COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS. 1. Introduction

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1 COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS PATRICIA HERSH AND CRISTIAN LENART Abstract. Ths work s part of a project on weght bases for the rreducble representatons of semsmple Le algebras wth respect to whch the representaton matrces of the Chevalley generators are gven by explct formulas. In the case of sl n, the celebrated Gelfand-Tsetln bass s the only such bass known. Usng the setup of supportng graphs developed by Donnelly, we present a smple combnatoral proof of the Gelfand-Tsetln formulas based on a ratonal functon dentty. Some propertes of the Gelfand-Tsetln bass are derved va an algorthm for solvng certan equatons on the lattce of semstandard Young tableaux. 1. Introducton Ths work s related to combnatoral constructons of weght bases for the rreducble representatons of semsmple Le algebras on whch the acton of the Chevalley generators s made explct. We wll use the setup ntroduced by Donnelly [3, 5, 6]. The man dea s to encode a weght bass nto an edge-colored ranked poset (called a supportng graph), whose Hasse dagram has ts edges labeled wth two complex coeffcents. Ths structure s known as a representaton dagram, and t explctly gves the acton of the Chevalley generators of the Le algebra on the weght bass. Verfyng that an assgnment of labels to an edge-colored poset s a representaton dagram amounts to checkng that the labels satsfy some smple relatons. Thus, constructng a bass of a representaton amounts to solvng a system of equatons assocated to a poset. The goal n the bass constructon s fndng supportng graphs wth a small number of edges, possbly edge-mnmal ones (wth respect to ncluson); ths amounts to fndng a bass for whch the acton of the Chevalley generators s expressed by a formula wth a small number of terms. It s often the case that the labels of an edge-mnmal supportng graph are essentally the unque soluton of the correspondng system of equatons. Ths property s known as the soltary property of the assocated bass. Another nterestng property of many supportng graphs constructed so far s that Kashwara s crystal graphs of the correspondng representatons [12, 13] are subgraphs. Thus, the theory of supportng graphs can be vewed as an extenson of the theory of crystal graphs, whch has attracted consderable nterest n the combnatorcs communty n recent years. Fnally, many supportng graphs are better behaved as posets than the correspondng crystal graphs/posets, beng lattces, modular lattces, or even dstrbutve lattces. In the case of rreducble representatons of sl n, the celebrated Gelfand-Tsetln bass s the only known bass wth respect to whch the representaton matrces of the Chevalley generators are gven by explct formulas. It turns out that the supportng graph of the Gelfand-Tsetln bass s edge-mnmal, soltary, and a dstrbutve lattce [6]; we wll call t the Gelfand-Tsetln lattce. Donnelly constructed soltary, edge-mnmal, and modular lattce supportng graphs for certan specal representatons, most notably: the fundamental representatons of sp 2n and so 2n+1 [3, 4, 5], the one-rowed representatons of so 2n+1 [8], and the adjont representatons of all smple Le algebras [7]. Molev constructed bases of Gelfand-Tsetln type (.e., whch are compatble wth restrcton to the Le subalgebras of lower P. H. was partally supported by Natonal Scence Foundaton grant DMS C. L. was partally supported by Natonal Scence Foundaton grants DMS and DMS

