Multi-indicial symmetric functions

ICMP A-MP A/revf/2 009/1 6 NITheP-09-10 Multi-indicial symmetric functions Joseph Ben Geloun a,b,c, ∗ and Mahouton Norb er t Hounkonnou b, † a National Institute for The or etic al Physics (NITheP ) Private Ba g X1, Matieland 7602, South Af ric a b International Ch air of Math ematic al Physics and Applic ations ICMP A–UN ESCO Chair, 0 72 B.P. 5 0 Cotonou, R epublic of Benin c D´ ep artement de Math ´ ematiques et Informatique F acult ´ e des Scie nc es et T e c hniqu e s, Universit ´ e Cheikh A nta Diop, Sene gal E-mails: ∗ b engeloun@su n .ac.za, † norb ert − hounko nnou@cimpa.net Abstract In this pap er, using the theory of categ ory , w e generalize known proper ties of symmetric po lynomials and functions and c har acterize the m ulti-indicial symmetric functions. Examples hav e been given on Sch ur functions. No v em b er 21, 2018 Keyw ord s: Symmetric p olynomials, symmetric fu nctions, Sc hur fun ctions. MCSs: 05E05, 13B25. 1 In tro d uction A great deal of atten tion has b een paid to the s ymmetric f u nctions and orthogonal p olynomials ([1, 2, 3] and r eferences therein). Indeed, symmetry is an inescapable f eature of most physical phenomena. F ollo wing [1], the theory of symmetric functions is one of the most classical parts of algebra, going b ac k to the 16 th and 17 th cen turies and attempts of mathematicians of that ep o c h to solve p olynomial equations of degree higher th an t wo. Generalizat ion of symmetric functions in sev eral sets of v ariables (the so c alled multisymmetric functions) wa s found b y McMahon in the b eginning of the past cen tury [4]. Still recen tly , McMahon symmetric p olynomials ha ve b een studied in d ifferent con texts [5]-[8]. F or instance in [5], the McMahon symm etric p olynomials in tw o sets of v ariables ha ve b een used to fin d explicit form u las and to p ro ve P -recursiv en ess for some ob jects suc h as Latin rectangles and 0 − 1 matrice s with zeros on the diag onal and giv en r ow and column sums. Th ereafter, using the approac h by McDonald [1], Dalb ec extended the t heory o f m ultisymmetric functions in t w o sets of v ariables to t he m ultihomogeneous ca se, the so called fa ctorizable forms, in c haracteristic 0 field and p r o vided with a MAPLE c o de for generating such ob jects [6]. V accarino [7 ] generalized the abov e results as w ell as those of [8] (dealing with c haracteristic 2 fields) to the r in g of multisymmetric fu nctions ov er a comm u tative ring. Among the v arious families of symmetric functions, the most significan t are und oubtedly the Sc hur functions, b ecause of their in timate relationship with the irreducible c h aracters of b oth the symmetric group and the general linear groups, and f or their combinatoria l applications. In this pap er, the McDonald f ormalism has b een extended u s ing the theory of category , in order to define multi-indicial symmetric functions including differen t sets of v ariables with sev er al tensorial indices. More sp ecifically , this pap er addresses results on t w o r emark able classes of symmetric functions with mixed t yp es of tensor indices and in tro duces their f ull c haracterizatio n . Illustration has b een giv en on Sc hur f unctions. In Section 2, w e give a generalizatio n of kno wn prop erties of the r ing of symmetric poly- nomials. The ring o f symm etric functions Λ whic h is an in v er s e limit i s defined as a universal ob ject. In Section 3, we deal with the stu d y of multi- indicial sym m etric p olynomials. Relev an t prop erties of the graded rings of suc h p olynomials are deriv ed. The multi-indicial symmetric functions are logically introdu ced. Section 4 is devot ed to the d efi nition of m ulti-indicial par- tition and the corresp onding definition of the Sc hur fu nction. W e end th e p ap er with some concluding remarks. 2 Symmetric p olynomials: main results In this section, we bu ild the theo retical framew ork of our study . F or that, we recall main prop erties of the ring of symmetric polynomials and giv e their generalizati on. The ring of symmetric functions is d efined as a un iv ersal ob ject. Let us in tro d uce the defi nition [1]: Definition 1. L et x 1 , x 2 , . . . , x n b e n indep endent indeterminates, S n b e the symmetric g r oup of p ermutations of a set with n elements acting on the p olynomial ring Z [ x 1 , x 2 , . . . , x n ] by p ermuting t he indeterminates, i.e: ∀ P = a I x I ∈ Z [ x 1 , x 2 , . . . , x n ] , (1) ∀ σ ∈ S n , σ P = σ a I x I = a I x I σ ( · ) , 1 wher e x = x 1 x 2 . . . x n , a I ∈ Z . I = ( i 1 , i 2 , . . . , i k ) , with 0 ≤ i k and 1 ≤ k ≤ n , denotes the usual multi- index nota tion (the implicit summation is use d). Then, Λ n := Z [ x 1 , x 2 , . . . , x n ] S n is th e subring of Z [ x 1 , x 2 , . . . , x n ] of symmetr ic p olynomials obtaine d b y p ermuting the x i . Remark 1. L et us p ay attention to the fact that this sum is glob al ly i nvariant under any p e r- mutation, inste ad of the monomial terms taken sep ar ately. F or example, x J may not b e e qual to x σJ . Example 1. A ssu me n=4, i.e the set of indeterminates is { x 1 , x 2 , x 3 , x 4 } . The fol lowing p oly- nomials b elong to Λ 4 : f 1 = x 1 + x 2 + x 3 + x 4 , ∀ r ∈ N , f r = x r 1 + x r 2 + x r 3 + x r 4 , f = x 1 x 2 + x 1 x 3 + x 1 x 4 + x 2 x 3 + x 2 x 4 + x 3 x 4 . If f ∈ Λ n , one can write f = P r ≥ 0 f r , where f r is t he homogeneous comp onen t of f of degree r . One can v erify that e ach of th e f r is i tself in v arian t under S n and hence, Λ n is a graded rin g. T h is statemen t can b e written as: Λ n = L r ≥ 0 Λ r n , where Λ r n is the ad d itiv e group of h omogeneous symm etric p olynomials in { x 1 , x 2 , . . . , x n } , pro vided t he fol lo wing con ve n tion: 0 is h omogeneous of any degree. On e r equires also that a p olynomial of degree 0 is nothing b u t an element of the co efficien t ring, i.e Λ 0 n = Z . Adding a new in determinate x n +1 , we can realize the ring Λ n +1 = Z [ x 1 , x 2 , . . . , x n , x n +1 ] S n +1 and the follo win g statemen t holds [1]. Lemma 1. L et π n +1 b e the mapping fr om Λ n +1 to Λ n define d by setting x n +1 = 0 . The mapping π n +1 is a surje ctive homomorph ism of gr ade d rings, i.e π n +1 : Λ n +1 → Λ n , ∀ r ∈ N , π n +1 r := π n +1 | Λ r n +1 : Λ r n +1 → Λ r n . The mapping π r n +1 is surje ctive ∀ r ≥ 0 a nd an isomor phism