FPSAC 204, Chcago, USA DMTCS proc. AT, 204, 44 452 Yamanouch topplng - Extended abstract Robert Cor Pasquale Petrullo 2 Domenco Senato 2 LaBRI Unversté Bordeaux, France 2 Unverstà degl Stud della Baslcata, Italy Abstract We study an extenson of the chp-frng game. A gven set of admssble moves, called Yamanouch moves, allows the player to pass from a startng confguraton α to a further confguraton β. Ths can be encoded va an acton of a certan group, the topplng group, assocated wth each connected graph. Ths acton gves rse to a generalzaton of Hall-Lttlewood symmetrc polynomals and a new combnatoral bass for them. Moreover, t provdes a general method to construct all orthogonal systems assocated wth a gven random varable. Résumé. On s ntéresse c à une varante du modèle combnatore du tas de sable. Un ensemble partculer de sutes d éboulements, les éboulements de Yamanouch est défn. Les éléments de cet ensemble permettent de passer d une confguraton à une autre, cec peut être représentépar l acton d un certan groupe, le groupe des éboulements que l on peut assocer à tout graphe connexe. Cette acton donne leu à une généralsaton de polynômes symétrques de Hall-Lttlewood et un nouveau champ combnatore pour ceux-c. Une extenson à la constructon d autres famlles de polynômes orthogonaux est proposée. Keywords: chp-frng game, Yamanouch words, Young tableaux, orthogonal polynomals, Hall-Lttlewood symmetrc polynomals. Introducton In [3] A. Björner, L. Lovász and P. Schor have studed a soltary game called the chp-frng game whch s closely related to the sandple model of Dhar [6], arsng n physcs. In more recent papers some developments around ths game were proposed. Musker [] ntroduced an unexpected relatonshp wth ellptc curves, Norne and Baker [] by means of an analogous game, proposed a Remann-Roch formula for graphs, for whch Cor and Le Borgne [5] presented a purely combnatoral descrpton. An algebrac presentaton of the theory can be found n [2] and [2]. In ths paper we nvestgate ths game, whch we refer to as the topplng game, by exhbtng a wde range of connectons wth classcal orthogonal polynomals and symmetrc functons. We focus our attenton on a noteworthy class of admssble moves, that we have called Yamanouch moves. These moves are bult up startng from sutable elementary moves, denoted by T, T 2,..., T n and called topplngs. A frst crucal fact s that Yamanouch moves are not nvertble, ths provdes a partal order on the set of confguratons and then prncpal order deals may Emal: cor@labr.u-bordeaux.fr Emal:p.petrullo@gmal.com Emal:domenco.senato@unbas.t 365 8050 c 204 Dscrete Mathematcs and Theoretcal Computer Scence (DMTCS), Nancy, France
442 R. Cor, D. Senato, P. Petrullo be consdered. A second nterestng pont s that topplngs can be seen as operators actng on the rng of formal seres Z[[x ±, x± 2,..., x± n ]], and n partcular on the rng of polynomals Z[x, x 2,..., x n ]. Hence, each prncpal order deal H α = {β α Y β}, whch stores all confguratons obtaned from a startng confguraton α by means of a Yamanouch move, may be dentfed wth the seres H α (x) = x β, α Y β where x β = x β xβ2 2 xβn n. Topplngs generate a group G, the topplng group, attached to any graph G. By settng T [] = T T 2 T we obtan algebracally ndependent elements spannng a subalgebra Z[G] of the group algebra Z[G]. We show there exsts an operator τ, wth τ Z[G], whch depends on G but not on α, such that τ x α = H α (x) and τ = n = T []. By settng T [,j] = T [] T [+] T [j ] we obtan a further set of generators of Z[G] yeldng a deformaton of τ and a weghted verson of H α (x). More precsely, we recover and Ĥα(x) = ˆτ x α, and also we obtan ˆτ = <j n Ĥ α (x) = α Y β T [,j] C(λ)x β, where C(λ) counts the number of parwse dstnct decompostons n terms of the generators T [,j] s of