2 2 PATRICIA HERSH AND CRISTIAN LENART rank) for all rreducble representatons of the symplectc and orthogonal Le algebras [17, 18, 19]. The correspondng representaton dagrams (.e., the acton of a system of Chevalley generators on the bass) are not explctly gven, but they can be derved from Molev s formulas for the acton of certan elements spannng the Le algebra. As posets, these supportng graphs are not lattces n general, and there are ndcatons that they are not edge-mnmal n general, ether. Our ultmate goal s fndng edge-mnmal supportng graphs for symplectc and orthogonal representatons, as well as studyng ther combnatorcs. As a frst step, n ths paper we revst the Gelfand-Tsetln bass for sl n, by studyng t n Donnelly s combnatoral setup. Ths allows us to show that the constructon of the Gelfand-Tsetln bass reles on nothng more than a smple ratonal functon dentty. Moreover, the correspondng soltary and edge-mnmal propertes, whch were derved va theoretcal consderatons n [6], are proved here n a very explct way, by a smple algorthm for solvng equatons on the Gelfand-Tsetln lattce. We envson that such algorthms and ratonal functon denttes wll play a crucal role n our future work. On the other hand, let us note that the proofs of the Gelfand- Tsetln formulas that appeared snce the orgnal paper [10] by Gelfand and Tsetln n the fftes (whch contaned no proof) use more sophstcated algebrac methods, based on: lowerng operators [21, 22, 23], boson-calculus technques [1], polynomal expressons for Wgner coeffcents [11], the theory of the Mckelsson algebras [24], and the quantum algebras called Yangans [16, 20]. In turn, Molev s constructons [17, 18, 19] of hs bases for orthogonal and symplectc representatons are based on complex calculatons related to Yangans. In terms of the combnatoral model for descrbng the supportng graph, we use semstandard Young tableaux rather than Gelfand-Tsetln patterns. By analogy, we expect to use Kashwara-Nakashma or De Concn tableaux [2, 14] for the representatons of the symplectc and orthogonal algebras. Note that these tableaux were already used n Donnelly s work mentoned above, whereas Molev s work s based on Gelfand-Tsetln patterns of type B D. Acknowledgement. We are grateful to Robert Donnelly for explanng to us hs work on supportng graphs for representatons of semsmple Le algebras. 2. Background 2.1. Supportng graphs. We follow [6, 9] n descrbng the setup of supportng graphs/representaton dagrams. We consder fnte ranked posets, and we dentfy a poset wth ts Hasse dagram, thus vewng t as a drected graph wth edges s t for each coverng relaton s t. These edges wll be colored by a set I, and we wrte s t to ndcate that the correspondng edge has color I. The connected components of the subgraph wth edges colored are called -components. Besdes a gven color, each edge s t s labeled wth two complex coeffcents, whch are not both 0, and whch are denoted by c t,s and d s,t. Gven the poset P, let V [P] be the complex vector space wth bass {v s } s P. We defne operators X and Y on V [P] for n I, as follows: (2.1) X v s := c t,s v t, Y v t := d s,t v s. t : s t s : s t For each vertex s of P, we also defne a set of ntegers {m (s)} I by m (s) := 2ρ (s) l (s), where l (s) s the rank of the -component contanng s, and ρ (s) s the rank of s wthn that component. Let g be a semsmple Le algebra wth Chevalley generators {X, Y, H } I. Let {ω } I and {α } I denote the fundamental weghts and smple roots of the correspondng root system, respectvely. Consder an edge-colored and edge-labeled ranked poset P, as descrbed above. Let us assgn a weght to each vertex by wt(s) := I m (s)ω. We say that the edge-colored poset P satsfes the structure condton for g f wt(s) + α = wt(t) whenever s t. We now defne two condtons on the pars of edge labels (c t,s, d s,t ). We call π s,t := c t,s d s,t an edge product. The edge-labeled poset P satsfes the crossng condton f for any vertex s and any color we

