function, in RS and N3 fo Ramanujan sums, and moe ecently in B3 fo the diviso function A unified account of these esults will be found in BN Since these quantities ae often difficult to evaluate, it is of inteest to have a desciption that enables us to compae them o to have an idea of thei foms With this aim we shall give fomulas fo the Diichlet seies of a convolution poduct of two functions in the context of the Hall algeba genealizing the esult of BR We shall finally use the diviso classsublattice coespondence to deal with the algeba of aithmetical functions as defined in N4 Note, howeve, that unlike in N4, we estict ou attention to integal matices with nonzeo deteminant to get a moe complete desciption We shall identify in a natual way the p-component of this algeba as the completion of the abstact Hecke algeba built ove In, p and the unimodula goup see K, whee In, p is the set of intege matices whose deteminant is a powe of p Fom this we shall deduce that this p-component is isomophic to the algeba of fomal powe seies with indeteminates ove As a futhe consequence we shall get THEOREM 16 The ing of aithmetical functions is isomophic as a -algeba to the ing of fomal powe seies in countably many unknowns oe Fom the above we infe that this ing does not have any zeo divisos and that it is factoial a popety shown by Cashwell and Eveett cf CE while studying the case 1 II NOTATIONS Let 1 be the dimension intege matices with coefficients in 4, 4 4 In M det M 0, U M det M 1

5 198 BHOWMIK AND RAMARÉ We need simila notations fo the p-pimay components when p is a pime numbe We will use, p to denote the set of matices of deteminant p,, and 4 In M det M 0, p We finally put M det M Any M In is equivalent ove to a unique diagonal matix diag a i, whee ai is a positive intege, and ai a i1, which we call its Smith Nomal fom o SNF We shall also wite SNF M a i If R is a fee -module of ank and e,,e 1 is a basis of R, then M In can be consideed as a mapping If M M R, then the cokenel GM RM is a finite abelian goup It has a unique decomposition of the fom Ł a with a a We shall call diag a i1 i i i1 i its Smith Nomal Fom o, when equied, we shall call it the SNF of M It is of couse equal to SNF M and does not depend eithe on R o on e,,e 1 Whee only p-goups ae concened, we have a p i, and we shall call i i1 the type of M o of GM o of M A patition is a sequence,,,, 0, 0, 1 2 t with i i1, Fo a patition we let i denote its conjugate patition and define n Ý i 1 i i We associate the following diagam with a patition: build a vetical line ie, a column of 1 squaes; to the ight of this column put a column of 2 squaes with its uppe end level with that of the pevious column, and so on We thus get a tiangula shape containing squaes 1 2, p III AN INTERPRETATION IN TERMS OF LATTICES In this section we intepet the divisibility of matices in tems of lattices and define the left GCD and ight LCM in this context We denote the categoy of ou objects by V, the set of all lattices submodules of ank of If V and V ae two elements of V 1 2, then the mophisms between V1 and V2 ae defined as the -linea mappings fom to of ank such that V 1 V 2 We let Ab denote the categoy of abelian tosion goups that ae fee poducts of cyclic goups equipped with the usual mophisms We conside the functo V Ab, which associates V with evey module V of V and tansfoms a mophism fom V1 to V2 into : V V 1 2 x x

6 ALGEBRA OF MATRIX ARITHMETIC 199 We have LEMMA 31 Eey mophism f: V V can be witten as f 1 2 with In and V 1 V 2 Moeoe, f is injectie if and only if V V 1 2 Poof Let e, e,,e be a basis of such that ae,,ae is a basis of V, and let,,, be a basis of such that b,,b is a basis of V 2 We can wite Ý f e m b j 1, i ij j j j1 fo some choice of m ij Let N be a lage intege paamete We modify the m into m by m ij ij ii mii bnif i N is taken lage enough, the deteminant of the map ei Ý j1mijj fo i 1, is asymptotic to N b1 b and thus is nonzeo if N N 0, say We take N N 0, and veify futhe that x f x Moeove, f 0 0, so that V 1 V 2 To descibe Hom, V we define Epi V, V as the subset of Hom V, V consisting of elements such that V V 2 We also let Iso V Epi V, V Evey element of Iso V induces an isomophism on V since its deteminant is 1 We have LEMMA 32 If Epi, V and Hom, V then thee exists Hom, such that Poof Fo any x in, x is in V Thus thee exists a unique y in such that y x LEMMA 33 Let Epi, V Then Hom, V Hom, 0 0 Poof One inclusion is clea Fo the othe let B be a basis of and Hom, V Then B can be expessed in tems of B 0 Hence the esult Let us fix a basis B e 1, e 2,,e of With any M we associate the linea mapping M of, whose matix is M in B and M M Wheneve convenient, we shall wite M instead of M We also define GM M It is well known that GM is in one-to-one coespondence with the SNF of M We now define divisos: Left Diisos: V1 is a left diviso of V2 iff V2 V 1 The set of left divisos of V will be denoted by LD V Note that LD V is natually odeed by inclusion