the unque element T λ = T λ T λ2 2 Tn λn n the topplng group G such that β = T λ (α). Parameters z, z 2, z 3, q may be ntroduced n order to keep track of certan statstcs l, l 2, l 3, d defned on the set of all decompostons of a gven T λ. Then, the followng seres can be consdered, Ĥ α (x; z z 2 z 3, q) = x β. α Y β T λ =T [,j ]T [2,j 2 ] Such a seres satsfes Ĥα(x; z, z 2, z 3, q) = ˆτ(z, z 2, z 3, q) x α, where ˆτ(z, z 2, z 3, q) = <j n z l zl2 2 zl3 3 qd ( q)t [,j] z 3 z j 2 z (j 2) ( 2). T [,j] z 3 z j 2 z (j 2) ( 2) Snce ˆτ(z, z 2, z 3, q) = ˆτ ( z, z 2, ( q)z 3, q(q ) ), then a combnatoral descrpton of ˆK α (x; z, z 2, z 3, q) = ˆτ(z, z 2, z 3, q) x α s obtaned for free. Moreover, l l 2 l 3 d 0
Yamanouch topplng 443 mples that the coeffcent of x β n ˆK α (x; z, z 2, z 3, q), as well as that n Ĥα(x; z, z 2, z 3, q), s a polynomal wth nteger coeffcents. Then, one can consder a projecton Π such that Πx β = x β f β N n, and Π x β = 0 otherwse. Hence both {Π Ĥα(x; z, z 2, z 3, q)} α and {Π ˆK α (x; z, z 2, z 3, q)} α, wth α rangng over N n, are bases of Z[z, z 2, z 3, q][x]. Our nterest n the topplng game s explaned by focusng the attenton on a very specal famly of graphs {L n } n. In ths case, we prove that the polynomal sequence {Π ˆK α (x; z, z 2, z 3, q)} α, va a sutable lnear functonal of umbral type [4], yelds to a parametrzed extenson of classcal orthogonal polynomals. More concretely, set α = (n, n,..., n ) and let X = X, X 2,..., X n denote..d. random varables. Then, denote by p n (z, z 2, z 2, q; t) the expected value of the polynomal obtaned by replacng x = t and x + = X n ˆK α (x; z, z 2, z 3, q): p n (t; z, z 2, z 3, q) = E ˆK α (t, X,..., X n ; z, z 2, z 3, q). The polynomal sequence {p n (t; z, z 2, z 3, q)} n generalzes the orthogonal polynomal system assocated wth X [4, 3]. Classcal cases, lke Hermte, Laguerre, Posson-Charler, Chebyshev arse for z = z 2 = z 3 = q = when sutable random varables are chosen. Moreover, f we replace x β wth complete homogeneous symmetrc functons h β (x) then an acton of the topplng group on symmetrc functons s obtaned. In ths context, when G = L n the acton of the T [,j] s turns out to be closely related to that of rasng operators [8] and ths leads to a generalzaton of Hall-Lttlewood symmetrc functons [9] {R λ (x; t)} λ. The topplng game not only provdes a new elementary combnatoral bass for symmetrc functons and orthogonal polynomals [0], but t also suggests further generalzatons to analogous envronments defned startng from further famles of graphs {G n } n. 2 The topplng game Here and n the followng, by a graph G = (V, E) we wll mean a connected graph, wth set of vertces V = {v, v 2,..., v n }, and whose set of undrected edges E contans at most one edge {v, v j } for each par (v, v j ). Also, we say that v and v j are neghbours whenever {v, v j } E. A confguraton on G s a map, α: v V α(v ) Z, assocatng each vertex v wth an ntegral weght α(v ). If we set α = α(v ) then we may dentfy any confguraton α wth the array (α, α 2,..., α n ). We set ɛ = (δ, δ 2,..., δ n ) so that ɛ s the confguraton assocatng v wth and v j wth 0 f j. A topplng of the vertex v s a map T G : Zn Z n defned by T G (α) = α d ɛ, + ɛ j () {v j,v } E and d = {vj {v, v j } E} s the degree of v. Roughly speakng, the map T G ncreases by the weght α j of each neghbour v j of v, and smultaneously decreases by d the weght α. As a trval consequence we deduce that the sze α = α + α 2 + + α n of any α Z n s preserved by each topplng T G. One may vsualze topplngs as specal moves of a sutable combnatoral game defned on the graph G. More precsely, fx a startng confguraton α = (α, α 2,..., α n ) on the graph and label each vertex v wth ts own weght α. By frng the vertex v we change the confguraton α nto a new confguraton β = T G (α). A move of the topplng game smply s a fnte sequence of fred vertces v v 2... v l, possbly not all dstnct. Now, let us fx two confguratons α and β on a graph G and let us look for all possble moves changng α nto β. Also, assume that certan moves are forbdden n the game, so that each player s forced to perform only those moves that are n a gven set M of