3 COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS 3 have (2.2) π r,s π s,t = m (s). r : r s t : s t A relaton of the above form s called a crossng relaton. The edge-labeled poset P satsfes the damond condton f for any par of vertces (s, t) of the same rank and any par of colors (, j), possbly = j, we have (2.3) c u,s d t,u = d r,s c t,r, u : s j u and t u r : r s and r j t where an empty sum s zero. If for gven pars (s, t) and (, j) there s a unque vertex u such that s j u and t u, as well as a unque vertex r such that r s and r j t, then the relaton (2.3) for these pars and for the reverse pars (t, s) and (j, ) reduce to (2.4) c u,s d t,u = d r,s c t,r, c u,t d s,u = d r,t c s,r ; these relatons mply (2.5) π s,u π t,u = π r,s π r,t. A relaton of the form (2.3), (2.4), or (2.5) s called a damond relaton. We want to defne a representaton of g on V [P] by lettng the Chevalley generators X and Y act as n (2.1), and by settng (2.6) H v s := m (s)v s. The followng proposton gves a necessary and suffcent condton on the edge labels. Proposton 2.1. [9, Lemma 3.1][6, Proposton 3.4] Gven an edge-colored and edge-labeled ranked poset P, the actons (2.1) and (2.6) defne a representaton of g on V [P] f and only f P satsfes the damond, crossng, and structure condtons. If the gven condtons hold, the set {v s } s P s a weght bass of the gven representaton, the edgecolored poset P s called a supportng graph of the representaton, and P together wth ts edge-labels s called a representaton dagram. Some general propertes of supportng graphs were derved n [Secton 3][6]. Many supportng graphs constructed so far have specal propertes, whch we menton below. A supportng graph s called edge-mnmal f no proper subgraph of t s the supportng graph for a weght bass of the correspondng representaton. Two weght bases related by a dagonal transton matrx are called dagonally equvalent. The supportng graph for a weght bass s called soltary f the dagonally equvalent bases are the only ones wth the same supportng graph. Hence, up to dagonal equvalence, a soltary weght bass s unquely determned by ts supportng graph. Two representaton dagrams are called edge product smlar f there s a poset somorphms between them whch preserves the edge colors and the edge products. The followng lemma hghlghts the mportance of edge products. Lemma 2.2. [9, Lemma 4.2] Let L be a representaton dagram for a weght bass B of V whch s connected (as a graph) and modular (as a poset). A representaton dagram K whch s edge product smlar to L s the representaton dagram of a dagonally equvalent bass to B The Gelfand-Tsetln bass. Let E j,, j = 1,...,n denote the standard bass of the general lnear Le algebra gl n over the feld of complex numbers. Consder a partton λ wth at most n rows, that s a weakly decreasng sequence of ntegers (λ 1 λ 2 λ n 0). Let V (λ) be the fnte-dmensonal rreducble representaton of gl n wth hghest

4 4 PATRICIA HERSH AND CRISTIAN LENART weght λ. A bass of V (λ) s parametrzed by Gelfand Tsetln patterns Λ assocated wth λ; these are arrays of nteger row vectors λ n1 λ n2 λ nn (2.7) λ n 1,1 λ n 1,n 1 λ 21 λ 22 such that the upper row concdes wth λ and the followng condtons hold: (2.8) λ k λ k 1,, λ k 1, λ k,+1, = 1,...,k 1 for each k = 2,..., n. Let us set l k = λ k + 1. λ 11 Theorem 2.3. [10] There exsts a bass {ξ Λ } of V (λ) parametrzed by the correspondng patterns Λ such that the acton of generators of gl n s gven by the followng formulas: ( k ) k 1 (2.9) E kk ξ Λ = λ k λ k 1, ξ Λ, (2.10) (2.11) E k,k+1 ξ Λ = E k+1,k ξ Λ = k k (l k l k+1,1 ) (l k l k+1,k+1 ) (l k l k1 ) (l k l kk ) (l k l k 1,1 ) (l k l k 1,k 1 ) (l k l k1 ) (l k l kk ) ξ Λ+δk, ξ Λ δk. The arrays Λ ± δ k are obtaned from Λ by replacng λ k by λ k ± 1. It s supposed that ξ Λ = 0 f the array Λ s not a pattern; the symbol ndcates that the zero factor n the denomnator s skpped. 