if and only if r ≤ n . This Lemma can b e generalized as follo ws. Corollary 1. L et n b e a nonne g ative inte ger. F or any p ∈ N , p 6 = 0 , the mapping Π n + p : Λ n + p → Λ n , define d by setting x n +1 = 0 , x n +2 = 0 , . . . , x n + p = 0 , is a surje ctive homo morphism of gr ade d rings. F urtherm or e, th e r estriction Π n + p | Λ r n + p := Π r n + p : Λ r n + p → Λ r n (2) is surje ctive for al l r ≥ 0 , and an isom orphism if and only if r ≤ n . In the f ollo wing, the notation A ≡ B means that the set A is in b ijection wit h B . Note that, her e, since the group homomorphism ( linearit y) is insured, group bijection means group isomorphism. So, in th e follo wing, w e will use one or other terminology to refer to the same prop erty . Pro of of Corollary 1. W e pro ceed b y induction on p . The order p = 1 corresp onds to Lemma 1 , i.e Π r n +1 ≡ π r n +1 and Λ r n +1 ≡ Λ r n . The surj ectivit y is then immediate ∀ p ∈ N , p 6 = 0, as r ≥ 0. F o r th e one to one prop ert y , su pp ose the statemen t holds for the order p − 1. F or p , setting n + p − 1 = n ′ and n + p = n ′ + 1 and using Lemma 1, Λ r n ′ +1= n + p ≡ Λ r n ′ = n + p − 1 ⇔ r ≤ n + p − 1 . Pro ceeding step by step, w e get Λ r n +1 ≡ Λ r n ⇔ r ≤ n ; Λ r n +2 ≡ Λ r n +1 ⇔ r ≤ n + 1; . . . ; Λ r n + p ≡ Λ r n + p − 1 ⇔ r ≤ n + p − 1 . Therefore, Λ r n + p ≡ Λ r n requires r ≤ min ( n + p − 1 , . . . , n + 1 , n ) = n . This ends the pro of of th e corollary .  2 Example 2. Give n n = 2 and r = 2 so that the set of indeterminates is { x 1 , x 2 } , then, the fol lowing p olynomials f i ar e symmetric and of de gr e e 2 , i.e b elong to Λ 2 2 : f 1 = x 1 x 2 , f 2 = x 2 1 + x 2 2 , ∀ p, q ∈ Z , f = pf 1 + q f 2 ∈ Λ 2 2 . A dding a new indeterminate x 3 , we have the c orr esp onding elements of Λ 2 3 f ′ 1 = x 1 x 2 + x 1 x 3 + x 2 x 3 , f ′ 2 = x 2 1 + x 2 2 + x 2 3 , ∀ p, q ∈ Z , f ′ = pf ′ 1 + q f ′ 2 ∈ Λ 2 2 . with π 3 f ′ i = f i , for i = 1 , 2 . F rom Lemma 1, th e follo wing stat emen t holds. Corollary 2. The se qu enc e of gr oups 0 i − → Λ r n +1 π n +1 − → Λ r n p − → 0 , wher e i is the c anonic al inje ction, and p the pr oje ction onto { 0 } , is exact if and only if r ≤ n . This corollary ma y b e of great imp ortance for r > n in the Homolo gy Th eory in v olving the groups of sym m etric p olynomials [10]. Definition 2. L et r b e a nonne gative inte ger. The pr oje ctive (or inv e rse) limit Λ r = lim ← − n Λ r n is the ad ditive gr oup of se quenc es of homo gene ous symmetric p olynomials of de gr e e r such tha t f r = ( f r 1 , f r 2 , . . . , f r n , . . . ) with ∀ n ∈ N 6 { 0 } , f r n ∈ Λ r n and π n +1 ( f r n +1 ) = f r n . (3) The elements of Λ r ar e c al le d pr oje ctive limits and Λ r is c al le d the homo gene ous gr oup of de gr e e r of pr oje ctive limits. B e sides, let Λ = L r ≥ 0 Λ r b e the gr ade d ring define d by the dir e ct sum of the homo gene ous gr oups Λ r . A n element f of Λ is a sum of pr oje ctive limits, namely f = P r ≥ 0 f r such t hat, for any de gr e e r , f r b elongs to the homo gene ous gr oup Λ r . An element of Λ is c al le d a symmetric function. It can b e sh o wn the follo wing statement [1]. Prop osition 2. With the ab ove notation, ther e is a su rje ctive ho momorphism of gr ade d rings Π n : Λ → Λ n define d by setting x p = 0 , ∀ p ≥ n + 1 . Example 3. Given two nonne gative inte gers r and n , the p artial sum f r n = P n i =1 x r i defines the se quenc e ( f r n ) n ∈ N as a pr oje ctive limit of Λ r . This symmetric f unction is of de gr e e r and define d by f r = P ∞ n =1 x r n . Remark 2. O ften in th e liter atur e, th er e is no distinction b etwe en the pr oje ctive limit which is a se quenc e and the limit of the c orr esp onding p artia l sum which is a func tion. In any c ase, given f n , th e expr ession lim ← − n f r n = f r c ontains al l informa tion gener ate d by t he e quation (3) . More r igorously , w e consider also the follo wing definition of the in verse limit [9]. Let I b e a set of indices. Supp ose a giv en r elation of partial orderin g in I . W e say that I is d irected if giv en i, j ∈ I , there is k ∈ I suc h that i ≤ k and j ≤ k . Ass u me that I is d irected. Let n o w consider A a category , and { A i } a family of o b jects in A . F or eac h pair i, j suc h that i ≤ j , let us consider a giv en morp hism: f ( j,i ) : A j → A i 3 suc h that, when ever i ≤ k ≤ j , one gets f ( j,k ) ◦ f ( k, i ) = f ( j,i ) and f ( i,i ) = id, where id is the identit y mapping of A i . Su c h a family is called a directed f amily of morp hisms. An in ve rse limit for the family ( f ( j,i ) ) is a u niv ersal ob ject of the fo llo wing category C . O b ( C ) consists of pairs ( A, ( f i )) wh ere A ∈ O b ( A ) and ( f i ) is a f amily of morph isms f i : A → A i , i ∈ I , suc h that, for all i ≤ j , the follo wing diagram is comm utative : A f j      ✠ A j ❅ ❅ ❅ ❅ ❅ ❘ f i ✲ A i f ( j,i ) Giv en t wo nonnegativ e in tegers n 1 ≤ n 2 , let (Π ( n 2 ,n 1 ) ) b e the family of homomo rphisms of graded r ings fr om Λ n 2 to Λ n 1 suc h that Π ( n 2 ,n 1 ) := Π n 1 +( n 2 − n 1 ) , where Π n is defin ed by Corollary 1. W e can easily c hec k that (Π ( n 2 ,n 1 ) ) is a directed family of ring h omomorphisms in the category of graded rings. Th e family ( Π n ) of Prop osition 2 defin es the follo wing comm u tativ e diag r am: ∀ m, n ∈ N , m ≥ n, Λ Π m      ✠ Λ m ❅ ❅ ❅ ❅ ❅ ❘ Π n ✲ Λ n Π ( m,n ) (Λ , Π n ), considered as a u niv ers al ob ject, is the in v erse limit of the d ir ected family ( Π ( n 2 ,n 1 ) ). So, w e agree with the prop erty that the inverse limit define d by the family of dir e cte d homo- morphism s (Π ( n 2 ,n 1 ) ) is e qual to the inverse limit define d by th e family of pr oje ction ( π n ) . Here and thereafter, defin ing in ve r se limit by the family { Λ n , Π ( n 1 ,n 2 ) } or b y the family { Λ n , π n } is e quivalent . 3 Multi-indicial symmetric functions In this section, w e define the symm etric function of infin ite n u m b er of en tries that we call m u lti-indicial symmetric function. Giv en m, n, k ∈ N , let us consider the follo wing s et of indep enden t indeterminates n a { 1 ≤ p ≤ m, [ µ ] 1 ≤ i ≤ k } o . See T able 1. The n otatio n [ µ ] i means an y m ulti-index of the form µ 1 µ 2 . . . µ i , with 1 ≤ µ i ≤ n , 1 ≤ i ≤ k . F or instance, a m [ µ ] p denotes in general a mµ 1 µ 2 ...