7 200 BHOWMIK AND RAMARÉ Weak Complementay Diisos: Let V1 and V2 be two submodules of We say that U V1 is a weak complementay diviso of V2 if thee exists Epi V, V 1 2 Such a definition would make sense only if V1 uv fo u U, and if fo a mophism of Epi V, V the map belongs to EpiV, V 1 2 The set of weak complementay divisos of V will be denoted by WCD V Futhemoe, fo M In, we let LD M be the set of left diviso classes of M This set is natually odeed by divisibility We now descibe the links between these thee sets of divisos To this end we define the thee following aows: : LD M, LD M,, and M U M, 1 1 : LD M, LD M,, V M U, 1 1 : LD M WCD M, M1 U U M 1 1 M, whee M is the matix in B of any u Epi, V 1 1 We show that these aows ae well defined This is clealy so fo and We note also that depends only on the HNF of M and that is well defined by Lemma 33 We check that Id and Id, so that LD M LD M 1 We show that is a sujection by taking an element of WCDM, U V, and one of EpiV, M 2 2 and then letting V1 1 M We then have M V 2, as equied Finally, we pove that and ae ode-peseving Let M M 1 2 M Then M MNfo some N In, which implies M M ,as equied Convesely, if M M 1 2, then M1 M2 by the shee fact that the mapping associated with M2 is well defined We have thus poved a shape fom of Theoem 11, ie, and ae one-to-one and ode-peseing and ae ineses of one anothe Futhemoe, is a sujection 1 We will hencefoth use instead of In fact, a bit moe can be said, since actually identifies the SNF of the complementay diviso THEOREM 34 Let V1 be a left diiso of V and choose V2 so that 1 V U V Then thee exists Epi V, V Epi, V Fo any such choice V,, we hae an exact sequence, V V V 0

8 ALGEBRA OF MATRIX ARITHMETIC 201 Poof With obvious notations we have U V2 U M 1 1 M Thus 1 1 V u M M and M u is a point of Epi V, V such that V Thus is injective and V 1 2 V1V, which poves the exactness of the above sequence Theoem 34 tells us that V1V complementay diviso is isomophic to the SNF of the COROLLARY 35 The numbe of diiso classes M1 of M such that 1 SNF M S and SNF M M S2 is equal to the numbe of subgoups of G M such that G S and G M GS 2 1 Futhemoe, we have Coollay 12 We now conside lattices V veifying V V fo evey Hom, It is easily seen that the lattices m veify this popety and that they ae the only ones to do so Futhemoe, given any lattice V, we have V det V coesponding to the fact that any algebaic extension of is contained in a Galois extension This popety extends to seveal lattices, so that the intesection of two lattices is yet anothe lattice We shall call such lattices diagonal lattices whose mee existence is enough to pove Coollay 14 Poof of Coollay 14 Fo this poof we need the mateial pesented in Section IV Let m 1 The HNF s we ae looking fo ae the m lattices L contained in p whose cotypes ae,, 1 It is thus m,,m the sum ove all possible types of g p cf Section IV, The fist pat of the theoem follows eadily To get the pecise expession, we fist use the fact that g g, and then use Bikhoff s esult,, Poposition 45 We note, finally, that this index is an impotant quantity in Hecke algebas Following Kieg K, it would aleady have been possible to show that the index is a polynomial in p One obtains that the index is a homomophism fom H to, and that H is a polynomial algeba geneated by some T,0,,T, fo which we know that ind T, j See Remak 73b, Poposition 72 and Theoem 81 j p of Chapte 5 as well as Coollay 45 of Chapte 1 of K Obtaining the degee of the polynomial by this appoach does not seem to be easy We shall give definitions of left GCD and ight LCM in tems of lattices Given A and B in In, thee exists a D l, unique up to multiplica- tion on the ight by a unimodula matix, such that A B D l which is, by the way, the analogue of the definition given in H, Theoem 97, Chap 14