444 R. Cor, D. Senato, P. Petrullo admssble moves. Henceforth, by wrtng α M β we wll mean that α can be changed nto β by means of a sutable admssble move n M. When useful, M wll be dentfed wth the correspondng set of monomals belongng to the free monod T generated by the alphabet T = {T G, T 2 G,..., T n G }. At ths pont a possble strategy of a player passes through the determnaton of those moves of mnmal length (.e. mnmal number of fred vertces) that are admssble and that change α nto β. Therefore, the topplng game starts once a par (G, M) s chosen, and two confguratons α, β Z n are gven so that α M β. Hence, the dea s to characterze, n an explct way, the subset M α,β of all admssble moves changng α nto β, then to determne all moves n M α,β of mnmal length. In the followng we wll focus our attenton on a specal set of admssble moves, whch we name Yamanouch moves, defned by Y = {T G T G 2 T G l 2... l s Yamanouch}. Recall that a sequence, or word, w = 2... l of postve ntegers s sad to be Yamanouch f and only f, for all k and for all, the number occ(, k), of occurrences of n 2... k, satsfes occ(, k) occ( +, k). If w = 2... l then we set Tw G = T G T G 2 T G l and wrte α w β f and only f β = Tw G (α). Assocated wth each Yamanouch word w = 2... l there s an nteger partton λ(w) = (λ, λ 2,...) whose th part λ equals the number occ(, l) of occurrences of n w. A sutable fllng of the Young dagram of λ(w) yelds a codng of w n terms of a standard Young tableau. More precsely, the Young tableau assocated wth w s the unque tableaux of shape λ(w) whose th row (of length λ ) stores all j such that w j =. Ths fxes a bjecton between the set of all Yamanouch words of l letters and the set of all standard Young tableaux of l boxes. Proposton If α w β and λ = λ(w) then for all w such that λ = λ(w ) we have α w β. If α w β then the set of all Yamanouch words extracted from all standard Young tableaux of shape λ(w) corresponds to moves n Y α,β. On the other hand the converse s not true. Hence, n order to get an explct characterzaton of Y α,β and of those moves n Y α,β of mnmal length we need a slghtly deeper nvestgaton. 3 The topplng group Assume a graph G = (V, E) s gven and wrte T, T 2,..., T n nstead of T G, T 2 G,..., T n G. The topplng group assocated wth G s the group G generated by T, T 2,..., T n. Let denote ts unty. By vrtue of () we have T T j = T j T for all, j, and then G s commutatve. In partcular, ths says that all g G may be expressed n terms of topplngs as a product of type T a = T a T a2 2 T n an, for a sutable array of ntegers a = (a, a 2,..., a n ) Z n. On the other hand, t s also easy to verfy that T T 2 T n (α) = α for all α, and then we have T T 2 T n =. As a consequence, all g G admt a presentaton T a wth a N n. In fact, f g = T a and f a / N n then let k = mn{a, a 2,..., a n } and set b = a k(ɛ +ɛ 2 + +ɛ n ). Then b N n and T b = T a (T T 2 T n ) k = T a = g. The followng lemma characterzes all presentatons of the unty n G. Lemma 2 For all graphs G we have T a = f and only f a = k(ɛ + ɛ 2 + + ɛ n ). A frst consequence of Lemma 2 s that apart from T T j = T j T and T T 2 T n = there are not further relatons satsfed by T, T 2,..., T n. Ths provdes a characterzaton of all dstnct presentatons of any element n the topplng group G. Theorem 3 For all graphs G and for all a, b N n we have T a = T b f and only f there exsts k Z such that b = a + k(ɛ + ɛ 2 + + ɛ n ).