µ p , f or a ny 1 ≤ µ i ≤ n . The s ets of 4 indeterminates ma y b e organized in the follo wing man n er: D 0 m = { a m } , D 0 ( m ) = [ 1 ≤ l ≤ m D 0 l , D k m = [ 1 ≤ µ 1 ,µ 2 ,...,µ k ≤ n a mµ 1 µ 2 ...µ k , (4) D k ( m ) = m [ l =1 D k l , D ( k ) m = k [ l =0 D l m , D ( m ) = ∞ [ k =0 D k ( m ) , D ( k ) = ∞ [ m =1 D ( k ) m , (5) D = D ( m ) ∪ D ( k ) . (6) The follo wing statemen t holds by a s imple combinato rics. Prop osition 3. L et m , n and k b e thr e e nonne gative inte g e rs. Then, | D ( k ) m +1 | = q k , | D k +1 ( m ) | = n k +1 m, | D ( k +1) ( m +1) | = q k + n k +1 ( m + 1) , wher e D ( k +1) ( m +1) = D ( k ) m +1 ∪ D k +1 ( m ) ∪  a ( m +1) µ 1 µ 2 ...µ k µ k +1  , q k = n k +1 − 1 n − 1 if n 6 = 1 and q k = k + 1 if n = 1 . Definition 3. Given m , n , k , th r e e nonne gative inte gers such that m, n ≥ 1 , then the p olyno- mial ring Z ( a 1 , a 2 , . . . , a m , a 1 µ , a 2 µ , . . . , a mµ , a 1 µ 1 µ 2 , a 2 µ 1 µ 2 , . . . , a mµ 1 µ 2 , . . . , (7) a 1 µ 1 µ 2 ...µ k , a 2 µ 1 µ 2 ...µ k , . . . , a mµ 1 µ 2 ...µ k ) , wher e µ, µ 1 , µ 2 , . . . , µ k − 1 and µ k take al l values in { 1 , 2 , . . . , n } , i s denote d by Z h a m a m [ µ ] k i . The numb er of indeterminates is mq k , wher e q k = n k +1 − 1 n − 1 if n 6 = 1; q k = k + 1 , if n = 1 . The symmetric gr oup S mq k defines the gr ade d ring of symmetric p olynomials of Z [ a m a m [ µ ] k ] : Λ m,k = Z [ a m a m [ µ ] k ] S mq k . Lemma 4. L et r , m and k b e two nonne gative inte gers. Ther e is a g r oup isomorph i sm Λ r m,k ≡ Λ r mq k le ading to a gr ade d ring isomorph i sm Λ m,k ≡ Λ mq k Pro of. The s et of ind eterminates n { a p } , { a pµ } p ; µ =1 ,..., n , . . . , { a pµ 1 µ 2 ...µ k } p ; µ 1 ,µ 2 ,...,µ k = 1 ,... ,n o p =1 ,..., m can b e view ed as the set of in d eterminates { x 1 , x 2 , . . . , x mq k } . The indep end ence of indetermi- nates requir es the a mµ 1 µ 2 ...µ p to corresp on d to a unique x i . Sin ce the t wo sets p ossess the same cardinal, a w ell defin ed bijection can b e built from one onto the other.  By con ve n tion, Λ m, 0 = Λ m , Λ 0 ,k = Λ q k − 1 , Λ 0 , 0 = Z . Definition 4. L et m and k b e nonne gative inte gers, m ≥ 1 . (i) The gr ade d ring homomor phism h m +1 ,k : Λ m +1 ,k → Λ m,k , such that ∀ r ∈ N , h m +1 ,k | Λ r m +1 ,k := h r m +1 ,k : Λ r m +1 ,k → Λ r m,k and define d by setting a m +1 = 0 , a ( m +1) µ = 0 , . . . , a ( m +1) µ 1 µ 2 ...µ k = 0 , ∀ 1 ≤ µ i ≤ n, is c al le d the ( m + 1 , k ) horizontal pr oje ction or simply the h - pr oje ction when no c onfusion o c cu rs; the r estriction h r m +1 ,k is c al le d th e ( m + 1 , k ) ho rizontal pr oje ction of de gr e e r . 5 (ii) The gr ade d ring homom orphism v m,k +1 : Λ m,k +1 → Λ m,k , such that ∀ r ∈ N , v m,k +1 | Λ r m,k +1 := v r m,k +1 : Λ r m,k +1 → Λ r m,k and d e fine d by setting a 1 µ 1 µ 2 ...µ k +1 = 0 , a 2 µ 1 µ 2 ...µ k +1 = 0 , a mµ 1 µ 2 ...µ k +1 = 0 , ∀ 1 ≤ µ i ≤ n, is c al le d ( m, k + 1) vertic al pr oje ction or simply the v-pr oje ction when no c onfusion o c curs; the r estriction v r m,k +1 is c al le d th e ( m, k + 1) vertic al pr oje ction of de gr e e r . (iii) The gr ade d ring homomorphism π m +1 ,k +1 : Λ m +1 ,k +1 → Λ m,k , such that ∀ r ∈ N , π m +1 ,k +1 | Λ r m +1 ,k +1 := π r m +1 ,k +1 : Λ r m +1 ,k +1 → Λ r m,k and d e fine d by setting a m +1 = 0 , a ( m +1) µ = 0 , . . . , a ( m +1) µ 1 µ 2 ...µ k = 0 , a 1 µ 1 µ 2 ...µ k +1 = 0 , a 2 µ 1 µ 2 ...µ k +1 = 0 , a mµ 1 µ 2 ...µ k +1 = 0 and a ( m +1) µ 1 µ 2 ...µ k +1 = 0 , ∀ 1 ≤ µ i ≤ n, is c al le d the ( m + 1 , k + 1) pr oje ction. The r e striction π r m +1 ,k +1 is c al le d the ( m + 1 , k + 1) pr oje ction of de gr e e r . Lemma 5. Given m a no nne gative inte ger, m ≥ 1 , the h-pr oje ction h r m +1 , 1 : Λ r m +1 , 1 → Λ r m, 1 is surje ctive for al l r ≥ 0 and bije ctive i f and only if r ≤ ( n + 1) m . Pro of. The results follo w from Lemma 4 and Corollary 1 . Th e su rjectivit y is immediate. F or the pro of of the bijectivit y , w e obtain usin g Lemma 4 Λ r m +1 , 1 ≡ Λ r ( m +1) q 1 and Λ r m, 1 ≡ Λ r mq 1 . F rom Corollary 1, the r.h.s expressions are bijectiv e i f and only if 0 ≤ r ≤ mq 1 .  Prop osition 6. (i) ∀ k ≥ 0 , the h-pr oje ction h r m +1 ,k : Λ r m +1 ,k → Λ r m,k is surje ctive for al l r ≥ 0 and bije ctive if and only if r ≤ mq k . (ii) ∀ m ≥ 1 , the v- pr oje ction v r m,k +1 : Λ r m,k +1 → Λ r m,k is surje ctive for al l r ≥ 0 and bije ctive if an d only if r ≤ mq k . (iii) ∀ k ≥ 0 , ∀ m ≥ 1 , the pr oje ction π r m +1 ,k +1 : Λ r m +1 ,k +1 → Λ r m,k is surje c tive for al l r ≥ 0 and bije ctive if and only if r ≤ mq k . Pro of. The pro ofs of the surj ections are immediate by the use of Lemma 4. So, let us pa y atten tion to the p ro ofs of the bijections. On e can sh o w ( i ) by ind u ction on k . Consider Lemma 5 as the order k = 1. The follo win g step is s imilar to the pro of o f Corollary 1, taking the min on d ifferen t v alues of r ≤ m in { ( m + 1) q k , m q k } = mq k . The s teps ( ii ) and ( iii ) can b e shown b y the same wa y . Indeed, w e can easily give th e pr escrib ed equiv alen t of Lemma 5 for k and for b oth m an d k .  Definition 5. Given r a nonne gative int e ger, then (i) We c al l horizontal (r esp. vertic al) se quenc e (m,k) of de gr e e r the inverse system denote d by ( Λ r m,k , h r m,k ) m ∈ N , (r esp. (Λ r m,k , v r m,k ) k ∈ N ); (ii) W e c al l se q u enc e (m,k) of de gr e e r the inverse system denote d by (Λ r m,k , π r m,k ) m,k ∈ N . The next prop osition can b e ded u ced fr om Pr op osition 6. Prop osition 7. With the ab ove notation, the fol lowing diagr am in which al l mappings ar e surje ctive for al l r ∈ N , a nd bije ctive if r ≤ mq k , is c ommuta tive 6 Λ r m +1 ,k +1 ✲ h r m +1 ,k +1 ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s π m +1 ,k +1 Λ r m,k +1 ❄ v r m +1 ,k +1 ✲ h r m +1 ,k Λ r m +1 ,k ❄ v r m,k +1 Λ r m,k and l e ads to the c orr e sp onding c ommutative diagr am with r esp e ct to the gr ade d ring structur e. The previous dev elopment leads to the follo wing consequence. Giv en a nonnegativ e in teger r , taking th e p ro jectiv e limit with resp ect to the h orizon tal (resp. v ertical) sequence ( m, k ) of degree r , we obtain ˜ Λ r .,k = lim ← − m Λ r m,k (resp. ˜ Λ r m,. = lim ← − k Λ r m,k ) th at w e ca ll the h orizon tal (resp. v ertical) pro jectiv e limit of deg ree r of the sequ ence (Λ r m,k ) m ∈ N (resp. (Λ r m,k ) k ∈ N ). ˜ Λ r .,k (resp. ˜ Λ r m,. ) is the additive group of horizon tal (resp. v ertical) symm etric functions of degree r . F urthermore, giv en m ∈ N (resp. k ∈ N ), for eac h k (resp. m ), there is a sur j ectiv e homomorphism H r m,k : Λ r .,k → Λ r m,k ( resp. V r m,k : Λ r m,. → Λ r m,k ) defined by a p>m = 0 and ∀ q ∈ N , a p>m ;[ µ ] 0 ≤ q ≤ k = 0 (resp . a 1 ≤ p ≤ m ;[ µ ] q ≥ ( k +1) = 0) wh ic h is bijectiv e iff. r ≤ mq k . Remark 3. The elements of ˜ Λ r m,. and ˜ Λ r .,k ar e se quenc es of symmetric functions of a given de gr e e r . The gr oups ˜ Λ r m,. and ˜ Λ r .,k ar e not isomorphic. Inde e d, one way to e asily r e alize this is to notic e that the ring Z h a m a m [ µ ] k i has not the same dep endenc e with r esp e c t to the indeterminates a m and a m [ µ ] k . Implicitly, a m [ µ ] k dep ends on n , while, in an obvious manner, a m do es not. Thus the elements of Λ r m,. , at the limit k → ∞ , do not involve the inte ger p ar ameter n at the opp osite of those of ˜ Λ r .,k as m → ∞ . This c onstruction of the ring Z h a m a m [ µ ] k i is differ ent fr om th e c onstruction of a p olyno mial ring in the indeterm inates [10] { a m,n,k } 0 ≤ m ≤ M ; 0 ≤ n ≤ N ;0 ≤ k ≤ K , given M , N , K ∈ N , wh i c h c onsists in assigning the fr e e entries of a 3 -tensor, for instanc e. The indep endenc e b etwe en the indeter minates, in this c ase, should c orr esp ond to the isomorphism of sets of se quenc es of symmetric f unctions in the r emaining indic es. We have Λ r m,n, ( . ) ≡ Λ r m, ( . ) ,k ≡ Λ r ( . ) ,n,k , wher e the p oint me ans that the c orr esp onding index tends to infinity. Summing o v er the d egrees, one obtains the graded rings ˜ Λ .,k = L r ≥ 0 ˜ Λ r .,k , ˜ Λ m,. = L r ≥ 0 ˜ Λ r m,. of h orizon tal sequences of symmetric fu nctions and v ertical sequences of symmetric functions, resp ectiv ely . Definition 6. L et r b e a nonne gative inte ger. Two symmetr ic functions P r and Q r of de gr e e r ar e said e qual if and onl y if ∀ n ∈ N , P r n = Q r n . 7 Let P r m,. ∈ ˜ Λ r m,. . F or an y k ∈ N , P r m,. = ( P r m, 0 , P r m, 1 , . . . , P r m,k , . . . ) suc h that, for any k , v r m,k +1 P r m,k +1 = P r m,k . W e also obtain, for an y m ∈ N , h r m +1 ,k P r m +1 ,k = P r m,k . Hence, the mapping h r m : ˜ Λ r m,. → ˜ Λ r m − 1 ,. defined b y h r m ( P r m,. ) = ( h r m, 0 P r m, 0 , h r m, 1 P r m, 1 , . . . , h r m,k P r m,k , . . . ) , allo ws to get h r m ( P r m,. ) = P r m − 1 ,. . This shows that h r m is a we ll d efi ned pro jection and defines the p r o jectiv e limit of degree r of the v er tical sequ ence  Λ r m,.  m ∈ N b y P r = lim ← − m P r m,. . W e call this inv erse limit the h(v)-limit of degree r . Besides, defining the mapp ing v r k : ˜ Λ r .,k → ˜ Λ r .,k , by v r k ( P r .,k ) = ( v r 0 ,k P r 0 ,k , v r 1 ,k P r 1 ,k , . . . , v r m,k P r m,k , . . . ) , w e get v r k ( P r .,k ) = P r .,k − 1 whic h shows that v r k is a well defined pro jection whic h defines the pro jectiv e limit of degree r of the horizonta l sequence (Λ r .,k ) k ∈ N b y P ′ r = lim ← − k P r .,k . W e ca ll this inv erse limit the v(h)-limit of degree r . P r and P ′ r are not a priori the same qu an tit y . But, they are actually isomorph ic. In deed from [9], the follo wing h olds. Theorem 1. L et M and K b e two dir e cte d sets, ( A m,k ) m ∈ M ; k ∈ K b e a family of Ab elian gr oups e quipp e d with ho momorphisms lab ele d by M × K , and defining an inverse limit. A ssigning the obvious or dering to the pr o duct M × K , i.e ( m, k ) ≤ ( m ′ , k ′ ) ⇔ m ≤ m ′ and k ≤ k ′ , the fol lowing inverse limits exist and ar e isomo rphic in a natur al way: lim ← − m lim ← − k A m,k = lim ← − k lim ← − m A m,k . (8) The inv erse systems ((Λ r m,k , v r m,k ) , h r m ) and ((Λ r m,k , h r m,k ) , v r k ), giving rise to the v(h)-limit and the h(v)-limit, resp ectiv ely , are equiv alen t. W e establish this equiv alence in the follo wing. Prop osition 8. L et r, k , m ( m ≥ 1 ) b e thr e e nonne gative inte gers. F or al l m 1 , m 2 , k 1 , k 2 ∈ N , such t hat 1 ≤ m 1 ≤ m 2 and 0 ≤ k 1 ≤ k 2 , the map pings φ r ( m 2 ,m 1 ) ,k : Λ r m 2 ,k → Λ r m 1 ,k and ψ m, ( k 2 ,k 1 ) : Λ r m,k 2 → Λ r m,k 1 , define d by φ r ( m 1 ,m 1 ) ,k ≡ I , ψ m, ( k 1 ,k 1 ) ≡ I , φ r ( m 2 ,m 1 ) ,k ≡ h r ( m 1 +1) ,k ◦ h r ( m 1 +2) ,k ◦ · · · ◦ h r m 2 ,k and ψ r m, ( k 2 ,k 1 ) ≡ v r m, ( k 1 +1) ◦ v r m, ( k 1 +2) ◦ · · · ◦ v r m,k 2 ar e wel l define d surje ctive gr oup hom omorph i sms. F urthermor e, given k (r esp. m ), ( φ ( m 2 ,m 1 ) ,k ) (r esp. ( ψ m, ( k 2 ,k 1 ) ) ) defines a dir e cte d f amily of h omomorphisms of gr ade d rings. Pro of. The sur jectivit y is given by ind uction from the d efinition of h r m,k and v r m,k . More- o ve r , one can easily c hec k, that giv en k , for an y m 1 ≤ p ≤ m 2 , φ r ( m 2 ,p ) ,k ◦ φ r ( p,m 1 ) ,k = φ r ( m 2 ,m 1 ) ,k . Giv en m, the similar prop ert y also holds for ψ r m, ( k 2 ,k 1 ) .  Lemma 9. With t he ab ove notation, φ r ( m,m − 1) ,k ≡ h r m,k , ψ r m, ( k, k − 1) ≡ v r m,k , φ r ( m,m − 1) ,k ◦ ψ r m, ( k, k − 1) = π r m,k , φ r ( m 2 ,m 1 ) ,k 1 ◦ ψ r m 2 , ( k 2 ,k 1 ) ≡ ψ r m 1 , ( k 2 ,k 1 ) ◦ φ r ( m 2 ,m 1 ) ,k 2 , (9) ∀ 0 ≤ q ≤ k 2 , φ r ( m 2 ,m 1 ) ,q ◦ φ r ( m 2 ,m 1 ) ,k 2 = φ r ( m 2 ,m 1 ) ,k 2 , 8 ∀ 1 ≤ p ≤ m 2 , ψ r m 2 , ( k 2 ,k 1 ) ◦ ψ r p, ( k 2 ,k 1 ) = ψ r m 2 , ( k 2 ,k 1 ) . (10) F urthermor e, one has, ∀ m 1 ≤ p ≤ m 2 and ∀ k 1 ≤ q ≤ k 2 , φ r ( m 2 ,p ) ,k ◦ φ r ( p,m 1 ) ,k = φ r ( m 2 ,m 1 ) ,k , and ψ r m, ( k 2 ,q ) ◦ ψ r m, ( q ,k 1 ) = ψ r m, ( k 2 ,k 1 ) Pro of. This is immediate f rom Pr op ositions 7 and 8.  