9 202 BHOWMIK AND RAMARÉ Similaly, A B M, whee M is a ight common multiple this intesection is yet anothe lattice by vitue of the emak following Coollay 35 Note that we have the exact sequence 0G M G A G B G Dl 0 x x, x x, y x y, which finally yields Coollay 13; although fist we need two auxiliay esults on the anks of finite abelian goups LEMMA 36 If H is a subgoup of a finite abelian goup G, thee exists H 0, a diect facto of G, of the same ank as H, such that H H0 G Poof We lift the situation in whee the ank of G is We take V such that V G and denote this sujection by s Now thee exists a submodule W of such that V W and sw H with ank W ank H We can find a basis e 1,,e of fo which thee exist integes a 1,,a such that W Ý i1 ae i i With a suitable e- odeing of the e -s we could assume that fo i t, t being the ank of H, i a 0 and that a a 0 We set W Ý t e and W i t1 0 i1 i 1 Ý it1 e i, so that W1 W0 and W W 0 Fom this we infe that G sw sw To pove that this sum is diect, we simply show that 1 0 W V W V 1 0 V Let w1 w0 Then 1 0 This gives w w 0 Hence w V Now H sw satisfies ou equiement LEMMA 37 Let G, H, K be finite abelian goups with ank G ank H ank K If thee exists an exact sequence f 0 H G K 0, then G H K Poof By Lemma 36 we can wite G H H, whee H f H and is of the same ank as H With obvious notations, let g k k 1 1 0, s s 1,g k k be a basis of K But k,,k s geneates H 1, fo the cadinality of the subgoup of H1 geneated by the above elements is the cadinality of K and hence of H ² 1 Let H2 be geneated by k1 1 k 0 1, s,k k s : 1 0 We check that H2 H0 G To pove that the above sum is diect, we use the fact that dim H2pG dim H0pG dim GpG, which gives H H Thus H2 H0 G By definition H2 K Hence g H 0, ie, H Ke g f H 0 0 g

10 ALGEBRA OF MATRIX ARITHMETIC 203 Poof of Coollay 13 We notice that Dl is a GCD of A and B if and only if pdl is a GCD of pa and pb The same is tue fo LCM We can thus assume the anks of both G A and GB to be equal to Since the anks of GD and GM ae at most, the conclusion follows l We note that othe poofs of Coollay 13 can be found in T1 and N1, and that by duality we also obtain G A GB GD GM l IV PARTITIONS AND MATRICES Thoughout this section p is a fixed pime numbe We study some links between patitions and matices and stat with a desciption of the LRsequence of a subgoup H of an abelian p-goup G By the stuctue theoem of finite abelian goups which can easily be deived fom the study above, G is isomophic to a goup p 1 p 1 2 Following K and M, we define the type of G as being the patition G,, 1 Given a subgoup H of G, we have access to its type H and to its cotype GH Futhe invaiants ae obtained by taking i Gp H fo i 0 We now need to ecall the definition of a LR-sequence A LR-sequence of type, ; is an inceasing sequence 1,, s of patitions with 1 and s A thid batch of popeties involving is also imposed To explain it we build the diagam associated with 1 cf Section II and wite 0 ove all of the squaes On top of this diagam, put the one fo 2, with the uppe left cones of the two diagams coinciding Wite 1 ove the new squaes ceated Continue in this way with 3,, s We end up with a diagam epesenting s, whee the squaes ae numbeed 0 to s 1 We assume the following popeties to hold tue: i When a hoizontal line ie, a ow is ead fom left to ight, the symbols ae weakly inceasing ii When a vetical line ie, a column is ead fom top to bottom, stating afte the last 0, the symbols if any ae stictly inceasing iii Fo any i 1 and any k 1, the numbe of symbols in the last k columns stating fom the left is not less than the numbe of symbols i 1 iv Fo any i 1, the numbe of symbols i in the whole diagam is i

