2 Anatol N. Kirillov and Arkadiy D. Berenstein points set, (K n ) Z, of the cone K n is in a one-to-one correspondence with the set STY (n) of standar

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1 GROUPS GENERATED BY INVOLUTIONS, GELFAND-TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX ANATOL N. KIRILLOV RIMS, Kyoto University, Kyoto 66, Japan Steklov Mathematical Institute, Fontanka 27, St.Petersburg, 19111, Russia and ARKADIY D. BERENSTEIN Council on Cybernetics, Russian Academy of Sciences, Moscow, Russia ABSTRACT We construct families of piecewise linear representations (cpl-representations) of the symmetric group S n and the ane Weyl group Sn e of type A (1) n;1 acting on the space of triangles X n. We nd a nontrivial family of local cpl-invariants for the action of the symmetric group S n on the space X n and construct one global invariant w.r.t. the action of the ane Weyl group Sn e (socalledcocharge). We nd the continuous analogs for the Kostka-Foulkes polynomials and for the crystal graph. We give an algebraic version of some combinatorial transformations on the set of standard Young tableaux. Introduction. In this paper we dene and study a new class of representations of the symmetric group S n, namely, the continuous piecewise linear representations (cplrepresentations) of S n in the space of triangles X n. By denition, a triangle x 2 X n is a triangular array of real numbers x = (x ij ) x ij 2 R 1 i j n. More precisely, following [GZ1],[GZ2],[BZ1], we consider the Gelfand-Tsetlin cone K n consisting of all triangles x 2 X n such that x ij x i+1 j+1 x ij x i j;1 1 i j n ; 1 1 i j n x ij 1 i j n: This is a nondegenerate convex polyhedral cone in the space X n = R n(n+1) 2,having 2 n generators (see Remark 2.2). It is well known (e.g. [GZ1]) that the integral

2 2 Anatol N. Kirillov and Arkadiy D. Berenstein points set, (K n ) Z, of the cone K n is in a one-to-one correspondence with the set STY (n) of standard Young tableaux, having all entries not exceeding n. The cplaction of the symmetric group S n on the space X n,given in our paper, is such that it conserves the Gelfand-Tsetlin cone K n and that on the set STY (n) it coincides with the action of the symmetric group on the set of standard Young tableaux given by A. Lascoux and M.-P. Schutzenberger [LS2], [LS3] (see Theorem 2.3). Our main observation is that a great many of combinatorial constructions on the set of standard Young tableaux, e.g. the Schutzenberger involution [Sch1],[EG], [Ki1], the dual Schutzenberger involution [Sch1], a promotion transformation [Sch1], [EG], the action of the symmetric group [LS2],[LS3], the crystal graph structure on the set STY ( n), [Ka1],[Ka2], the construction of cocharge [LS1],[Ki1], may be transfered to the Gelfand-Tsetlin cone K n, and even to the whole space of triangles X n. Our constructions are based on a consideration of \elementary transformations" t j 1 j n ; 1, and T 2 R. Denition.1 Assume that x 2 X n, then t j (x) =ex ex ik = x ik if k 6= j T (x) =ex, where ex ij = min(x i j+1 x i;1 j;1 )+max(x i j;1 x i+1 j+1 ) ; x ij (:1) and we presuppose that x j := +1 x j j;1 := ;1 1 j n ; 1 ex ik = x ik if (i k) 6= (1 1) ex 11 = x 11 ; : We denote by G n =<t 1 t n;1 > the group generated by t i 1 i n ; 1. The transformations t i 1 i n ; 1, satisfy the following relations (see Corollary 1.1): where q i := t 1 t 2 t 1 (i) t 2 i =1 t i t j = t ji if jj ; ij 2 (ii) (t 1 t 2 ) 6 =1 (:2) (ii) (t 1 q i ) 4 =1 {z} t 3t 2 t {z 1 } t it i;1 t {z 1 } : if 3 i n ; 1 The restriction of the involutions t i to the set of standard Young tableaux STY ( ) of a given shape and content admits a simple combinatorial interpretation. It is these restrictions that are ordinary used in order to prove that Schur functions are the symmetric, e.g. [BK],[SW],[Sa] and Section 2A. We assume that the relations (.2) are the dening relations for the group G n. By any way, the group G n seems to be very interesting. It is easy to see that the order of the group G 3 is equal to 12. But if n 4 then G n is innite and for any N there exist an epimorphism of the group G 4 on the symmetric group S N (see comments after Corollary 1.3). The group G n admits an extension Gn e by means of R 1 : eg n :=< t 1 t n;1 T 2 R >:

3 Combinatorics of the Gelfand-Tsetlin patterns 3 We have the following relations between the generators in the group e Gn : (i) T T = T + (ii) (t 1 T ) 2 =1 t j T t j = T 3 j n ; 1 2 R (iii) t i t j = t j t i if jj ; ij 2 (iv) (T t 2 t 1 T t 2 t 1 ) 3 =1 for any 2 R (:3) (v) T t 2 T ;+ t 2 T ; t 2 = t 2 T t 2 T ;+ t 2 T ; (vi) (t 1 T q j t 1 T q j ) 2 =1 (vii) T q j T q j = q j T q j T 3 j n ; 1 3 j n ; 1 2 R where transformation q j is dened in (.2). The proofs of the relations (i)-(v) are based on direct computations. The main diculties arise in the proof of the (vi). In order to understand better the relations (.2) and (.3), let us consider the following elements in the group e Gn : s i := q i t 1 q i ;1 1 i n ; 1 (:4) s i = q i T ; q i ;1 1 i n ; 1 (:5) The main result of Section 1 is Theorem 1.1, which is equivalent to the relations (i)-(vi), and asserts that 1. The involutions s i 1 i n ; 1, satisfy the relations of the symmetric group S n, i.e. a) s 2 i =1 1 i n ; 1 b) (s i s i+1 ) 3 =1 1 i n ; 2 (:6) c) s i s j = s j s i if ji ; jj 2: 2. The transformations s () i 1 i n ; 1 2 R, satisfy the colored braid relations, i.e. for any 2 R we have a) s () i b) s () i c) s () i s () i = s (+) i 1 i n ; 1 s (+) i+1 s () i = s () i+1 s(+) i s () i+1 1 i n ; 2 (:7) s () j = s () j s () i if ji ; jj 2: The relation (.3),(vi) is equivalent to the statement that the transformations s () i and s () j commute if ji ; jj 2. In oder to prove the last statement, we use another expression for s () i 1 i n;1, as a product of the Lusztig involutions [Lu2]. We recall the corresponding denitions, because we use not exactly the same involutions that contained in [Lu2], but their analogs for the space of triangles X n.

4 4 Anatol N. Kirillov and Arkadiy D. Berenstein Denition.2. For each triple of integers (ijk) 1 i < j < k n, let us dene a transformations R ijk : X n! X n in the following manner R ijk (x) =ex x 2 X n where ex i j = x i j + x i k ; x i k;1 ; min(x j k ; x j k;1 x i j ; x i j;1 ) ex j j = x j j ; x i k + x i k;1 + min(x j k ; x j k;1 x i j ; x i j;1 ) (:8) ex = x if ( ) 6= (i j) or (j j): Let us denote by L n the group generated by all R ijk with 1 i < j < k n. We have the following relations between the generators in the group L n : (i) (R ijk ) 2 =1 (ii) R ijk R i j k = R i j k R ijk if j(ijk) \ (i j k )j 6= 2 (:9) (iii) (R ijk R ijl R ikl R jkl ) 2 =1 (iv) (R ijl R ikl ) 3 =1 if 1 i<j<k<l n: We assume that the relations (.9) are the dening ones for the group L n. To go further, let us dene the transformations T (i) acting on the space X n : The statement that s () i T (i) = ex x 2 X n where ex ii = x ii ; ex = x if ( ) 6= (i j): and s () j expression for s () i 1 i n ; 1, (Theorem 1.2): s () i commute if ji ; jj 2, follows from the following = R i;1 i R i;2 i R 1 i T (i) R 1 i R i;2 i R i;1 i (:1) where R jk := R jkk+1 1 j <kn ; 1. The proof of identity (.1) is based on induction and on the recurence formula for s () i (see (1.29)). In order to obtain the corresponding results for the involutions s i (see (1.28a)), we use Crucial Lemma. Assume x 2 X n (x) =( 1 n ), then s i (x) =s ( i; i+1 ) i (x) (:11) where (x) is a weight of the triangle x 2 X n, i.e. i (x) := jx (i) j;jx (i;1) j 1 i n x () := : The relations (.3) between generators of the group Gn e allow us to construct many interesting subgroups in Gn e. For example 1. Let 2 R be xed, then the elements s 1 s () s 1 2s () s 2 n;1s () n;1 are the standard generators of the symmetric group S n.