Remark 4. (9) c an b e viewe d as the data of a c ommutative diagr am. Giv en nonnegativ e inte gers k , m ≥ 1, the dir ected f amilies ( φ r ( m 2 ,m 1 ) ,k ) and ( ψ r m, ( k 2 ,k 1 ) ) de- fine a directed family of homomorphisms in b oth th e ind ices m and k as follo ws. ∀ m 1 , m 2 , k 1 , k 2 ∈ N , such that 1 ≤ m 1 ≤ m 2 and 0 ≤ k 1 ≤ k 2 , let Φ r ( m 2 ,m 1 ) , ( k 2 ,k 1 ) : Λ r m 2 ,k 2 → Λ r m 1 ,k 1 b e the mapp ing d efi ned by Φ r ( m 1 ,m 1 ) , ( k 2 ,k 1 ) ≡ ψ r m 1 , ( k 2 ,k 1 ) , Φ r ( m 2 ,m 1 ) , ( k 1 ,k 1 ) ≡ φ r ( m 2 ,m 1 ) ,k 1 , (11) Φ r ( m 2 ,m 1 ) , ( k 2 ,k 1 ) ≡ φ r ( m 2 ,m 1 ) ,k 1 ◦ ψ r m 2 , ( k 2 ,k 1 ) = ψ r m 1 , ( k 2 ,k 1 ) ◦ φ r ( m 2 ,m 1 ) ,k 1 . (12) W e deduce, f rom Lemm a 9, with m 1 ≤ p ≤ m 2 and k 1 ≤ q ≤ k 2 , Φ r ( m 2 ,p ) , ( k 2 ,k 1 ) ◦ Φ r ( p,m 1 ) , ( k 2 ,k 1 ) ≡ Φ r ( m 2 ,m 1 ) , ( k 2 ,k 1 ) , (13) Φ r ( m 2 ,m 1 ) , ( k 2 ,q ) Φ r ( m 2 ,m 1 ) , ( q, k 1 ) ≡ Φ r ( m 2 ,m 1 ) , ( k 2 ,k 1 ) . Applying Theorem 1 w ith M = N 6 { 0 } and K = N whic h are obviously d irected s ets,  Φ r ( m 2 ,m 1 ) , ( k 2 ,k 1 )  m 1 , 2 ∈ M ; k 1 , 2 ∈ K is a d irected family of homomorphisms lab eled b y M × K whic h allo ws to wr ite, by a nalogy with (8) lim ← − m lim ← − k Λ r m,k = lim ← − k lim ← − n Λ r m,k . The follo wing statemen t is v alid. Theorem 2. (i) L et r, m 1 , m 2 , k 1 , k 2 b e nonne gative inte gers such that 1 ≤ m 1 ≤ m 2 and 0 ≤ k 1 ≤ k 2 . The mappings φ ( m 2 ,m 1 ) : Λ m 2 ,. → Λ m 1 ,. and ψ ( k 2 ,k 1 ) : Λ .,k 2 → Λ .,k 1 define d by φ r ( m 2 ,m 1 ) = h r m 1 +1 ◦ h r m 1 +2 ◦ · · · ◦ h r m 2 , ψ r ( k 2 ,k 1 ) = v r k 1 ◦ v r k 1 +1 ◦ · · · ◦ v r k 2 ar e surje ctive hom omorph i sms of gr ade d rings, define dir e cte d families ( φ ( m 2 ,m 1 ) ) and ( ψ ( k 2 ,k 1 ) ) of homomorphisms of gr ade d rings. F urthermor e, the inverse limits induc e d by th e se families a r e e q ual, i. e lim ← − m Λ m,. = Λ = lim ← − k Λ .,k . (ii) Give n a nonne gative inte ger m (r esp. k ), m > 0 , the mapping H m : Λ → Λ m,. (r esp. V k : Λ → Λ .,k ) define d by setting a p>m = 0 and ∀ q ∈ N , a p>m ;[ µ ] q = 0 (r esp. a p> 0 = 0 and a p> 0;[ µ ] q>k = 0 ) is a surje ctive h omomor phism of gr ade d rings. 9 Pro of. W e p ro ve that the t wo in v erse limits coincide. The statemen t, mainly obtained b y the defin ition of any un iv ersal ob ject of a category , h olds in general b y T heorem 1. L et us illustrate, here, this statemen t b y a p articular case. W e consider that k and m are t wo nonnegativ e integ ers with m > 0. Moreo v er, homomorphism means surjectiv e h omomorphism of graded rin gs. Giv en t w o nonnegativ e intege rs m > 0 and k , there are f our directe d families of homomorphisms ( φ ( m 1 ,m 2 ) ,k ) , ( ψ m, ( k 1 ,k 2 ) ) , ( φ ( m 1 ,m 2 ) ) and ( ψ ( k 1 ,k 2 ) ) , generating four kin d s of categories C k , C m , C 1 and C 2 whose th e s ets of ob jects are giv en by O b ( C k ) = { ( R , ( H m,k ) m ) } , H m,k : R → Λ m,k , (14 ) O b ( C m ) = { ( R , ( V m,k ) k ) } , V m,k : R → Λ m,k , (15) O b ( C 1 ) = { ( R , ( H m ) m ) } , H m : R → Λ m,. , (16) O b ( C 2 ) = { ( R , ( V k ) k ) } , V k : R → Λ k ,. , (17) resp ectiv ely , w h ere R is any graded ring. T he c ategorie s of (14)-(1 7 ) generate , up to a unique isomorphism, unive rsal ob j ects g iv en by (Λ .,k , ( H m,k ) m ) , (Λ m,. , ( V m,k ) k ) , ( ˜ Λ 1 , ( H m ) m ) and ( ˜ Λ 2 , ( V k ) k ) , resp ectiv ely . Let us consider the follo wing diagram ˜ Λ r 2 ✛ f ✲ g ˜ Λ r 1 ✲ H r m ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s A m,k Λ r m,. ❄ V r k ✲ H r m,k Λ r .,k ❄ V r m,k Λ r m,k Giv en k , for all m , it comes A m,k := V m,k ◦ H m : ˜ Λ 1 → Λ m,k . Moreo v er, ( ˜ Λ 1 , ( A m,k ) m ) ∈ O b ( C k ) and there is a unique h omomorphism ϕ k : ˜ Λ 1 → Λ .,k suc h that: V m,k ◦ H m = H m,k ◦ ϕ k . It f ollo ws that ( ˜ Λ 1 , ( ϕ k ) k ) ∈ O b ( C 2 ) and thus, there exists a un ique homomorph ism g : ˜ Λ 1 → ˜ Λ 2 suc h that ϕ k = V k ◦ g . Hence, we get V m,k ◦ H m = H m,k ◦ V k ◦ g . (18) Moreo v er, in the same manner, giv en m and the homomorphism H m,k ◦ V k : ˜ Λ 2 → Λ m,k , for all k , we hav e, thr ough the u niv ersal ob ject prop erty of (Λ m,. , ( V m,k ) k ), the uniqu e h omomorp hism  m : ˜ Λ 2 → Λ m,. suc h that H m,k ◦ V k = V m,k ◦  m . 10  m induces, b y the univ ersal ob ject prop erty of ( ˜ Λ 1 , ( H m )), the factorization  m, 2 = H m ◦ f , where f : ˜ Λ 2 → ˜ Λ 1 is u niquely d efined. Consid er H m,k ◦ V k = V m,k ◦ H m ◦ f . (19) F rom (18) and (19), we dedu ce f ◦ g = I . Conv ersely , w e can also sho w th at g ◦ f = I .  Theorem 3. The diagr am define d by the T able 2 is c ommutative in the sense that any of its squar es is c ommuta tiv e. Pro of. Giv en four nonnegativ e in tegers m, k , p, q , with m > 0, any of the internal dia- grams, i.e any d iagram of the form Λ r m + p,k + q ✲ φ r ( m + p,m ) ,k + q ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ s Φ ( m + p,m ) , ( k + q , k ) Λ r m,k + q ❄ ψ r m + p, ( k + q ,k ) ✲ φ r ( m + p,m ) ,k Λ r m + p,k ❄ ψ r m, ( k + q ,k ) Λ r m,k is comm u tativ e from Pr op osition 9 and the p rop erties (13). Let u s pa y atten tion to the d iagrams in volving th e inv erse limits. There are three kind s of such diagrams. (i) T he fi rst inv olv es t w o i n verse limits in m : Λ r .,k 2 ✲ ψ r ( k 2 ,k 1 ) Λ r .,k 1 ❄ H r m,k 2 ✲ ψ r m, ( k 2 ,k 1 ) Λ r m,k 2 ❄ H r m,k 1 Λ r m,k 1 Suc h a d iagram is comm utativ e for (Λ .,k 1 , ( H m,k 1 )) is a universal ob j ect. ψ m, ( k 2 ,k 1 ) ◦ H m,k 2 : Λ .,k 2 → Λ m,k 1 can b e facto rized b y the unique ring homomorphism Λ .,k 2 ψ ( k 2 ,k 1 ) − → Λ .,k 1 as ψ m, ( k 2 ,k 1 ) ◦ H m,k 2 = H m,k 1 ◦ ψ ( k 2 ,k 1 ) . (ii) T he second inv olv es t w o inv erse limits in k : 11 Λ r m 2 ,. ✲ φ r ( m 2 ,m 1 ) Λ r m 1 ,. ❄ V r m 2 ,k ✲ φ r ( m 2 ,m 1 ) ,k Λ r m 2 ,k ❄ V r m 1 ,k Λ r m 1 ,k The diagram is also comm utativ e by the usual defin ition of u niv ersal ob ject (Λ m 1 ,. , ( V m 1 ,k )) b y a n alogy with the pr o of of the case ( i ). (iii) The third inv olv es the inv erse limit Λ: Λ r ✲ H r m Λ r m,. ❄ V r k ✲ H r m,k Λ r .,k ❄ V r m,k Λ r m,k The comm utativit y results fr om the same argumen t.  The comm utativit y of any of the d iagrams defined by T able 2 represents the inv erse limit defined b y the three directed families ( φ ( m 1 ,m 2 ) ), ( ψ ( k 1 ,k 2 ) ) and (Φ ( m 2 ,m 1 )( k 2 ,k 1 ) ). T h u s, {{ Λ m,k , v m,k } , h m } ⇔ {{ Λ m,k , h m,k } , v k } ⇔  Λ m,k , Φ ( m 2 ,m 1 ) , ( k 2 ,k 1 )  that leads to ˜ Λ = lim ← − m lim ← − k Λ r m,k = lim ← − m ˜ Λ m,. = lim ← − k ˜ Λ .,k . (20) Finally , the set of usu al symmetric fu n ctions is reco ve red, i.e. ˜ Λ ≡ Λ . 4 Multi-partition and m ulti-indicial Sch u r fun ctions In this section, we deal w ith the definition of multi-partitio n and study the corresp onding inter- esting family of symmetric fu nctions kn own as the Sc hur fu nctions [1 ]. Definition 7 . A p artition λ is a finite or i nfinite se quenc e of inte gers ( λ 1 , λ 2 , . . . , λ i , . . . ) , w i th λ 1 ≥ λ 2 ≥ · · · ≥ 0 and | λ | := P i λ i < ∞ , so that, fr om a c ertain p oint onwar ds (if λ is infinite), al l the λ i ar e 0 . The non zer o λ i ar e c al le d the p arts of λ . The numb er of p arts is th e length l ( λ ) of λ . 