3. The representaton dagram of the Gelfand-Tsetln bass We restrct ourselves to sl n, for whch we have the standard choce of Chevalley generators H k := E k,k E k+1,k+1, X k := E k,k+1, and Y k := E k+1,k, for k n I := [n 1] = {1,..., n 1}. We dentfy parttons wth Young dagrams, so we refer to the cells (, j) of a partton λ. There s a natural bjecton between the Gelfand-Tsetln patterns assocated wth λ and semstandard Young tableaux (SSYT) of shape λ wth entres n [n], see e.g. [15]. A pattern Λ can be vewed as a sequence of parttons λ (1) λ (2) λ (n) = λ, wth λ (k) = (λ k1,..., λ kk ). We let λ (0) be the empty partton. Condtons (2.8) mean that the skew dagram λ (k) /λ (k 1) s a horzontal strp. The SSYT T assocated wth Λ s then obtaned by fllng the cells n λ (k) /λ (k 1) wth the entry k, for each k = 1,...,n. We now defne a representaton dagram for sl n wth edge colors I on the SSYT of shape λ wth entres n [n]. We have an edge S k T whenever the tableau T s obtaned from S by changng a sngle entry k + 1 nto k; necessarly, ths s the leftmost entry k + 1 n a row. The correspondng poset, whch s known to be a dstrbutve lattce, wll be called the Gelfand-Tsetln lattce, and wll be denoted by GT(λ). To defne the edge labels on GT(λ), fx a SSYT T n ths lattce, and let the correspondng Gelfand- Tsetln pattern Λ be denoted as n (2.7). The labels on the ncomng/outgong edges to/from T whch are colored k wll only depend on the correspondng parttons λ (k 1), λ (k), and λ (k+1). The outer rm R of λ (k) conssts of all cells (, j) not n λ (k) such that at least one of the cells (, j 1), ( 1, j), ( 1, j 1) belongs to λ (k). A cell (, j) of R s called an outer corner f (, j 1) and ( 1, j) belong to R, and an nner corner f nether (, j 1) nor ( 1, j) belongs to R. The nner and outer corners are nterleaved,

5 COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS 5 and the number of the former exceeds by 1 the number of the latter. Number the cells of R from northeast to southwest startng from 1. Let a 1 <... < a p be the numbers attached to the nner corners, and a 1 <... < a p 1 the numbers attached to the outer corners. Furthermore, f r 1 = 1 < r 2 <... < r p are the rows of the nner corners, we denote by b the length (whch mght be 0) of the component of λ (k+1) /λ (k) n row r, for = 1,...,p; smlarly, we denote by b the length of the component of λ (k) /λ (k 1) n row r +1 1, for = 1,...,p 1. Note that row r of T contans both k and k + 1 precsely when r +1 1 = r, b > 0, and b > 0. The notaton ntroduced above s llustrated n the fgure below; the Young dagram wth a bold boundary s that of λ (k), whle the ndcated cells are those n λ (k+1) /λ (k) and λ (k) /λ (k 1). b r b + 1 _ r +1 1 a r a Assume that we have T k U, and that the entry k + 1 n T changed nto k s n some row r for 1 p (ths s always the case). Thus, the Gelfand-Tsetln pattern correspondng to U s Λ + δ kr. Then let 1 ( (3.1) c U,T := b 1 + b ) p ( j 1 b ) j. a j=1 a j a j=+1 j a Smlarly, assume that we have S k T, and that the entry k n T changed nto k + 1 s n some row r +1 1 for 1 p 1 (ths s always the case). Thus, the Gelfand-Tsetln pattern correspondng to S s Λ δ k,r+1 1. Then let ( ) 1 p 1 ( ) (3.2) d S,T := b 1 b j a 1 + b j a j a j. a j=1 The horzontal strp condtons on λ (k+1) /λ (k) and λ (k) /λ (k 1) guarantee that c U,T and d S,T are nonnegatve ratonal numbers whch only become 0 when b = 0, respectvely b = 0. j=1 j=+1 It s not hard to calculate the edge products usng (3.1) and (3.2). For nstance, we have 1 ( (3.3) π T,U = b 1 b ) p ( j 1 b ) 1 ( ) j p 1 ( ) 1 + b j a j a a j a a j a 1 + b j a j a and a smlar formula for π S,T. j=+1 Proposton 3.1. The edge-colored and edge-labeled poset defned above s edge product smlar, as a representaton dagram, to that of the Gelfand-Tsetln bass. j=1 j=,