11 204 BHOWMIK AND RAMARÉ As a useful additional popety, one checks that v The kth ow stating fom the top contains some o all of the symbols 0, 1,, k Given such a diagam, its noth is the top of the page, its east is the ight-hand side of the page, and so on Hee is an example of an LR-sequence of type 3, 2, 1, 6, 4, 3, 1 ; 7, 6, 6, 2 : The LR-sequence itself is the sequence of patitions 3,3,1, 4,4,2,1, 5,4,3,2, 6,5,4,2, 6,6,5,2, 6,6,6,2, 7,6,6,2 We define a sting of length k as being a line linking symbols k, k 1,,1, with exactly one of these symbols, each segment of this line being oiented between noth and notheast We now have LEMMA 41 GienaLR-sequence of type, ;, it is possible to build 1 stings such that each symbol 1 belongs to a sting and that two distinct stings do not intesect Poof We pove this lemma by ecusion on the highest symbol k Fo k 1, it is tivial Let us suppose ou stings to be built until the symbol k, and let us add the symbol k 1 Take the symbol k 1, which is the most east, say squae S Then thee exists a symbol k noth and east of S by popety iii Such a symbol is focibly noth of S by i and ii We take the eastenmost of such symbols We continue in a simila fashion Geen cf G has shown that the sequence S H GH, GpH,, Gp s H G

12 ALGEBRA OF MATRIX ARITHMETIC 205 fo a lage enough s is a LR-sequence of type H, GH ; G We shall count subgoups accoding to thei LR-sequence It would be inteesting to descibe such classes In this diection we have CONJECTURE 42 Let p be a pime numbe Let G be a finite abelian p-goup and H and K be two of its subgoups Then thee exists an automophism of G such that H K if and only if H and K have the same LR-sequence Note that in the Appendix we pove this conjectue fo some special cases We ecall the following theoem announced by P Hall in the 1950s it was pobably aleady stated by Fobenius in the beginning of the centuy and poved by T Klein in 1969 K THEOREM 43 T Klein Let p be a pime numbe Let G be a finite abelian p-goup of type The numbe of subgoups of G haing a gien LR-sequence and being of type and cotype is a monic polynomial in p with intege coefficients and of degee n n n Summing ove all possible LR-sequences having a fixed type and a fixed cotype tells us that the numbe of subgoups of G having type and cotype is a polynomial in p, which we denote by g p, of degee n n n, and the leading coefficient of which is c,, the numbe of LR-sequences of type and cotype Summing ove all possible types and cotypes and ecalling Coollay 35, we get that the numbe of divisos of a matix In, p is a polynomial in p, a fact that is poved in a vey shot way in B3 Note: In T3 Thompson also consides LR-sequences Howeve, the eade should be awae of the fact that his definition is not compatible with McDonald s o with ous; what Thompson calls an LR-sequence is what we call the type of an LR-sequence, ie, the tiple,, up to some eodeing Thee might be seveal LR-sequences having the same type Having descibed LR-sequences and thei elations with subgoups, we tun towad thei use The concept of stings will tun out to be helpful We pesent a vey simple poof of an inequality elating the invaiant factos of A, B, and C AB T2 We denote these factos by s M s M s M 1 1 PROPOSITION 44 Let C AB, whee A and B ae intege nonsingula matices Then s C s C s C s A s A s A s B s B s B l1 l2 lt i1 i2 it j1 j2 jt