Yamanouch topplng 445 Now, once a N n s chosen, we may set k = mn{a, a 2,..., a n } and defne b = a k(ɛ +ɛ 2 + +ɛ n ). Clearly T a = T b and b s the unque array n N n of mnmal sze wth ths property. In other terms, T b s the unque reduced decomposton of g = T a. Theorem 4 Let (G, Y) be a topplng game. If α Y β then there exsts a unque partton λ = (λ, λ 2,..., λ l ) wth l n parts such that T λ (α) = β. Hence, Y α,β conssts of exactly all Yamanuoch moves assocated wth standard Young tableaux of shape µ = λ + k(ɛ + ɛ + + ɛ n ), for a sutable k N. In partcular, all mnmal moves n Y α,β are the Yamanuoch moves assocated wth standard Young tableaux of shape λ. 4 A partal order on Z n Our next goal s the followng: we want to characterze n an explct way the sets H α = {β Z n α Y β} defned for all α Z n. To ths am, we denote by I n the set of all a = (a, a 2,..., a n ) N n such that a = 0 for some. Note that, for all g G there s exactly one a I n such that g = T a. In partcular, T a s the reduced decomposton of g. Ths gves the followng characterzaton of the group algebra Z[G]. Theorem 5 We have { } Z[G] = c a T a c a Z and the sum nvolves a fnte number of non-zero terms, a I n and n partcular Z[G] = Z[x, x 2,..., x n ] x x 2 x n, wth x x 2 x n denotng the prncpal deal generated by x x 2 x n. We consder not only elements n Z[G] but we also admt formal seres n T, T 2,..., T n. In partcular we set { } Z[[G]] = c a T a c a Z and the sum nvolves a possbly nfnte number of non-zero terms. a I n If we assocate each α Z n wth a monomal x α = x α xα2 2 xαn n then we may translate the acton of G on Z n nto an acton of Z[[G]], and n partcular of Z[G], on the rng of formal seres { Z[[x ±, x± 2,..., x± 2 ]] = c αx α }. cα Z and the sum nvolves a possbly nfnte number of non-zero terms α Z n Ths s done by settng T a x α = x T a (α) for all a N n and for all α Z n. Clearly, knowng the set H α = {β Z n α Y β} s equvalent to knowng the formal seres H α (x) = x β. α Y β In the followng we wll determne a specal element τ Z[[G]] satsfyng τ x α = H α (x). Before dong that we ntroduce a new relaton G on Z n : set β G α f and only f β = T λ (α) wth λ λ 2... λ n 0. Clearly, f λ n 0 then we may defne µ = λ λ n (ɛ + ɛ 2 + + ɛ n ) so that T λ (α) = T µ (β)
446 R. Cor, D. Senato, P. Petrullo and T µ s a reduced decomposton for T λ. Note that µ s an nteger partton wth at most n parts. Henceforth we wll denote by P n the set of all nteger parttons wth at most n parts or, equvalently the subset of all λ s n I n such that λ λ 2... λ n. The followng proposton makes explct the relaton between the relaton G and the topplng game. Proposton 6 For all graphs G the relaton G s a partal order on Z n. Moreover, G and Y agree on Z n. Proposton 6 assures us that the set H α s nothng but the order deal generated by α, that s H α = {β β G α}. Let us consder the followng element n Z[[G]]: τ = λ P n T λ, and let t act on the monomal x α. Beng β = T λ (α) for at most one λ P n, then we recover τ x α = T λ x α = x β = H α (x). λ P n β G α The crucal pont s that a closed expresson of τ s avalable by ntroducng sutable elements of G. More precsely, we defne T [] = T T 2... T for all =, 2,..., n. Note that T [] may be thought of as a Yamanouch move assocated wth a Young standard tableau of one column flled wth, 2,...,. Theorem 7 Let λ P n and denote by λ = (λ, λ 2,..., λ l ) the conjugate of λ. Then we have Corollary 8 We have Now, we may wrte T λ = T [λ ]T [λ 2 ] T [λ l ]. τ = H α (x) = n = n = T []. T [] x α. Moreover, straghtforward computatons wll prove the followng theorem. Theorem 9 For each graph G there exsts a formal seres τ(x) Z[[x ±, x± 2,..., x± n ]] such that H α (x) = τ(x)x α, for all α Z n. 5 A specal set of Yamanouch moves If G s the topplng group assocated wth a graph G then we set { } Z[G] = c λ T λ c λ Z and the sum nvolves a possbly nfnte number of non-zero terms. λ P n