6 6 PATRICIA HERSH AND CRISTIAN LENART Proof. Observe frst that, f j < and λ (p) j > λ (m), then the dfference l pj l m s the length of the + 1). Stll assumng > j, ths means that hook wth rghtmost cell (j, λ (p) j ) and bottom cell (, λ (m) l krj l kr = a a j and l k+1,rj l kr = a a j + b j ; thus, by parng these two factors, we obtan l kr l k+1,rj l kr l krj = 1 + b j a a j. The other three types of brackets n (3.1) and (3.2) are obtaned n a smlar way from the Gelfand- Tsetln formulas (2.10) and (2.11). However, there are exceptons when r p = k + 1 (see below), namely the bracket correspondng to j = p n (3.1), and the bracket correspondng to j = p 1 n (3.2). Fnally, note that the quotent n (2.10) correspondng to the rows dfferent from r j s 1; a smlar statement holds for (2.11). Let us now dscuss the excepton mentoned above. Assume that r p = k + 1. The coeffcent c U,T dffers from the correspondng coeffcent n (2.10) by a factor 1/(a p a ). Smlarly, the coeffcent d S,T dffers from the correspondng coeffcent n (2.11) by a factor a p 1 a + b p 1 = a p a. The last equalty holds because the frst λ (k) k entres n row k of T are equal to k. It s clear from (3.3) and the dscusson related to the slght dscrepancy between (2.10)-(2.11) and (3.1)-(3.2) that the representaton dagram of the Gelfand-Tsetln bass s edge product smlar to the one descrbed n ths secton. 4. A proof of the Gelfand-Tsetln formulas Based on Proposton 2.1, the proof of the Gelfand-Tsetln formulas amounts to the frst statement n the theorem below. Theorem 4.1. The edge-colored and edge-labeled poset defned n Secton 3 satsfes the damond, crossng, and structure condtons. Hence, t s the representaton dagram of the rreducble representaton of sl n wth hghest weght λ. Proof. We use the notaton n Secton 3 related to the SSYT T, as well as the notaton n Secton 2.1. Let µ k := b b p 1 be the number of entres k n T. We have m k(t) = µ k µ k+1 by defnton. Furthermore, wt(t) = µ 1 ε µ n ε n snce, by defnton, wt(t), α k = m k (T) (recall that α k = ε k ε k+1 s the dual bass element to ω k under the scalar product ε, ε j = δ j ). Hence, gven T k U, the structure condton wt(t) + α k = wt(u) s verfed. We now address the crossng condton. We have m k (T) = b b p 1 b 1... b p. In order to smplfy formulas, we make the substtuton x 2 1 := a, y 2 1 = b for = 1,...,p, and x 2 := a, y 2 := b for = 1,...,p 1. Based on (3.3), the crossng relaton (2.2) for vertex T and color k can be wrtten as follows: N ( (4.1) y 1 + y ) N j = y, x j x 1 j N j where N := 2p 1. Note that we can take the above sum over all from 1 to 2p 1 because when we attempt an llegal change of a k + 1 nto k or vceversa, the correspondng term s 0. Indeed, assume for nstance that we ntend to change a k + 1 n row r, but ths s llegal because we have a k mmedately above t. Ths means that a a 1 = b 1, and therefore the rght-hand sde of (3.3) cancels. Now let us prove the ratonal functon dentty (4.1). It s not hard to see that ts left-hand sde s nvarant under the dagonal acton of the symmetrc group S N on the varables x 1,..., x N and y 1,...,y N. Let us expand and extract the coeffcent of y 1 y 2... y k. For k = 1 t s clearly 1. For 2 k N, t s

7 COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS 7 the followng symmetrc ratonal functon n x 1,...,x k : k 1 j k j 1 x j x. The common denomnator s the Vandermonde determnant n x 1,...,x k. Thus the numerator s an antsymmetrc polynomal n the same varables, but ts degree s ( ) ( k 2 (k 1) = k 1 ) 2, so t has to be 0. Ths concludes the proof of (4.1) by the symmetry of ts left-hand sde under the dagonal acton of S N. Snce the Gelfand-Tsetln lattce GT(λ) s a dstrbutve lattce, the damond relatons take the smpler form (2.4). Assume that we have T k U, S l U, R l T, and R k S. We use the same