13 206 BHOWMIK AND RAMARÉ whenee i i, j j, l i j m, 1 m t 1 t 1 t m m m Poof It is enough to assume that the deteminant of C is a powe of a pime p Let GC, GB, and GC be the associated patitions Let We will pove m imjmm m t Ý m1 jm t Ý m1 The case t 1 is obvious Fo t 2, we conside the 1th nonzeo element in column and the j 1 i 1 elements to the ight of it in the 1 same ow Let these j elements compise what we call s subow We similaly constuct s with j 1 elements 2 2 Let k elements of s be included in the stings passing though the 1 2 elements of s By definition, the elements of s ae at least equal to We let the minimum value be 1 u 1 Thus j 1 u 1, u1 0 Now 1 thee ae j 1 k elements of s that have not been counted in these stings and which, theefoe, give at least j 1 k 2 1 stings of length at least 2 Fo 0 k1 j 1, we aleady have j2 j1 stings of length at least 2, and hence j 2 2 Fo k1 j 1, we conside the column epesenting i 1 Since a sting passing though the th column into s can utilize at most u i elements of the column, thee is an element at least 2 u1 on this column, which makes up fo the missing sting, and we have u j We now use induction on t Let j n un1 un fo 1 n t Fo n n t 1, we use the same easoning as above to find, in the wost case ie, when k j t t t, one additional sting of length t1 u t, given by an element of at least this value, which can be found on the column epesenting i t Thus j t1 u t, t 1 which concludes the ecusion In fact, the condition satisfied by the indices in Poposition 44 can be genealized, as was done by Thompson T3 We need the concept of a ow LR-sequence, to be diffeentiated fom the column LR that we have used up to now To the best of ou undestanding, the definition of a ow LR-sequence can be dastically simplified fom that of Thompson s T3 which we do hee m

14 ALGEBRA OF MATRIX ARITHMETIC 207 DEFINITION The sequence of inceasing patitions a 0, a 1,, a s is a ow LR-sequence of type a, b, c if and only if the sequence of 0 1 s conjugate patitions a, a,, a is a column LR-sequence of type a, b, c, whee n denotes the conjugate patition of n witten in inceas- ing ode Note that the diagam of a ow LR-sequence a 0, a 1,,a s and a 0 1 s column LR-sequence a, a,, a ae the same; thei types ae mutual conjugates EXAMPLE The column LR-sequence of the last example is a ow LR-sequence of type a, b, c, with a 2, 2, 3, b 1, 1, 2, 3, 3, 4, and c 1, 3, 3, 3, 3, 4, 4 Thompson s poof fo the condition of divisibility in tems of ow and column LR-sequences is vey long, and we believe that it can be simplified along the lines of the poof of Poposition 44 In 1933 G Bikhoff had aleady established a patial esult in the diection of Theoem 43, and his esult cf B has the advantage that the involved polynomials ae moe explicit His fomulation is athe complicated, and we state his esult in the fom given in Bu PROPOSITION 45 G Bikhoff Let G be a finite abelian p-goup of type, and let be the type of a subgoup of G The numbe of subgoups of G haing type is gien by Ł i1 i i i1 i1 p i1 i i, whee is the gaussian polynomial p Using Coollay 12 togethe with the above poposition, we get an explicit expession fo the numbe of divisos of an intege matix In B3, the fist named autho has given anothe way of evaluating this function The poof is shot and thus gives a simple method of obtaining the total numbe of subgoups of a finite abelian goup in fact, the numbe of subgoups having a given cadinality is also obtained p V THE STRUCTURE OF THE ALGEBRA OF ARITHMETICAL FUNCTIONS The value of an aithmetical function is invaiant on matices A, 0 0 ž 0 A/

15 208 BHOWMIK AND RAMARÉ and 1 0 ž / 0 A Futhemoe, this value is the same on all matices equivalent to A Thus we can confine ou attention to nonsingula matices We conside the set of functions that depend only on the double cosets, ie, Hˆ f : U In U 4 Altenatively, Hˆ can be seen as the set of functions In that depend only on the SNF o as the set of fomal infinite linea combinations of double cosets H ˆ,, is, of couse, a vecto space ove The convolution poduct N4 ove all diviso classes H of S is given by which we wite as Ý 1 f g S f H g SH, HS Ý S 1, S2 in SNF f g S f S g S S, S ; S, whee the weight S, S ; S 1 2 is the numbe of H in HNF that divide S and whose SNF is S 1, and such that the SNF of H 1 S is S 2 With this poduct H ˆ,,, is a commutative -algeba We etace the definition of this poduct in the context of an abstact Hecke algeba H Since GL, U is a Hecke pai see K, we simply have a poduct ove the set 4 H f Hˆ f S 0 except in finitely many points The poduct is defined as follows Let and dl S U SU U H disjoint with d S U U SU l 1 d T U T U L U disjoint with d T U T U U 1