12 Remark 5. Two p artitio ns λ 1 and λ 2 which differ only by a se quenc e of 0 at the end ar e e qual. F or instan c e, ( 1 , 2) and (1 , 2 , 0 , 0 , . . . ) ar e r e gar de d as th e sa me p artition. Definition 8. Given two nonne g ative i nte gers m and k , we c al l a [ m, k ] -p artition (or a multi- p artitio n when no c onfusion o c cu rs) th e o r der e d se quenc e λ [ m,k ] = ( λ [ m, 0] , λ [ m, 1] , . . . , λ [ m,k ] ) define d by a set of k + 1 p artitions such that: λ [ m, 0] = λ [ m, 0] 1 ≤ i = ( λ [ m, 0] 1 , λ [ m, 0] 2 , . . . , λ [ m, 0] p , . . . ) , λ [ m, 1] = ( λ [ m, 1] pµ ) 1 ≤ p,µ , λ [ m,k ] = ( λ [ m,k ] pµ 1 µ 2 ...µ k ) 1 ≤ p,µ 1 ,µ 2 ,...,µ k , (21) with 1 ≤ µ l ≤ n, f or 1 ≤ l ≤ k , so that th e fo l lowing pr op erty is satisfie d: λ [ m, 0] 1 ≥ λ [ m, 0] 2 ≥ · · · ≥ λ [ m, 0] p ≥ · · · ≥ λ [ m, 0] m ≥ · · · ≥ λ [ m, 1] 11 ≥ λ [ m, 1] 12 ≥ · · · ≥ λ [ m, 1] 1 µ ≥ · · · ≥ λ [ m, 1] 21 ≥ · · · ≥ λ [ m, 1] 2 µ ≥ · · · ≥ λ [ m, 1] 2 n ≥ . . . ≥ λ [ m, 1] pµ ≥ . . . λ [ m, 1] m 1 ≥ λ [ m, 1] m 2 ≥ · · · ≥ λ [ m, 1] mn ≥ · · · ≥ λ [ m,k ] mµ 1 ...µ k ≥ λ [ m,k ] mµ 1 ... ( µ k +1) ≥ · · · ≥ 0 . (22) λ [ m,p ] , for any 0 ≤ p ≤ k , is c al le d a sub-p artition of λ [ m,k ] . F urthermor e, we identify | λ [ m,k ] | = X 0 ≤ p ≤ k | λ [ m,p ] | . The length of the [ m, k ] -p artition is define d by the sum of the lengths of its sub-p artitions, namely l ( λ [ m,k ] ) = X 0 ≤ p ≤ k l ( λ [ m,p ] ) . One can easily see that the so defin ed [ m, k ]-p artition is ’exhaustiv e’ relativ ely to the n u m b er of indeterminates, i.e it assigns an exp onent to ea c h of them. F urth er m ore, a [ m, 0]- partition is, by con ven tion, a partition in the sense of Definition 7 . Eac h of the λ [ m,k ] , tak en separately , defines a partition suc h that the ordered sequence (21) wh ic h defines λ [ m,k ] remains a p artition. One can d efine the monomial symmetric function in th e mq k indeterminates by the sum of all distinct monomials that can b e obtained f rom a λ [ m,k ] = a λ [ m, 0] 1 1 . . . a λ [ m, 0] m m Y m ; µ a λ [ m, 1] mµ mµ · · · Y m ; µ 1 ,µ 2 ,...,µ k a λ [ m,k ] mµ 1 µ 2 ...µ k mµ 1 µ 2 ...µ k , (23) b y p erm u tation of the a ’s. I n p articular, f or any i ∈ [0 , k ], λ [ m,k ] = ( λ [ m, 0] = (0) , . . . , λ [ m,i ] = (1 , 1 , . . . , 1 | {z } r − times , 0 , 0 , . . . ) , . . . , λ [ m,k ] = (0)) . One readily r eco v ers the defin ition of classical symmetric m on omial e 1 = m (1 r ) [1]. It is then immediate that Z -basis of ˜ Λ .,k and ˜ Λ m,. can b e obtained as a function of the monomial symmetric functions corresp ond ing to (23), when the [ m, k ]-p artition runs through all m u lti-partitions. Let us come bac k to the usual theory . Let n b e a nonnegativ e in teger. In the follo wing, δ is the partition defined by ( n − 1 , n − 2 , . . . , 1 , 0). The follo wing stat emen t holds [1]. 13 Prop osition 10. Given a nonne gative inte ger n , for e ach p artition α = ( α 1 , α 2 , . . . , α n ) , of nonne g ative i nte gers such that α 1 > α 2 > · · · > α n ≥ 0 , the homo gene ous p olyno mial define d by a α = det( x α j i ) 1 ≤ i , j ≤ n (24) is d ivisible by the V andermo nde determ inant a δ in Z [ x 1 , x 2 , . . . , x n ] . The p artition α can b e chosen as α i = λ i + ( n − i ), for 1 ≤ i ≤ n , s o that α = λ + δ , where λ is a partition of length at most n . T he quotien t s λ ( x 1 , . . . , x n ) = a λ + δ 6 a δ is a sym metric p olynomial, homogeneous of degree | λ | . Passing to n + 1 v ariables, we ha ve s λ ( x 1 , . . . , x n , x n +1 ) | x n +1 =0 = s λ ( x 1 , . . . , x n , 0) = s λ ( x 1 , . . . , x n ) . The uniqu ely defined quotien t s λ ∈ Λ, that redu ces to s λ ( x 1 , . . . , x n ) when x p ≥ n +1 = 0, for an y n ≥ l ( λ ), is the Schur function corresp onding to the partition λ . Let us c onsider the mq k indeterminates with n ≥ 1 (see T able 1). W e d efine th e [ m, k ] -partition δ [ m,k ] =  δ [ m, 0] , δ [ m, 1] , . . . , δ [ m,k ]  , by δ [ m, 0] = ( δ [ m, 0] p = mq k − p ) 1 ≤ p ≤ m , δ [ m, 1] = ( δ [ m, 1] pµ = mnq k − 1 − ( p − 1) n − µ ) (1 ≤ p ≤ m );(1 ≤ µ ≤ n ) , and, for any 0 ≤ d ≤ k , 1 ≤ p ≤ m and 1 ≤ µ i ≤ n , δ [ m,d ] pµ 1 µ 2 ...µ d = δ [ m,d − 1] mn...n − ( p − 1) n d − X 1 ≤ l ≤ d − 1 ( µ l − 1) n d − l − µ d = n d ( mq k − d − ( p − 1)) − X 1 ≤ l ≤ d − 1 ( µ l − 1) n d − l − µ d , where the ind ex mn . . . n cont ains ( d − 1) times the index n . Exp licitly , it can b e written δ [ m, 0] = ( δ [ m, 0] 1 = mq k − 1 , δ [ m, 0] 2 = mq k − 2 , . . . , δ [ m, 0] p = mq k − p, . . . , δ [ m, 0] m = mnq k − 1 ) , where the identit y q k − 1 = n q k − 1 has b een used. δ [ m, 1] = ( δ [ m, 1] 11 = mnq k − 1 − 1 , δ [ m, 1] 12 = mnq k − 1 − 2 , . . . , δ [ m, 1] 1 µ = mnq k − 1 − µ, δ [ m, 1] 1 n = n ( mq k − 1 − 1) , δ [ m, 1] 21 = n ( mq k − 1 − 1) − 1 , . . . , δ [ m, 1] 2 n = n ( mq k − 1 − 2) , . . . , δ [ m, 1] pµ = mnq k − 1 − ( p − 1) n − µ, . . . , δ [ m, 1] mn = mn ( q k − 1 − 1) = mn 2 q k − 2 ) , δ [ m, 2] = ( δ [ m, 2] 1 , 1 , 1 = mn 2 q k − 2 − 1 , . . . , δ [ m, 2] pµ 1 µ 2 = mn 2 q k − 2 − ( p − 1) n 2 − ( µ 1 − 1) n − µ 2 , . . . , δ [ m, 2] mnn = mn 3 q k − 3 ) , . . . Finally , δ [ m,k ] 111 ... 1 = mn k − 1 and δ [ m,k ] mnn...n = 0. Hence, δ [ m,k ] realizes a partition s uc h that a δ [ m,k ] = det  a δ [ m,t ] qν 1 ν 2 ...ν t pµ 1 µ 2 ...µ d  (1 ≤ p,q ≤ m );(0 ≤ t,d ≤ k );(1 ≤ µ i ,ν j ≤ n ) corresp onds to th e V andermond e d eterminan t of the matrix A , s ee T able 3. Th e follo wing statemen t h olds . 14 Prop osition 11. L e t λ [ m,k ] b e a multi-p artition of length l ( λ [ m,k ] ) ≥ m q k such that the ine qual- ities (22 ) ar e strict. Ther e exists a [ m, k ] -p artition ℓ [ m,k ] of length at most mq k such t hat ∀ 1 ≤ p ≤ m, ∀ 0 ≤ d ≤ k , ∀ 1 ≤ µ 1 , µ 2 , . . . , µ d ≤ n, λ [ m,d ] pµ 1 µ 2 ...µ d = ℓ [ m,d ] pµ 1 µ 2 ...µ d + ( mq k − ( p − 1) n d − d X l =1 ( µ l − 1) n d − l + 1 !) (25) Pro of. One h as to consider th e previous construction of δ [ m,k ] whose le ngth is mq k − 1. Eac h of the sub-partitions of δ [ m,k ] , namely the δ [ m,d ] , has a length less th an or equal to the length of the sub-partition λ [ m,d ] of λ [ m,k ] . Indeed, if the inequalities (22) are strict, for an y 0 ≤ d ≤ k , 1 ≤ p ≤ m , ℓ [ m,d ] pµ 1 µ 2 ...µ d = λ [ m,d ] pµ 1 µ 2 ...µ d − δ [ m,d ] pµ 1 µ 2 ...µ d ≥ 0 , that allo ws to defin e the [ m, k ]-p artition ℓ [ m,k ] whose length is at most mq k .  