Yamanouch topplng 447 Clearly Z[G] s a subalgebra of Z[G] and Theorem 7 assures us t s exactly the subalgebra generated by T [], T [2],..., T [n ]. Let us consder a wder set of generators of Z[G] defned by T [,j] = T [] T [+] T [j ] for all < j n. Note that each T [,j] s a Yamanouch move assocated wth a tableau whose shape conssts of consecutve ntegers. Obvously T [] = T [,+] so that the T [,j] s generate the whole Z[G]. On the other hand, the presentaton of any g G as a product of these generators s n general not unque. In order to fnd a reduced decomposton of g = T λ, that s a presentaton that nvolves a mnmal number of generators, we wrte g = T [λ ]T [λ 2 ] T [λ m ], then rearrange and assocate so that T λ = (T []T [2] T [j ]) (T [k ](T [k2 ] T [kjk ]), and each product (T [h ](T [h2 ] T [hjh ]) conssts of an ncreasng sequence of consecutve ntegers. Thus, a reduced decomposton of T λ s gven by T λ = T [, j +]T [2, 2j2 +] T [k, kjk +]. Example If λ = (8, 7, 4, 3, 2, 2, ) then λ = (7, 6, 4, 3, 2, 2, 2, ). We recover T λ = T [7] T [6] T [4] T [3] T [2] T [2] T [2] T [] = (T [] T [2] T [3] T [4] )(T [2] )(T [2] )(T [6] T [7] ) = T [,5] T 2 [2,3] T [6,8]. A reduced decomposton of T λ s gven T [,5] T 2 [2,3] T [6,8]. A decomposton T λ = T [,j ]T [2,j 2] T [l,j l ] s sad to be square free f and only f each generator occurs at most once. For nstance, T [,2] T [,3] T [2,5] s square free but T [,2] T [,2] T [2,3] T [2,5] s not square free. Also, note that T [,2] T [,3] T [2,5] = T [,2] T [,2] T [2,3] T [2,5] so that a gven element n Z[G] may have both square free and not square free decompostons. Now, consder the operator ˆτ n Z[[G]] defned by ˆτ = <j n T [,j]. (2) Note that ˆτ s the analogue of τ for the generators T [,j] s. Moreover, we recover ˆτ = λ P n C(λ)T λ, wth C(λ) beng the number of parwse dstnct presentatons of g = T λ n terms of the generators T [,j] s. Fnally, we may wrte Ĥ α (x) = ˆτ x α = C(λ)x β, β G α wth λ denotng the unque element n P n such that β = T λ (α). 6 The graph L n and classcal orthogonal polynomals In ths secton we restrct our attenton to a specal famly of graphs, denoted by {L n } n, and defned by L n = (V n, E n ), wth V n = {v, v 2,..., v n } and E n = {{v, v 2 }, {v 2, v 3 },..., {v n, v n }}, for all n. Now, assume a lnear functonal L: C[t] C s gven. An orthogonal polynomal system assocated wth L s a polynomal sequence {p n (t)} n, wth deg p n = n for all n N, such that L p n (t)p m (t) = 0 f and only f n m. We refer to [4] for a background on ths subject. The topplng game performed on