16 ALGEBRA OF MATRIX ARITHMETIC 209 Note that d S d S l because of the tansposition It is the numbe of HNF of SNF S and is denoted by ind S K, p 11 We have U SU U T U S, T ; R U RU, Ý RUUSU T UU whee 4 S, T ; R, H L U R ½ 5 U LU L U T U, RL 1 U SU, and this last expession is exactly how we have defined ealie, so that H is a subalgeba of H ˆ The convolution poduct of aithmetical functions then gives a natual definition fo the Hecke poduct on H We could use this definition to pove, fo example, the associativity of the Hecke poduct, an othewise difficult execise On the othe hand, this coespondence helps us to pove below that Hˆ is factoial When p is a pime numbe, we define the pimay component H by 4 H f : U In U, f S 0 except in finitely many points, p, p In K, it is shown that H is isomophic as a -algeba to X,, X, p 1 and that H is isomophic to the tenso poduct of the H s Defining Hˆ, p, p in a simila way, we thus see that Hˆ, p is isomophic as a -algeba to the algeba of fomal powe seies in vaiables, which we denote by X,, X, fom which we deduce that Hˆ 1, p is a local noetheian factoial ing with no zeo divisos As fo H ˆ, we see that it is isomophic to the ing of fomal powe seies in countably many unknowns X i, p, 1 i, p pime The factoiality of this latte ing has been shown by Cashwell and Eveett in 1959 cf CE To pove Theoem 16 it is useful to intoduce a nom ove H ˆ Fo any nonzeo f in H ˆ, put 4 f sup M 1, fo M In, f M 0 and extend it by 0 0 We then veify that f g max f, g, f g f g, f 1 iff f Id 0 It is then a outine matte to identify the invetible elements: denoting by the function defined by Id 1 and M 0 wheneve M 1, we check that g is invetible if and only if g 1, which is obtained in anothe way in N4, Theoem 312, p

17 210 BHOWMIK AND RAMARÉ VI APPLICATION: ZETA FUNCTION OF THE CONVOLUTION PRODUCT The intepetation of diviso classes in tems of lattices has anothe impotant coollay, which is that the pimay component H, p of H is isomophic to the Hall algeba see M It is inteesting to ealize that when studied fom an algebaical point of view this algeba is called a Hecke algeba, and when studied fom a combinatoial viewpoint it is called a Hall algeba, although often thee is not much inteaction between eseaches in these two aeas Hee we indicate an application of the isomophism of algebas If we wish to associate a zeta function with the convolution poduct of aithmetical functions mentioned in Section V, we ealize that the zeta function is not a simple poduct of those of the components of the convolution Because of the sublattices involved, thee is a weight attached that is the sum of g p,, the Hall polynomials mentioned in Section IV Fom the foegoing discussions we find that this weight is exactly the sum of coefficients of the Hecke poduct of Section V, S, S ; S 1 2, whee S1 is of type, S is of type, and S is of type Moe pecisely, we have Z S 1 2, s Ý s S SSNF Ý 1 S1 2 S2 S S Ý S, S ; S s s 1 2 S 1, S2 SNF 1 2 SSNF 1 S1 2 S2 Ý s s Ýg, p S S S 1, S2 SNF 1 2 To detemine Z, s 1 2 in the two-dimensional case, we have computed the constant S, S ; S Thus g p fo m k, k 1 2,, n l, l, and t m k l, k l n t is known, and we have g, p n p 1 1p if t 0, m n, n p if t 0, m n, nt p 1 1p if 0 t n m, p if t n m 0, m BR, Poposition 1 Howeve, a geneal esult of this kind is not yet accessible