5 Combinatorics of the Gelfand-Tsetlin patterns 5 2. Let 1 n;1 be the real numbers, 6=,and es i := es ( i) i = s i s ( i) i i =1 n; 1. Let us put es := es ( ) = es n;1 es n;2 es 1 s ( ) 1 es 2 es n;1 : Then es es 1 es n;1 are the standard generators of the ane Weyl group of the type A (1). It is important thatthecocharge c n;1 n(x) of a triangle x 2 X n (Section 3) is an invariant w.r.t. the action of any involutions es ( i) i i n ; 1. Let us summarize the content of the Section 1 of our paper. We construct a family of the cpl-representations of the symmetric group, nd a family of the cplrepresentations of the ane Weyl group of the type A (1) n;1 (see item after Corollary 1.3) and construct a cpl-representation of the colored braid relations. We developed a geometric techniques for proving some (non trivial!) identities between piecewise linear functions (see Theorem 1.3). Our next step is to give a combinatorial interpretation of the transformations under consideration and to study the continuous piecewise linear invariants w.r.t. the action of the symmetric group generated by s 1 s n;1,onthespace of triangles X n. In Section 3 we prove (see Theorem 3.2) that the following cpl-functions on the space of triangles X n are s j -invariants: where 2 i j n ; 1 1j(x) =min(x 1 j+1 ; x 1 j x j j ; x j+1 j+1 ) ij(x) =min(x i;1 j ; x i;1 j;1 x i j+1 ; x i j ) (:12) 1j(x) =(min(x j;1 j + x j j ; x j;1 j;1 ; x j j+1 x 1 j+1 + x j+1 j+1 ; x 1 j ; x j j )) + 2j(x) =(min(x 1 j + x j j ; x 1 j+1 ; x j+1 j+1 (:13) x 1 j;1 + x 2 j+1 ; x 1 j ; x 2 j )) + ij(x) =(min(x i;2 j + x i;1 j ; x i;2 j;1 ; x i;1 j+1 x i;1 j;1 + x i j+1 ; x i;1 j ; x i j )) + where 3 i j n ; 1, and for any a 2 R (a) + := max(a ). We recall that a cpl-function ' dened on the space X n is said to be an s j -invariant if'(s j (x)) = '(x) for all x 2 X n. It seems a very interesting task to nd a fundamental system I j of cpl-invariants for s j, that is a system such that any cpl-invariant under the action of s j is a min ; max linear combination of that from I j. It is not clear whether or not this set is nite, because we have the trivial examples of the following kind x ik if k 6= j or min('(x) '(s j (x))) where '(x) isany cpl-function on X n. However, the number of fundamental cpl-s j - invariants having a complexity bounded by some integer k, is nite. Here we dene for