parameters a, b, a, r for the SSYT T as n Secton 3. We need to show that c U,T d S,U = d R,T c S,R. Ths s trval except for l = k and l = k + 1, because then c U,T = c S,R and d S,U = d R,T. Now consder the case l = k. Let r be the row contanng the changed k+1 n T and R (for obtanng S), and let r j+1 1 be the row contanng the changed l = k n U (for obtanng S) and T. Assume that j, the other case beng completely smlar. Then, by (3.1) and (3.2), we have d S,U d R,T = 1 1 a j a = c S,R c U,T. Fnally, consder the case l = k+1. Let r be the row contanng the changed k+1 n T (for obtanng U) and R, and let r j be the row contanng the changed l = k +1 n U and T (for obtanng R). Assume that j >, the other case beng completely smlar. Then, by (3.1) and (3.2), we have c U,T c S,R = d R,T d S,U = ( 1 b j ( 1 a j a ) ( 1 b j 1 a j a b a j b j a + b ) ( 1 ) 1, A smple calculaton shows that the two quotents are equal. b 1 a j b j a + b ) 1. By Proposton 2.1, the edge-colored and edge-labeled poset defned n Secton 3 encodes a representaton of sl n. Consder the bass vector correspondng to the maxmum of the poset GT(λ), namely to the SSYT of shape λ wth all entres n row equal to. Ths s clearly a hghest weght vector, and relaton (2.6) shows that the gven representaton s the rreducble one wth hghest weght λ. 5. Edge-mnmalty and the soltary property In ths secton, we gve an algorthm for determnng the edge products n a Gelfand-Tsetln lattce from the vertex weghts {m (s)}. The dea s to show frst how the edge products of color 1 are forced by the vertex weghts; then as an nductve step, we show that after edge products have been determned for all edges colored 1, 2,...,k 1, then there s an algorthm forcng the edge products for all of the edges colored k. Remark 5.1. If the edge products for three of the four coverng relatons comprsng a damond are known, then the damond relaton (2.5) wll determne the fourth. We begn by descrbng the algorthm for the edges colored 1. Notce that the 1-components are chans, or n other words for each poset element T there s at most one U coverng T such that the coverng relaton T U s colored 1 and lkewse there s also at most one R covered by T such that the coverng relaton R T s colored 1. Ths s mmedate from the fact that the value 1 may only appear n the frst row of a semstandard Young tableau, so that any semstandard Young tableau has at most

8 8 PATRICIA HERSH AND CRISTIAN LENART one copy of the value 1 whch may be ncremented to a 2 stll yeldng a semstandard Young tableau, snce the only canddate s the rghtmost 1 n the frst row. For any mnmal or maxmal element T n a 1-component, the crossng condton forces the edge product for the unque coverng nvolvng T. We may keep repeatng ths dea,.e. at each step usng the crossng condton to force one addtonal edge product, namely one at a poset element S such that all other edge products for coverng relatons nvolvng S that are colored 1 have already had ther values forced. In ths manner, we determne all edge products for coverng relatons colored 1. Before turnng to the nductve step, we make a few observatons about semstandard Young tableaux that wll be used n our upcomng algorthm for determnng edge products of color k once the edge products are known for colors 1, 2,...,k 1. We now thnk of an element T of the Gelfand-Tsetln lattce as a semstandard Young tableau, usng the earler descrpton of coverng relatons T U as pars of SSYT n whch U s obtaned from T by ncrementng an entry from to + 1. Recall that ths coverng relaton T U s then sad to have color. Remark 5.2. The value k may only appear wthn the frst k rows of a SSYT, snce the column mmedately above any occurance of the value k must consst of a strctly ncreasng sequence of postve ntegers. Consequently, there are at most k upward edges