18 ALGEBRA OF MATRIX ARITHMETIC 211 The undestanding of this zeta function should thow light on the Hall algeba In paticula, if we conside both 1 and 2 to be the constant function M 1 fo all M, then the zeta function associated has the easie fomulation, g, p Z, s Ý s, S about which some infomation is aleady available fo example, in BW APPENDIX: ENDOMORPHISMS OF GM This appendix povides some evidence in suppot of ou belief that fo two subgoups H and K of a finite abelian p-goup, thee exists an automophism such that H K if and only if H and K have the same LR-sequence Conjectue 42 A step towad undestanding this situation is the study of the automophisms of G We define an invaiant y of an element y of G such that y z if and only if thee exists an automophism f of G such that f y z This chaacteization, in tun, eveals pat of the stuctue of G as a module ove HG M, its ing of endomophisms, and enables us to pove the above conjectue when H is an HG M -submodule of G in this case the poof of the conjectue educes to showing that no two distinct HG M -submodules of G have the same LR-sequence Conside the set of classes of matices T such that TM MH fo some H, ie, such that M 1 TM IfT1 and T2 ae conguent modulo M and if T1 is a homomophism, then so is T 2 Note that we have a multiplication ove HG M induced by composition, so that HG M is a ing with unity We fist descibe it though coodinates Let us take M in the fom diag p i, whee Then T t 1 2 i, j is a homomophism if and only if j t 0 p i i, j j i Note also that t is to be taken modulo p j i, j In this way we easily get the type of the abelian goup HG M to be min, i j i, j Note futhe that fo any subgoup H G thee exists a homomophism T such that TG H obtained by taking a diviso class Let us see a counteexample The condition Im f Im g does not ensue that thee exists hf gh Fo example, assume p 2 e1 pe2 0 and define f by f e e and f e pe e and g e pe1 and g e e We veify that h does not exist 2 2

19 212 BHOWMIK AND RAMARÉ On applying standad esults we see the following THEOREM A1 Let y GM Then HG M y z Ý ze i i i y p z 4 Futhemoe, the type of this subgoup is y i k k k and its cotype is y k k We pesent anothe inteesting counteexample: it is false to say that if y and z have the same ode and the depth of y is equal to that of z the depth being defined as the lagest h such that x with y p h x, then 3 2 thee exists f HG M such that f y z To see this, take pe1pe2 pe1 0, y pe1 pe2 e 3, and z pe1 e2 pe 3 We veify that this popety can be violated only if 3 Befoe going any futhe, we study this sequence y y k k asso- ciated with y THEOREM A2 The sequence y eifies y k k1 k y 0 fo 1 k 1, o equialently, the sequences y k1 and y ae deceasing Recipocally, fo any sequence eifying these popeties thee exists a y with y Note that as a coollay of this theoem, we have that y y k k1 as soon as k k1 Poof All of that follows fom the definition It is woth mentioning that y is a fixed point fo ž / R: min max 0, k k i i 1i We now give anothe chaacteization of the y k s k k y LEMMA A3 Hom GM, p y p p k k y Poof Since f HG M with f y p e k, we get one inclusion Fo the evese assume that thee exists F Hom GM, p k such k y1 that F y p Then f z F z e is in HG M k hence the con- tadiction Using the invaiant y, we can be even moe pecise THEOREM A4 Thee exists an automophism f such that f y z if and only if y z Poof It is, of couse, a necessay condition To pove that it is sufficient, we shall send y by an automophism to z Ý p k y e k Let y Ý xe By using mappings of the type i1 i i e e t p max0, h k e, e e i k k k k h i i