448 R. Cor, D. Senato, P. Petrullo the L n s gves rse to a nce combnatoral framework for orthogonal polynomal systems. Consder the unque p α (x) such that x α = ˆτ p α (x), or equvalently Observe that ˆτ = <j n p α (x) = ˆτ x α. (3) ( T [,j] ) = ( ) l T [,j ]T [2,j 2] T [l,j l ], where the sum s taken over all square free decompostons n G. Because the sze of a confguraton s preserved by the topplng game, p α (x) s a homogeneous polynomal of degree α. Moreover, we have p α (x) = ( ) l( T [,j ]T [2,j 2] T [l,j l ] x α) = β p α,β x β, so that β ranges over all confguratons that can be obtaned from α by means of a square free Yamanouch move T [,j ]T [2,j 2] T [l,j l ]. Also, the coeffcent p α,β of x β n p α (x) has the followng combnatoral descrpton, p α,β = ( ) l, (4) where l ranges over all lengths of all square free Yamanouch moves changng α nto β. In order to manpulate polynomals wth an arbtrary large number of varables at the same tme we set C[x, x 2,...] = n C[x, x 2,..., x n ]. Now, let E : C[x, x 2,...] C denote a map such that. for all n the restrcton E : C[x, x 2,..., x n ] C s a lnear functonal; 2. for all n, for all p C[x, x 2,..., x n ] and for all σ S n, we have Ep(x σ(), x σ(2),..., x σ(n) ) = Ep(x, x 2,..., x n ). Henceforth, we wll call any functonal of ths type a symmetrc functonal. Once a symmetrc functonal E s gven then we may defne condtonal operators E : C[x, x 2,...] C[x ] for all. Each E s defned by E x α = x α Ex α x α for all α N n. Roughly speakng, each E acts on C[x,..., x, x +,...] as E acts, and fxes each polynomal n C[x ]. Remark If X, X 2,... s an nfnte sequence of..d. random varables, and f E s the expectaton functonal, then E : C[X, X 2,...] C gves an example of symmetrc functonal. Set G = L n and choose α Z n. In ths case t s not too dffcult to see that T [] x α = x α x + x all =, 2,..., n. Hence, by means of straghtforward computatons we recover ˆτ x α = <j n ( x j x ) x α. If p n (x) s defned as n (3) wth α = (n, n,..., n ), then we recover p n (x) = x 2 x 2 3 x n n (x x j ). <j n for
Yamanouch topplng 449 Thus, f E : C[x, x 2,...] C[x ] s a condtonal operator assocated wth a symmetrc functonal E : C[x, x 2,...] C, then we have n p n (x ) = E p n (x) = x k E x 2 x 2 3 x n n e n k (x 2, x 3,..., x n ) k=0 <j n (x x j ), wth e (x 2, x 3,..., x n ) denotng the th elementary symmetrc polynomal n x 2, x 3,..., x n. Ths mples that p n (x ) s a polynomal n x of degree at most n. In partcular, we have deg p n = n f and only f Ex 2 x 2 3 x n n (x x j ) 0 for all n. (5) 2 <j n On the other hand, snce E s symmetrc then we may replace each x wth x n (5) wthout changng ts value. Then we obtan Ex x 2 2 x n n (x x j ) = E (ˆτ (x x 2 x n ) n) 0. <j n Ths last condton, f satsfed for all n, wll assures deg p n = n. Theorem 0 (Orthogonal polynomal systems) Let E : C[x, x 2,...] C be a symmetrc functonal, set G = L n, and assume E ˆτ (x x 2 x n ) n 0 for all n. Then, the polynomal sequence {p n (t)} n defned by p n (x ) = E ˆτ (x x 2 x n ) n 0 for all n, satsfes E p n (x )p m (x ) = 0 f and only n m. Theorem 0 gves us a general method to construct all orthogonal systems assocated wth a gven lnear functonal L, provded they exst. In fact, gven L: C[t] C, we consder the unque symmetrc functonal E : C[x, x 2,...] C such that E x k = L t k, for all and for all k 0. Also, we set a α = E x α for all α Z n. Hence, an orthogonal polynomal system {p n (t)} n assocated wth L exsts f and only f E ˆτ (x x 2 x n ) n 0 for all n. Moreover, t s obtaned by settng p n (x ) = E ˆτ (x x 2 x n ) n. So, the followng combnatoral formula for p n (t) s provded: p n (t) = β p α,β a β2,...,β n t β, where α = (n, n,..., n ), β ranges over all confguratons that can be obtaned from α by means of square free Yamanouch move, and p α,β s the coeffcent (4). 7 Statstcs on the topplng group In ths secton we focus our attenton on the dstrbuton of certan statstcs l, l 2, l 3, d defned on the decompostons of elements n the topplng group n terms of the three famles of generators {T =, 2,..., n}, {T [] =, 2,..., n } and {T [,j] < j n}. More precsely, each T λ G,