6 6 Anatol N. Kirillov and Arkadiy D. Berenstein the complexity of a cpl-function as the least number of min and max needed for its representation. [Remark. In a previous denition of complexity we considered min and max as functions of two variables, e.g. min : R R! R. Thus, the complexity ofthe function min(x 1 x 2 x m ) is equal to m ; 1]. By any way, we may construct (see the proof of Theorem 3.2) other nontrivial examples of cpl-s j - invariants and it is not clear whether or not they are independent (in cpl-sense) from the invariants (.12) and (.13). The next interesting problem is to describe the fundamental system of continuous piecewise polynomial functions (cpp-functions) on the space X n, which areinvariant w.r.t. the action of the subgroup S n G n. Of course, we P have such trivial example as: x in 1 i n or minf '((x)) j 2 S n g, or 2S n '((x)), where '(x) is any cpp-function on X n. But the question is the following one: does there exist nontrivial cpl-invariants under the action of the symmetric group S n on the space X n? In Section 3wegive an armative answer to this question (see Theorem 3.1). In fact we dene a stable (ibid), cpl-invariant w.r.t. the action of a family of the ane Weyl groups of type A (1) n;1, namely, e Sn :=< es es 1 es n;1 >, see Corollary 1.3, which takes nonnegative values on the Gelfand-Tsetlin cone K n. Note that the complexity of the invariant under consideration is equal to (n ; 1)! ; 1, if n 3. An origin of such invariant is very clear. We have a distinguished S n - invariant function on the set STY (n), namely, the cocharge c n (T ) of a tableau T 2 STY (n) as dened by A. Lascoux and M.-P. Schutzenberger [LS1]. Note that it is possible (at least for n 5 and hypothetically for all n) to reformulate the denition of the cocharge of a tableau T 2 STY (n) in terms of the corresponding Gelfand-Tsetlin pattern x(t ) and obtain some function c n on the set (K n ) Z. We consider an extension of the function c n to the whole space X n in a natural manner, changing its domain of denition from (K n ) Z to X n. This gives us the desired cples n -invariant, c n (x) x 2 X n, which we still call cocharge. Details are contained in Section 3. We dene another cpl-s n -invariant (x) (whichdoes'te Sn -invariant!) by setting (x) =c n (q n;1 (x)) x 2 X n, where the involution q n;1 is dened in (.2). As an example we give an expression for cocharge c 4 : c 4 (x) =min(x 13 ; x 12 x 22 ; x 33 )+x 14 ; x 13 + x 33 ; x min(x 23 ; x 34 x 24 ; x 23 x 13 ; x 12 x 22 ; x 33 4 (x) ; x 34 x 24 ; 4 (x)): It is known (e.g. [H]) that the volume of the convex polytope K () consisting of all Gelfand-Tsetlin patterns with highest weight and weight (see Section 1) may be considered as a continuous analog of the weight multiplicity K := dimv (). We dene a continuous analog of the Kostka-Foulkes polynomial (q-analog of the weight multiplicity) [LS1],[Ma],[Ki1],by means of following integral K (q) = Z K () exp(hc n (x))dx q = exp(h): (:14) We expect to study the properties of the integral (.14) and the toric variety corresponding to the Gelfand-Tsetlin cone K n in a separate publication. In Remark

7 Combinatorics of the Gelfand-Tsetlin patterns we give a generalization of the Lascoux-Schutzenberger algorithm for computing the charge of a dominant weight standard Young tableau to the case of a standard Young tableau with an arbitrary weight. In Section 2 we study the restrictions of the transformations considered in Section 1 to the integral points set (K n ) Z of the Gelfand-Tsetlin cone K n. We show that the involution q i : STY (n)! STY (n) 1 i n ; 1, (see (.2)) coincides with the partial Schutzenberger involution S i : STY (n)! STY (n). Here for a given tableau T 2 STY (n), the involution S i acts non trivially only on the part T i+1 of the tableau T lling by thenumbers 1 i+1, and on this part coincides with the ordinary Schutzenberger's involution [Sch1] or Section 2C. Let us note that the group generated by theinvolutions q i 1 i n ; 1, coincides with the group G n =<t 1 t n;1 >, and thus we may describe the relations between the partial Schutzenberger involutions (see Remark 1.3). Further we have the following relation between the partial Schutzenberger involution q i and the action of the symmetric group S n on the set STY (n) as dened by A. Lascoux and M.-P. Schutzenberger [Sch2],[Sch3]: s i = q i q 1 q i 1 i n ; 1 (:15) where s i := (i i +1)2 S n is a simple transposition. The Schutzenberger involution S posesses many interesting properties in connection with the Robinson-Schensted correspondence [Sch2], with a rigged congurations [Ki1] and so on. In addition we explain in Remark 2.4 that the involution S allows to give a simple pure combinatorial proof of the following symmetry property of the Littlewood-Richardson numbers (see Proposition 2.8) c = c : (:16) All other symmetries of the LR-numbers follow from (.16) and the symmetries of the Berenstein-Zelevinsky triangles (see [BZ3] and Remark 2.4). Finally, it is interesting to note the following connection between our transformations s (1) i 1 i n ; 1, restricted on the integral points set of the convex polytope K (see Section 1), and the crystal graph corresponding to the irreducible representation V of the Lie algebra gl n with the highest weight, [Ka1],[Ka2], [KN]. Namely, let us denote by f i (respectively e i ) the restriction of the map s (;1) i (resp. s (+1) i ) on the GT-polytope K [Remark: the transformations s (1) i does't conserve GT-cone K n, so for x 2 K n we dene f i (x) = s (;1) i (x), if s (;1) i (x) 2 K n and f i (x) =, if s (;1) i (x) 62 K n ]. Let further (L() B()) be the crystal base of an irreducible gl n -module V with highest weight and Fi e Ei e 1 i n ; 1, be the deformed generators of the Hopf algebra U q (gl n ) as dened by M. Kashiwara [Ka1], [Ka2]. Then there exist a bijection N : B()! K \ (X n ) Z