and at most k downward edges colored k orgnatng at any specfed element T n the Gelfand-Tsetln lattce, snce only the rghtmost k n a row may yeld an upward coverng relaton colored k and only the leftmost k may yeld a downward one. Let us call a value k +1 n a SSYT decrementable f t may be replaced by k wth the result stll beng a SSYT. Smlarly, call a value k n a SSYT ncrementable f t may be replaced by the value k + 1 to yeld a SSYT. Remark 5.3. Let d(k) be the number of rows n a SSYT whch contan a decrementable copy of the value k + 1. Then d(k) s at most one more than the number of rows contanng a value strctly smaller than k whch s ncrementable. Ths s because each decrementable k + 1 whch s not n the frst row has a value < k mmedately above t; ths may be ncremented to obtan a SSYT unless t has another to ts mmedate rght, but n that case the rghtmost n ts row must be ncrementable (snce the entry mmedately below t n the SSYT, f any, wll be at least as large as k + 1). In dscussng any partcular SSYT below, let r j be the j-th row n ths SSYT whch contans a decrementable copy of k + 1. Remark 5.4. Consder a row r j of a SSYT whch s not the lowest such row. Let < k be the entry mmedately above the leftmost copy of k + 1 n row r j+1. Then we may decrement the leftmost k + 1 n row r j and ncrement the rghtmost n row r j+1 1 at the same tme to obtan a SSYT. Ths s because + 1 k, so that we preserve the property of rows weakly ncreasng from left to rght even n the case that r j = r j+1 1; preservaton of column strctness s clear. We wll use these pars of coverng relatons colored k and to provde damond relatons to be used n the algorthm below. Now we are ready to complete the descrpton of the algorthm. The dea for handlng color k s to proceed from top to bottom through each k-component. More precsely, we repeatedly choose a maxmal element S n the remanng part of a k-component and determne the edge products for all edges orgnatng at S whose edge products have not yet been determned; these must necessarly all proceed downward from S. At each stage,.e. at each poset element S encountered, we frst force the edge products for all but one of ts downward edges colored k; essentally, we use the damond condton for the damond nvolvng a downward coverng relaton R S whch s colored k and an upward coverng relaton S U whch s colored for some < k, as follows. Assume that the coverng relaton R S corresponds to decrementng the leftmost k + 1 n a row r j of S whch s not the lowest such row. Now lettng be the entry mmedately above the leftmost copy of k +1 n row r j+1 of S, lke n Remark 5.4, obtan U from S by ncrementng the rghtmost n row r j+1 1. Obtan the fourth poset element T

9 COMBINATORIAL CONSTRUCTIONS OF WEIGHT BASES: THE GELFAND-TSETLIN BASIS 9 comprsng the damond by takng U and decrementng the leftmost k + 1 n row r j, cf. Remark 5.4. Notce that three of the four edge products n ths damond wll have already been determned, by vrtue of ether beng colored for some < k or else nvolvng a poset element T whch s strctly greater than U n the k-component of the Gelfand-Tsetln lattce; thus, (2.5) forces the last edge product (cf. Remark 5.1). Ths leaves exactly one edge product yet to be determned among the downward edges colored k whch orgnate at S; t s the product for the edge that corresponds to decrementng the rghtmost k + 1 n the lowest row r j. But ths edge product s determned by the crossng condton. Theorem 5.5. The above algorthm shows explctly how to determne all the edge products of the Gelfand-Tsetln lattce, and thus mples that ths lattce has the soltary property. It also has the edge mnmalty property. Proof. Unqueness of edge products, or n other words the soltary property, s proven above wthn the descrpton of the algorthm. The pont s that we