W e write λ [ m,k ] = δ [ m,k ] + ℓ [ m,k ] . In an ob vious manner, a λ [ m,k ] is d ivisible b y the V ander- monde determinan t a δ [ m,k ] . Th e quotien t S ℓ [ m,k ] = a λ [ m,k ] 6 a δ [ m,k ] is of course a symmetric p olynomial, homogeneous of degree | ℓ [ m,k ] | . F or the sake of simplicit y , let us set | ℓ | := | ℓ [ m,k ] | . P assin g to m + 1 (resp . k + 1), w e se t a λ [ m +1 ,k ] 6 a δ [ m +1 ,k ] = S ℓ [ m +1 ,k ] , (resp. a λ [ m,k +1] 6 a δ [ m,k +1] = S ℓ [ m,k +1] ). Un der th e horizon tal (resp. ve r tical) pro jection h | ℓ | m +1 ,k (resp. v | ℓ | m,k +1 ), w e ha ve S ℓ [ m +1 ,k ] → S ℓ [ m,k ] (resp. S ℓ [ m,k +1] → S ℓ [ m,k ] ). T his horizon tal (resp. v ertical) sequence defines a n uniqu e horizon tal (resp. v ertical) inv erse limit S ℓ [ .,k ] ∈ Λ .,k (resp. S ℓ [ m,. ] ∈ Λ m,. ) whic h redu ces to S ℓ [ m,k ] setting ∀ p ≥ 1 , ∀ k ≥ 0 , a m + p = 0 , . . . , a ( m + p )[ µ ] k = 0 (resp . ∀ m ≥ 0 , ∀ p ≥ 1 , a m [ µ ] k + p = 0), for any mq k ≥ l ( ℓ [ m,k ] ). W e call S ℓ [ .,k ] (resp. S ℓ [ m,. ] ) the horizonta l (resp. v ertical) Sc hur fun ction corresp onding to ℓ . T aking th e inv erse limit with resp ect to the other index, one reco vers the usual Sc hur f u nction corresp onding to ℓ . 5 Concluding r emarks In this paper, we ha ve extended t he symmetric functions to the multi-indicial sym m etric func- tions. The multi-indicial s y m metric fu nctions ca n b e view ed as the elemen ts of the un iv ersal ob jects in the sense of the inv erse limits of the c atego ries C m ; k (14)-(17). By the construction of the corresp onding partitio n , called m ulti-partition, w e ha ve defined the m u lti-indicial Sc hur functions. F urther prop erties of multi-indicial symm etric functions as well as th e relation to the symmetric functions P λ ( q , t ) [1] w ill b e discussed in the forthcoming pap er [10]. Ac kno wledgmen ts M.N.H. thanks the National Institute for Theoretical Ph ysics (NITheP) a n d its Director Prof. F rederik G. Sc holtz for hospitalit y during a p leasan t stay in Stellen b osch where a part of this w ork h as b een finalized. This wo r k w as sup p orted un der a gran t of the National Researc h F oundation of S outh Africa and by th e ICTP through the O EA-ICMP A-Prj-15. The ICMP A is in p artnership with th e Daniel Iagolnitzer F oundation (DIF), F rance. 15 References [1] I. G. Mcdonald, Symmetric F unctions and Ortho gonal P olynomials , De an Jac queline B. L ewis M e morial L e ctur es , Univ er s it y Lecture Series , V ol. 12, Ru tgers Univ ersity , American Mathematica l So ciet y , 1998. [2] I. Gelfand, D . Krob, A. Lascoux, B. Leclerc, V. S . R etakh and J . Y. T h ib on, Nonc om- mutative Symmetric F unctions , Adv. in Math. 112 2 (1995), 218–348. [3] I. Gelfand, S. Gelfand, V. Retakh and R. Lee Wilson, Quasideterminants , to ap p ear in Adv. in Math. 193 1 (2005) , 56–14 1; e-prin t arX iv: math.QA/ 0208146 . [4] P . A. McM ahon, Combinatory Ana lysis , Cambridge Universit y Press 1915 , 1916; Chelsea reprint 1960. [5] I. M. Gessel, Enumer ative A pplic ations of Symmetric F u nctions , Actes 17 e Sem. Lothar. Com b in . (1987), 5–21. [6] J. Dalb ec, Multisymmetric functions , Beitrage Algebra Geom. 40 1 (1999), 27–51. [7] F. V accarino, The ring of multisymmetric fu nctions , Ann. Inst. F ourier 55 3 (2005), 717–7 31. [8] M. F esc hbac h, The mo d 2 c ohomo lo gy rings of the symmetric gr oup and invariants , T op ology 41 (2002), 57–84. [9] S. Lang, Algebr a , 2 nd Ed., Ad dison W esley P u blishing C ie inc., Y ale Un iv. New Ha ven, Connecticut, U.S.A., 1984. [10] J. Ben Geloun and M. N. Hounko n nou, in pr o gr ess . App endix: Examples of m ulti-indicial Sc h u r p olynomials with n = 2 v an mk denotes the V andermonde determinan t and spol y mk the Sc hur p olynomial associated with ℓ [ m,k ] . Example 1: m = 1 , k = 1 ℓ [1 , 1] =  ℓ [1 , 0] =  3  , ℓ [1 , 1] =  2 , 1   ; v an 11 = ( Y 1 , 1 − Y 1 , 2 ) ( X 1 − Y 1 , 2 ) ( X 1 − Y 1 , 1 ) ; spol y 11 = Y 1 , 1 X 1 Y 1 , 2  X 1 2 + Y 1 , 2 X 1 + Y 1 , 1 X 1 + Y 1 , 2 2 + Y 1 , 1 2 + Y 1 , 2 Y 1 , 1  . Example 2: m = 2 , k = 1 ℓ [2 , 1] =  ℓ [2 , 0] =  3 , 2  , ℓ [2 , 1] =  2 , 1 , 1 , 1   ; v an 21 = ( − Y 2 , 1 + Y 1 , 1 ) ( − Y 2 , 1 + X 2 ) ( X 2 − Y 1 , 1 ) ( − Y 2 , 1 + X 1 ) ( X 1 − Y 1 , 1 ) ( X 1 − X 2 ) × ( − Y 2 , 1 + Y 1 , 2 ) ( Y 1 , 1 − Y 1 , 2 ) ( − Y 1 , 2 + X 2 ) ( X 1 − Y 1 , 2 ) ( Y 2 , 1 − Y 2 , 2 ) × ( − Y 2 , 2 + Y 1 , 1 ) ( − Y 2 , 2 + X 2 ) ( − Y 2 , 2 + X 1 ) ( − Y 2 , 2 + Y 1 , 2 ) ; spol y 21 = X 1 X 2 Y 1 , 1 Y 1 , 2 Y 2 , 1 Y 2 , 2 × 16 ( 3 ( Y 2 , 1 X 2 Y 1 , 2 Y 2 , 2 + Y 2 , 1 Y 1 , 2 X 1 Y 1 , 1 + Y 2 , 1 Y 1 , 2 X 1 X 2 + Y 1 , 2 X 1 X 2 Y 1 , 1 + Y 1 , 2 X 2 Y 1 , 1 Y 2 , 2 + X 1 X 2 Y 1 , 1 Y 2 , 2 + Y 1 , 2 X 1 Y 1 , 1 Y 2 , 2 + Y 2 , 1 Y 1 , 2 X 1 Y 2 , 2 + Y 2 , 1 X 2 Y 1 , 2 Y 1 , 1 + Y 2 , 1 Y 1 , 1 Y 1 , 2 Y 2 , 2 + Y 2 , 1 X 1 Y 1 , 1 Y 2 , 2 + Y 2 , 1 X 2 Y 1 , 1 Y 2 , 2 + Y 2 , 1 X 1 X 2 Y 2 , 2 + Y 2 , 1 X 1 X 2 Y 1 , 1 + Y 1 , 2 X 1 X 2 Y 2 , 2 ) + Y 1 , 2 X 1 2 X 2 + Y 1 , 2 2 Y 1 , 1 Y 2 , 2 + Y 1 , 2 2 X 2 Y 2 , 2 + Y 1 , 2 2 X 2 Y 1 , 1 + Y 1 , 2 2 X 1 Y 2 , 2 + Y 1 , 2 2 X 1 Y 1 , 1 + Y 1 , 2 2 X 1 X 2 + Y 1 , 2 Y 2 , 2 2 Y 1 , 1 + Y 1 , 2 Y 1 , 1 2 Y 2 , 2 + Y 1 , 2 Y 2 , 2 2 X 2 + Y 1 , 2 Y 1 , 1 2 X 2 + Y 1 , 2 X 2 2 Y 2 , 2 + Y 1 , 2 X 2 2 Y 1 , 1 + Y 1 , 2 X 1 Y 2 , 2 2 + Y 1 , 2 X 1 Y 1 , 1 2 + Y 1 , 2 X 1 X 2 2 + Y 1 , 2 X 1 2 Y 2 , 2 + Y 1 , 2 X 1 2 Y 1 , 1 + Y 2 , 1 2 Y 1 , 1 Y 2 , 2 + Y 2 , 1 2 X 1 X 2 + Y 2 , 1 2 X 1 Y 2 , 2 + Y 2 , 1 2 X 2 Y 2 , 2 + Y 2 , 1 2 X 1 Y 1 , 1 + Y 2 , 1 2 X 2 Y 1 , 1 + Y 2 , 1 2 Y 1 , 2 Y 1 , 1 + Y 2 , 1 2 Y 1 , 2 X 2 + Y 2 , 1 2 Y 1 , 2 Y 2 , 2 + Y 2 , 1 Y 1 , 2 2 Y 2 , 2 + Y 2 , 1 Y 1 , 2 2 Y 1 , 1 + Y 2 , 1 Y 1 , 2 2 X 2 + Y 2 , 1 Y 1 , 1 2 X 2 + Y 2 , 1 Y 2 , 2 2 X 2 + Y 2 , 1 Y 1 , 1 2 Y 2 , 2 + Y 2 , 1 Y 2 , 2 2 Y 1 , 1 + Y 2 , 1 X 2 2 Y 1 , 1 + Y 2 , 1 X 2 2 Y 2 , 2 + Y 2 , 1 2 Y 1 , 2 X 1 + Y 2 , 1 Y 1 , 2 X 1 2 + Y 2 , 1 X 1 2 X 2 + Y 2 , 1 X 1 2 Y 1 , 1 + Y 2 , 1 X 1 2 Y 2 , 2 + Y 2 , 1 X 1 X 2 2 + Y 2 , 1 X 1 Y 2 , 2 2 + Y 2 , 1 X 1 Y 1 , 1 2 + Y 2 , 1 Y 1 , 2 X 2 2 + Y 2 , 1 Y 1 , 1 2 Y 1 , 2 + Y 2 , 1 Y 2 , 2 2 Y 1 , 2 + Y 2 , 1 Y 1 , 2 2 X 1 + X 1 2 X 2 Y 1 , 1 + X 1 2 X 2 Y 2 , 2 + X 1 2 Y 1 , 1 Y 2 , 2 + X 1 Y 1 , 1 2 X 2 + X 1 Y 2 , 2 2 X 2 + X 1 Y 1 , 1 2 Y 2 , 2 + X 1 X 2 2 Y 2 , 2 + X 2 2 Y 1 , 1 Y 2 , 2 + X 2 Y 1 , 1 2 Y 2 , 2 + X 1 X 2 2 Y 1 , 1 + X 2 Y 2 , 2 2 Y 1 , 1 + X 1 Y 2 , 2 2 Y 1 , 1 ) . Example 3: m = 1 , k = 2 ℓ [1 , 2] =  ℓ [1 , 0] =  3  , ℓ [1 , 1] =  2 , 1  , ℓ [1 , 2] =  1 , 1 , 1 , 1   ; v an 12 = ( − Z 1 , 2 , 2 + Z 1 , 1 , 2 ) ( X 1 − Z 1 , 1 , 2 ) ( X 1 − Z 1 , 2 , 2 ) ( Z 1 , 1 , 1 − Z 1 , 1 , 2 ) × ( Z 1 , 1 , 1 − Z 1 , 2 , 2 ) ( − Z 1 , 1 , 1 + X 1 ) ( Y 1 , 2 − Z 1 , 1 , 2 ) ( Y 1 , 2 − Z 1 , 2 , 2 ) × ( X 1 − Y 1 , 2 ) ( Y 1 , 2 − Z 1 , 1 , 1 ) ( Y 1 , 1 − Z 1 , 1 , 2 ) ( Y 1 , 1 − Z 1 , 2 , 2 ) × ( X 1 − Y 1 , 1 ) ( Y 1 , 1 − Z 1 , 1 , 1 ) ( Y 1 , 1 − Y 1 , 2 ) ( Z 1 , 1 , 2 − Z 1 , 2 , 1 ) × ( Z 1 , 2 , 1 − Z 1 , 2 , 2 ) ( − Z 1 , 2 , 1 + X 1 ) ( − Z 1 , 2 , 1 + Z 1 , 1 , 1 ) ( − Z 1 , 2 , 1 + Y 1 , 2 ) × ( − Z 1 , 2 , 1 + Y 1 , 1 ) ; spol y 12 = Y 1 , 1 Y 1 , 2 Z 1 , 1 , 1 Z 1 , 1 , 2 Z 1 , 2 , 1 Z 1 , 2 , 2 X 1 × ( X 1 2 Y 1 , 1 + X 1 2 Y 1 , 2 + Y 1 , 1 2 X 1 + Y 1 , 1 2 Y 1 , 2 + Y 1 , 2 2 X 1 + Y 1 , 2 2 Y 1 , 1 + Z 1 , 2 , 1 2 Z 1 , 1 , 1 + Z 1 , 2 , 1 2 Z 1 , 2 , 2 + Z 1 , 2 , 2 2 Z 1 , 1 , 1 + Z 1 , 1 , 1 2 Z 1 , 2 , 2 + Y 1 , 2 Z 1 , 2 , 1 2 + Y 1 , 2 Z 1 , 2 , 2 2 + Y 1 , 2 Z 1 , 1 , 2 2 + Y 1 , 2 2 Z 1 , 2 , 1 + Y 1 , 2 2 Z 1 , 2 , 2 + Y 1 , 2 2 Z 1 , 1 , 1 + Y 1 , 2 2 Z 1 , 1 , 2 + Z 1 , 1 , 2 Z 1 , 2 , 1 2 + Z 1 , 1 , 2 Z 1 , 2 , 2 2 + Z 1 , 1 , 2 Z 1 , 1 , 1 2 + Y 1 , 2 Z 1 , 1 , 1 2 + Z 1 , 1 , 2 2 Z 1 , 2 , 1 + Z 1 , 1 , 2 2 Z 1 , 1 , 1 + Z 1 , 2 , 2 2 Z 1 , 2 , 1 + Y 1 , 1 Z 1 , 2 , 2 2 + Y 1 , 1 Z 1 , 1 , 1 2 + Y 1 , 1 Z 1 , 1 , 2 2 + Z 1 , 1 , 2 2 Z 1 , 2 , 2 + Y 1 , 1 2 Z 1 , 1 , 1 + Y 1 , 1 2 Z 1 , 1 , 2 + Y 1 , 1 2 Z 1 , 2 , 2 + Y 1 , 1 2 Z 1 , 2 , 1 + X 1 Z 1 , 1 , 2 2 + X 1 Z 1 , 1 , 1 2 + X 1 Z 1 , 2 , 2 2 + Y 1 , 1 Z 1 , 2 , 1 2 + X 1 Z 1 , 2 , 1 2 + X 1 2 Z 1 , 1 , 2 + X 1 2 Z 1 , 1 , 1 + X 1 2 Z 1 , 2 , 2 + X 1 2 Z 1 , 2 , 1 + Z 1 , 1 , 1 2 Z 1 , 2 , 1 +2 ( Y 1 , 1 Y 1 , 2 Z 1 , 1 , 1 + X 1 Y 1 , 1 Z 1 , 2 , 1 + X 1 Y 1 , 1 Z 1 , 2 , 2 + X 1 Y 1 , 1 Z 1 , 1 , 1 + X 1 Y 1 , 1 Z 1 , 1 , 2 + Y 1 , 1 Y 1 , 2 Z 1 , 1 , 2 + Y 1 , 1 Y 1 , 2 Z 1 , 2 , 1 + Y 1 , 1 Y 1 , 2 Z 1 , 2 , 2 + X 1 Y 1 , 2 Z 1 , 2 , 2 + X 1 Y 1 , 2 Z 1 , 1 , 1 + X 1 Y 1 , 2 Z 1 , 1 , 2 + X 1 Y 1 , 2 Z 1 , 2 , 1 + X 1 Y 1 , 1 Y 1 , 2 + X 1 Z 1 , 2 , 1 Z 1 , 2 , 2 + X 1 Z 1 , 1 , 1 Z 1 , 2 , 1 + X 1 Z 1 , 1 , 2 Z 1 , 2 , 2 + X 1 Z 1 , 1 , 1 Z 1 , 2 , 2 + Z 1 , 1 , 2 Z 1 , 2 , 1 Z 1 , 2 , 2 + Z 1 , 1 , 2 Z 1 , 1 , 1 Z 1 , 2 , 1 + Z 1 , 1 , 2 Z 1 , 1 , 1 Z 1 , 2 , 2 + Y 1 , 1 Z 1 , 2 , 1 Z 1 , 2 , 2 + Y 1 , 1 Z 1 , 1 , 1 Z 1 , 2 , 1 + Y 1 , 1 Z 1 , 1 , 1 Z 1 , 2 , 2 + Y 1 , 1 Z 1 , 1 , 2 Z 1 , 2 , 2 17 + Y 1 , 1 Z 1 , 1 , 2 Z 1 , 2 , 1 + Y 1 , 1 Z 1 , 1 , 2 Z 1 , 1 , 1 + X 1 Z 1 , 1 , 2 Z 1 , 2 , 1 + X 1 Z 1 , 1 , 2 Z 1 , 1 , 1 + Z 1 , 1 , 1 Z 1 , 2 , 1 Z 1 , 2 , 2 + Y 1 , 2 Z 1 , 1 , 1 Z 1 , 2 , 1 + Y 1 , 2 Z 1 , 2 , 1 Z 1 , 2 , 2 + Y 1 , 2 Z 1 , 1 , 2 Z 1 , 1 , 1 + Y 1 , 2 Z 1 , 1 , 2 Z 1 , 2 , 1 + Y 1 , 2 Z 1 , 1 , 2 Z 1 , 2 , 2 + Y 1 , 2 Z 1 , 1 , 1 Z 1 , 2 , 2 ) ) . 18 T able 1: Data of indeterminates relativ e to m u lti-indicial symmetric fu nctions. n a { 1 ≤ p ≤ m, [ µ ] 1 ≤ i ≤ k } o − → Adding m + 1 ← − h -pro jection on Λ m,k D 0 ( m ) a 1 a 2 · · · a m - a m +1 D 1 ( m ) a 1 µ a 2 µ . . . a mµ 1 ≤ µ ≤ n a ( m +1) µ D 2 ( m ) a 1 µ 1 µ 2 a 2 µ 1 µ 2 . . . a mµ 1 µ 2 1 ≤ µ 1 , µ 2 ≤ n a ( m +1) µ 1 µ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . D k ( m ) a 1 µ 1 µ 2 ...µ k a 2 µ 1 µ 2 ...µ k . . . a mµ 1 µ 2 ...µ k 1 ≤ µ 1 , µ 2 , . . . , µ k ≤ n a ( m +1) µ 1 µ 2 ...µ k v -Pro jection ↑ - D ( k ) 1 D ( k ) 2 . . . D ( k ) m - D ( k ) m +1 on Λ m,k Adding k + 1 ↓ D k +1 ( m ) a 1 µ 1 µ 2 ...µ k µ k +1 a 2 µ 1 µ 2 ...µ k µ k +1 . . . a mµ 1 µ 2 ...µ k µ k +1 1 ≤ µ 1 , µ 2 , . . . , a ( m +1) µ 1 µ 2 ...µ k µ k +1 µ k , µ k +1 ≤ n 19 T able 2: Commutativ e diagram of general in verse limits Λ H m,. · · · − → Λ m,. Λ Λ · · · > Λ m + p,. φ ( m + p,m ) · · · − → Λ m,. h m − → Λ m − 1 ,. Λ m − 1 ,. . . . ∨ . . . . . . V k . . . ↓ Λ ., ( k + q ) · · · − → Λ m + p,k + q · · · − → Λ m,k + q h m,k + q − → Λ m − 1 ,k + q . . . ↓ V m − 1 ,k ψ ( k + q ,k ) . . . ↓ ψ m, ( k + q ,k ) . . . ↓ . . . ↓ ψ m − 1 , ( k + q , k ) Λ .,k Λ .,k · · · − → Λ m + p,k · · · − → Λ m,k h m,k − → Λ m − 1 ,k Λ m − 1 ,k v k ↓ v m,k ↓ ↓ v m − 1 ,k Λ .,k − 1 · · · − → Λ m + p,k − 1 · · · − → Λ m,k − 1 h m,k − 1 − → Λ m − 1 ,k − 1 Λ .,k − 1 ···− → H m,k − 1 Λ m,k − 1 20 T able 3: Matrix A generating the [ m, k ] V anderm on d e determin ant. f ( p, 0) = p ; 1 ≤ p ≤ m ; f ( p, [ µ ] t ) ≡ f ( p, µ 1 , µ 2 , . . . , µ t ) = mq t − 1 + ( q − 1) n t + P t l =1 ( ν l − 1) n t − l + ν q , t ≥ 1 , 1 ≤ ν l ≤ n, q k = n k +1 − 1 n − 1 , n 6 = 1; q k = k + 1 , n = 1 . Ro w Column Column Column 1 ≤ q ≤ m f ( q , [ µ ] t ) f ( m, n . . . n ) = mq k f (1 , 0) = 1 : a δ [ m, 0] 1 1 , a δ [ m, 0] 2 1 , . . . , a δ [ m, 0] m 1 , a δ [ m, 1] 11 1 , . . . , a δ [ m, 1] 1 n 1 , a δ [ m, 1] 21 1 , . . . , a δ [ m, 1] 2 n 1 , . . . , a δ [ m,t ] qν 1 ν 2 ...ν t 1 , . . . , a δ [ m,k ] mnn...n 1 . . . f ( p, 0) = p : a δ [ m, 0] 1 p , a δ [ m, 0] 2 p , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t p , . . . , a δ [ m,k ] mnn...n p . . . f ( m, 0) = m : a δ [ m, 0] 1 m , a δ [ m, 0] 2 m , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t m , . . . , a δ m,k mnn...n m f ( m, 1 , 0 , . . . ) = m + 1: a δ [ m, 0] 1 11 , a δ [ m, 0] 2 11 , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t 11 , . . . , a δ [ m,k ] mnn...n 11 . . . f ( m, µ, 0 , . . . ) = m + µ : a δ [ m, 0] 1 1 µ , a δ [ m, 0] 2 1 µ , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t 1 µ , . . . , a δ [ m,k ] mnn...n 1 µ . . . f ( p, [ µ ] 1 ) = ( p − 1) n + µ + m : a δ [ m, 0] 1 pµ , a δ [ m, 0] 2 pµ , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t pµ , . . . , a δ [ m,k ] mnn...n pµ . . . f ( p, [ µ ] 2 ): a δ [ m, 0] 1 pµ 1 µ 2 , a δ [ m, 0] 2 pµ 1 µ 2 , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t pµ 1 µ 2 , . . . , a δ [ m,k ] mnn...n pµ 1 µ 2 . . . f ( p, [ µ ] d ): a δ [ m, 0] 1 pµ 1 µ 2 ...µ d , a δ [ m, 0] 2 pµ 1 µ 2 ...µ d , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t pµ 1 µ 2 ...µ d , . . . , a δ [ m,k ] mnn...n pµ 1 µ 2 ...µ d . . . f ( m, [ µ ] k ): a δ [ m, 0] 1 mµ 1 µ 2 ...µ k , a δ [ m, 0] 2 mµ 1 µ 2 ...µ k , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t mµ 1 µ 2 ...µ k , . . . , a δ [ m,k ] mnn...n mµ 1 µ 2 ...µ k . . . f ( m, n . . . n ) = mq k : a δ [ m, 0] 1 mnn...n , a δ [ m, 0] 2 mnn...n , . . . , . . . a δ [ m,t ] qν 1 ν 2 ...ν t mnn...n , . . . , a δ [ m,k ] mnn...n mnn...n 21