20 ALGEBRA OF MATRIX ARITHMETIC 213 max0, h k fo k h, we can change x modulo gcd x p, k h h k Putting x p i with 0 and, p i i i i i 1, we see that xi can be taken t modulo p with t min max 0,, k h k h k Now, if h y, we do not move it, else t y, and we get y h h h h ie, equality without changing the othe k s Using this pocess epeatedly, we each a point y p k Ý y k e k, k, p 1 Note that we have only used tansfoms of deteminant 1 It is then easy to conclude the poof We say that a subgoup H GM is chaacteistic if f H H fo any f HG M Such subgoups ae the submodules of GM fo its stuctue of the HG M -module The smallest of these subgoups ae the HG M y, and any chaacteistic subgoup is a sum of such subgoups min 1 y, 1 z min y, z Since p e,, p e is a basis of HG M 1 y HG M z, we see that such a sum is yet anothe HG M x, so that any chaacteistic subgoup H is in fact a HG M x, and that we can define H to be x We easily pove that a chaacteistic subgoup is chaacteized by its function H Fom the above we see that in case f H K, H is a chaacteistic subgoup if and only if K is one Futhemoe, f H K implies that H K, which in tun gives that the cotypes of H and K ae equal Theoem A4 Thus a necessay and sufficient condition fo H and K to be equal is that H K, and we have poved the conjectue fo ou esticted H We ae now in a position to chaacteize ideals of HG M THEOREM A5 The coespondences between ideals of HG M and chaacteistic subgoups of G M gien by I ff H 04, H xi x 04 ae one to one and ae ecipocals of one anothe Poof The set I H ff H 04 is, of couse, an ideal ie, a left and ight ideal Recipocally, let I be an ideal of HG M, and let H be the intesection of the kenels of points of I It is a chaacteistic subgoup since if y H, g HG M, and f I, then fg y 0, since fg I Thus HG M H H Then I I H Moeove, fo any y H, max0, k H 1 thee exists f I with f y 0 Take y p e k Then compos- ing with a pojecto, we can find f I such that f Ý xe f x e k k i i k k k and k H f x e 0 if and only if p x k k k k since thee exists an f with max0, k H 1 f p e 0 Futhemoe, we can take f Ý xe k k, h i i

21 214 BHOWMIK AND RAMARÉ max0, h k H x p e Then any f in I H k h is a linea combination of these f ; we get the inclusion I H I as equied k, h COROLLARY A6 Ideals of HG M ae pincipal k k H Poof Put f e p e and look at the ideal J HG M k k f HG M We have f H 0 and thus hfg H 0, which means that J I H Since Ke f H, we get the evese inclusion and hence the esult REFERENCES B0 G Bhowmik, Completely multiplicative aithmetical functions of matices and cetain patition identities, J Ind Math Soc , 7383 B1 G Bhowmik, Diviso functions of intege matices: evaluations, aveage odes and applications, Asteisque , B2 G Bhowmik, Aveage odes of cetain functions connected with aithmetic of matices, J Ind Math Soc , B3 G Bhowmik, Evaluation of the diviso function of matices, Acta Aith , BN G Bhowmik and V C Nanda, Aithmetic of matices, manuscipt BR G Bhowmik and O Ramae, Aveage odes of multiplicative aithmetical functions of intege matices, Acta Aith , 4562 BW G Bhowmik and J Wu, Zeta functions of subgoups of abelian goups, pepint, 1997 B G Bikhoff, Subgoups of abelian goups, Poc London Math Soc , Bu L M Butle, A unimodality esult in the enumeation of subgoups of a finite abelian goup, Poc Ame Math Soc , CE E D Cashwell and C J Eveett, The ing of numbe-theoetic functions, Pacific J Math , C U Chistian, Ube teilefemde symmetische Matizenpaae, J Reine Angew Math , 4349 G J A Geen, Symmetic Functions and p-modules, Lectue Notes, Mancheste Univ Pess, Mancheste, 1961 H L K Hua, Intoduction to Numbe Theoy, Spinge-Velag, BelinNew Yok, 1982 K T Klein, The Hall polynomial, J Algeba , 6178 K A Kieg, Hecke Algebas, Mem Ame Math Soc M I G Macdonald, Symmetic Functions and Hall Polynomials, Oxfod Univ Pess, Oxfod, 1979 N1 V C Nanda, On GCD and LCM of matices, J Ind Math Soc, to appea N2 V C Nanda, Aithmetic functions of matices and polynomial identities, Colloq Math Soc Janos Bolyai , N3 V C Nanda, Genealizations of Ramanujan s sum to matices, J Ind Math Soc , N4 V C Nanda, On aithmetical functions of integal matices, J Ind Math Soc , Ne M Newman, Integal Matices, Academic Pess, New YokLondon, 1972

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