show how each edge product n turn s forced by earler ones. Edge mnmalty then follows from the fact that none of the resultng edge products are 0, cf. (3.3). The fact that the Gelfand-Tsetln lattce has the soltary and edge mnmalty propertes was prevously proven n [6, Theorem 4.4], but not n a constructve manner, so not n a way whch provdes an algorthm to determne all of the edge products. It would be nterestng now to relate the algorthm we have just gven to the known edge labels. References [1] G. Bard and L. Bedenharn. On the representatons of the semsmple Le groups. II. J. Mathematcal Phys., 4: , [2] C. de Concn. Symplectc standard tableaux. Adv. Math, 34:1 27, [3] R. Donnelly. Explct Constructons of Representatons of Semsmple Le Algebras. PhD thess, Unversty of North Carolna at Chapel Hll, [4] R. Donnelly. Symplectc analogs of the dstrbutve lattces L(m, n). J. Combn. Theory Ser. A, 88: , [5] R. Donnelly. Explct constructons of the fundamental representatons of the symplectc Le algebras. J. Algebra, 233:37 64, [6] R. Donnelly. Extremal propertes of bases for representatons of semsmple Le algebras. J. Algebrac Combn., 17: , [7] R. Donnelly. Extremal bases for the adjont representatons of the smple Le algebras. Comm. Algebra, 34: , [8] R. Donnelly, S. Lews, and R. Pervne. Constructons of representatons of o(2n + 1, C) that mply Molev and Rener- Stanton lattces are strongly Sperner. Dscrete Math., 263:61 79, [9] R. Donnelly, S. Lews, and R. Pervne. Soltary and edge-mnmal bases for representatons of the smple Le algebra G 2. Dscrete Math., 306: , [10] I. Gelfand and M. Cetln. Fnte-dmensonal representatons of the group of unmodular matrces. Doklady Akad. Nauk SSSR (N.S.), 71: , [11] M. Gould. On the matrx elements of the U(n) generators. J. Math. Phys., 22:15 22, [12] M. Kashwara. Crystalzng the q-analogue of unversal envelopng algebras. Commun. Math. Phys., 133: , [13] M. Kashwara. On crystal bases of the q-analogue of unversal envelopng algebras. Duke Math. J., 63: , [14] M. Kashwara and T. Nakashma. Crystal graphs for representatons of the q-analogue of classcal Le algebras. J. Algebra, 165: , [15] I. G. Macdonald. Symmetrc Functons and Hall Polynomals. Oxford Mathematcal Monographs. Oxford Unversty Press, Oxford, second edton, [16] A. Molev. Gelfand-Tsetln bass for representatons of Yangans. Lett. Math. Phys., 30:53 60, [17] A. Molev. A bass for representatons of symplectc Le algebras. Comm. Math. Phys., 201: , [18] A. Molev. Weght bases of Gelfand-Tsetln type for representatons of classcal Le algebras. J. Phys. A, 33: , [19] A. Molev. A weght bass for representatons of even orthogonal Le algebras. In Combnatoral Methods n Representaton Theory, volume 28 of Adv. Studes n Pure Math., pages , [20] M. Nazarov and V. Tarasov. Yangans and Gelfand-Zetln bases. Publ. Res. Inst. Math. Sc., 30: , [21] Hou Pe-yu. Orthonormal bases and nfntesmal operators of the rreducble representatons of group U n. Sc. Snca, 15: , 1966.

10 10 PATRICIA HERSH AND CRISTIAN LENART [22] D. Zhelobenko. The classcal groups. Spectral analyss of ther fnte-dmensonal representatons. Russ. Math. Surv., 17:1 94, [23] D. Zhelobenko. Compact Le groups and ther representatons, volume 40 of Translatons of Mathematcal Monographs. Amercan Mathematcal Socety, Provdence, RI, Translated from the Russan by Israel Program for Scentfc Translatons. [24] D. Zhelobenko. An ntroducton to the theory of S-algebras over reductve Le algebras. In Representaton of Le groups and related topcs, volume 7 of Adv. Stud. Contemp. Math., pages Gordon and Breach, New York, Department of Mathematcs, Rawles Hall, Indana Unversty, Bloomngton, IN E-mal address: phersh@ndana.edu Department of Mathematcs and Statstcs, State Unversty of New York at Albany, Albany, NY E-mal address: lenart@albany.edu