450 R. Cor, D. Senato, P. Petrullo λ P n, admts a unque expresson, up to order of the T s, of type g = T λ. Ths expresson nvolves l generators, and n partcular l equals the sze of the partton λ. Analogously, T λ can be wrtten n a unque way, up to order, n terms of the T [] s. In partcular, we have T λ = T [λ ]T [λ 2 ] and ths nvolves l 2 = λ generators. On the other hand, there are C(λ) ways of wrtng T λ as a product of generators T [,j] s. Each of such expressons nvolves a certan number, say l 3, of generators. Among these d l 3 are parwse dstnct. Theorem Set ˆτ(z, z 2, z 3, q) = <j n ( q)t [,j] z 3 z j 2 z (j 2) ( 2). (6) T [,j] z 3 z j 2 z (j 2) ( 2) Then we have ˆτ(z, z 2, z 3, t) = z l zl2 2 zl3 3 qd T [,j ]T [2,j 2], where the sum ranges over all parwse dstnct decompostons T [,j ]T [2,j 2] n Z[G]. We may defne a parametrzed verson of the seres Ĥα(x) by settng Ĥ α (x; z, z 2, z 3, q) = ˆτ(z, z 2, z 3, q) x α. (7) We recover Ĥ α (x; z, z 2, z 3, q) = α Y β ( z l zl2 2 zl3 3 qd )x β, where the coeffcent of x β stores the value of l, l 2, l 3, d relatve to all parwse dstnct decompostons of the unque T λ such that β = T λ (α). As a consequence we obtan a parameterzed extenson of orthogonal polynomals. In fact, for all n and for all α Z n set ˆK α (x; z, z 2, z 3, q) = ˆτ(z, z 2, z 3, q) x α. Snce ˆτ(z, z 2, z 3, q) = ˆτ ( z, z 2, ( q)z 3, q(q ) ) then the followng combnatoral descrpton s obtaned: ˆK α (x; z, z 2, z 3, q) = ( z l zl2 2 zl3 3 ( q)l3 d ( q) d) x β. α Y β Snce l l 2 l 3 d 0 then the coeffcent of x β n ˆK α (x; z, z 2, z 3, q) s a polynomal wth nteger coeffcents. By settng z = z 2 = z 3 = q = we have a non-zero contrbuton only for decompostons such that l 3 d = 0, that s square free decompostons. At ths pont, one can consder a projecton Π such that { Πx β x β f β N n = 0 otherwse, and then both {Π Ĥα(x; z, z 2, z 3, q)} α and {Π ˆK α (x; z, z 2, z 3, q)} α, wth α rangng over N n, are bases of Z[z, z 2, z 3, q][x]. By defnng a sutable symmetrc functonal E, and by settng p n (x ; z, z 2, z 3, q) = E ˆτ(z, z 2, z 3, q) (x x 2 x n ) n we obtan a polynomal sequence {p n (t; z, z 2, z 3, q)} n that generalzes classcal orthogonal polynomals.
Yamanouch topplng 45 Example 2 Hermte polynomals {H n (t)} n forms an orthogonal polynomal system wth respect to the lnear functonal L defned by L t k = E X k, where X s a Gaussan random varable. Assume X = X, X 2,... are ndependent Gaussan random varables and set x = t and x = X. Then the polynomal sequence {p n (t; z, z 2, z 3, q)} n defned by p n (t; z, z 2, z 3, q) = EΠĤα ( t, X, X 2,..., X n ; z, z 2, ( q)z 3, wth α = (n, n,..., n ) generalzes Hermte polynomals. 8 New combnatorcs for H-L symmetrc polynomals ) q, q The topplng group G acts on the rng Λ(x; z, z 2, z 3, q), of symmetrc polynomals n x, x 2,..., x n wth coeffcents n Z[z, z 2, z 3, q]. In fact, let h (x) denote the th complete homogeneous symmetrc polynomal, and for all α N n set h α (x) = h α (x)h α2 (x) h αn (x). By assumng h (x) = 0 for < 0 and h 0 (x) =, we set T h α (x) = h T(α)(x). (8) Fnally, f α s a partton and f s α (x) s a Schur polynomal then the symmetrc polynomals Ĥα(x; z, z 2, z 3, q) = ˆτ(z, z 2, z 3, q) s α (x) and ˆK α (x; z, z 2, z 3, q) = ˆτ (z, z 2, z 3, q) s α (x) may be consdered. Theorem 2 For any graph G the sets {Ĥα(x; z, z 2, z 3, t)} α and { ˆK α (x; z, z 2, z 3, t)} α, wth α ragng over all nteger parttons, are bases of the rng Λ(x; z, z 2, z 3, q). Agan, the specal case G = L n leads to some nterestng consequences. In fact, we recover T [,j] h α (x) = h α ɛ+ɛ j (x), so that T [,j] acts as a lowerng operator R j (see [9], p.24). In partcular ths mples Ĥ α (x; z, z 2, z 3, q) = ( q)z 3z j 2 z 2) ( 2) (j R j s <j n z 3 z j 2 z 2) ( 2) (j α (x). R j and ˆK α (x; z, z 2, z 3, q) = z 3z j 2 z 2) ( 2) (j R j s <j n ( q)z 3 z j 2 z 2) ( 2) (j α (x). R j Now, recall that the Hall-Lttlewood symmetrc polynomal R α (x; t), α beng a partton, satsfes R α (x; t) = ( tr j ) s α. Ths mples <j n ( ) R α (x; t) = ˆK q α (x;,, t, ) = lm Ĥ α x;,, ( q)t,. q q Remark 2 Hence, ˆK α (x; z, z 2, z 3, q) extends to four parameters Hall-Lttlewood symmetrc polynomals, and then several classcal bases of symmetrc functons such as Schur functons (t = 0). From ths pont of vew they share some analogy wth Macdonald symmetrc polynomals. Moreover, the topplng game also yelds several famles of symmetrc functons, each one attached to a graph G and then a general theory extendng the classcal case (G = L n ) could be carred out.
452 R. Cor, D. Senato, P. Petrullo Remark 3 Lowerng operators R j s have a counterpart R j s called rasng operators [8, 9]. In our settng they wll arse as generators, say the T [j,] s, of the subalgebra Z[G] spanned over Z by all T λ, wth 0 λ λ 2... λ n. References [] M. Baker, S. Norne, Remann-Roch and Abel-Jacob theory on a fnte graph. Advances n Mathematcs 25, 766 788 (2007). [2] N. Bggs Chp-frng and the crtcal group of a graph J. of Algebrac Comb. 9, 25 45 (999). [3] A. Björner, L. Lovász, P. Schor, Chp-frng game on graphs. Europ. J.of Comb. 2, 283 29 (99). [4] T.S. Chahara, An ntroducton to Orthogonal polynomals. Mathematcs and ts applcatons - A seres of monographs and texts 3, (978). [5] R. Cor, Y. Le Borgne, The Remann-Roch theorem for graphs and the rank n complete graphs. Preprnt, pp. 35 (203). [6] D. Dhar, Self-organzed crtcal state of the sandple automaton models. Phys. Rev. Lett., 64, 63 66 (990.) [7] D. Dhar, P. Ruelle, S. Sen, and D. Verma. Algebrac aspects of abelan sandple model. J. Phys. A A28, 805 83 (995). [8] A. Garsa, Orthogonalty of Mlne s polynomals and rasng operators. Dscrete Mathematcs 99, 247 264 (992). [9] I.G. Macdonald, Symmetrc functons and Hall polynomals. Oxford Mathematcal Monograph. Oxford Unversty Press (995). [0] I.G. Macdonald, Affne Hecke algebras and orthogonal polynomals. Cambrdge Tracts n Mathematcs 57, Cambrdge Unversty Press (2003). [] G. Musker, The crtcal groups of a famly of graphs and ellptc curves over fnte felds. Journal of Algebrac Combnatorcs 30, 255 276 (2009). [2] D. Perknson, J. Perlman, and J. Wlmes. Prmer for the algebrac geometry of sandples. preprnt, arxv:2.663, (20). [3] P. Petrullo, D. Senato, R. Smone, Orthogonal polynomals through the nvarnat theory of bnary forms. Preprnt (204). [4] G.-C. Rota and B. Taylor, The classcal umbral calculus. SIAM Journal of Mathematcal Analyss 25(2), 694 7 (994).