8 8 Anatol N. Kirillov and Arkadiy D. Berenstein such that (i) b 2 B() := B() \ V () i N(b) 2 K () \ (X n ) Z (ii) if Fi e (b) 2 B (() Fi e (b) 6= ) then N( Fi e (b)) = s (;1) i (N(b)) (iii) if Ei e (b) 2 B (() Ei e (b) 6= ) then N( Ei e (b)) = s (+1) i (N(b)): The classical representation theory of the symmetric and general linear groups is based on dierent combinatorial constructions among which the ones with Young tableaux play an essential role. The main reason is that Young tableaux in a natural way parametrize a basis of irreducible representation of the symmetric or general linear groups [Ru],[JK],[St]. In some sense the choice of a concrete realization of an irreducible representation predetermine the corresponding combinatorial structures. If we consider the realization of irreducible representations of the general linear group in the space of the Gelfand-Tsetlin patterns, [GZ1], then we deal with combinatorics of convex polytopes of a special kind. If we consider the realization of representations of the symmetric (or general linear) group by means of Specht (orweyl) module, [JK], then we deal with combinatorics of Young tableaux. While the combinatorics of Young tableaux is extensivly developed (see e.g. [Sa]), the combinatorics of the Gelfand-Tsetlin patterns seems only began to be studied [GZ1],[GZ2],[BZ1]. The main goal of our paper is to show that the majority of combinatorial constructions over Young tableaux admits an \algebraization" and may be dened on the space of triangles. Of course we did not exhaust the subject and we believe that other combinatorial constructions, e.g. Robinson-Schensted correspondence (e.g. [Sch2]), also admit natural continuous analogs. Acknowledgements. We are very grateful to many people who encouraged us on the dierent step of this work. We are obliged to L.D. Faddeev, I.M. Gelfand, M.-P. Schutzenberger, A.V. Zelevinsky, I.V. Cherednik, D. Foata, A. Lascoux, M.Kashiwara, P. Mathieu, T. Miwa and I. Pak for interesting discussions and useful comments. One of us (AK) gratitude goes to Dr.A. Schnizer for help in a computation of the examples of the action t 2 t 3 s 4 s 5, which lead to better understanding of the transformations under consideration, J.H.H. Perk for help in preparing English text, and also thank with much gratitude the colleagues at Kyoto University and RIMS for their invitation, their hospitality which made it possible to nish this work. I would like to acknowledge my special indebtedness to Dr. N.A. Liskova for the inestimable help in preparing the manuscript to publication.

9 x1. Groups acting on the space of triangles. Combinatorics of the Gelfand-Tsetlin patterns 9 Let n be a positive integer. In this section we dene the group G n generated by involutions and Gn e its extension by means of R 1, which acts on a space of triangles X n. By denition the space X n consists of all sequences x =(x (n) x (n;1) x (1) ) where x (j) = (x 1j x jj ) 2 R j. As a vector space X n ' R n(n+1) 2. We will call the vector x (n) 2 R n the highest weight of the triangle x 2 X n and denote it by (x) := x (n). Let us dene a weight := (x) of a triangle x 2 X n as a vector =( 1 n ) 2 R n such that j = jx (j) j;jx (j;1) j 1 j n (1:1) where jx (j) j := P j i=1 x ij x () =. For given vectors 2 R n and 2 R n we dene the following subspaces of the space X n : X = fx 2 X n j (x) =g X () =fx 2 X j (x) =g: (1:2) Now we dene the Gelfand-Tsetlin patterns [GZ1],[GZ2],[BZ1],[BZ2]. By definition a triangle x 2 X n is called a Gelfand-Tsetlin pattern (GT-pattern) i the following inequalities are satised x ij x i+1 j+1 1 i j n ; 1 x ij x i j;1 1 i<j n (1:3) x ij 1 i j n: We denote by the set of all GT-patterns K = K n X n and those which lie in the subspaces (1.2) correspondingly by K and K (). It is clear that K and K () are convex, compact polytopes in the space R n(n+1) 2 +. It is well known (see x2, or [GZ2]), that if is a partition, and is a composition, then the number of integral points in the convex polytope K (respectively in K ()) is equal to the dimension of the irreducible representation V of the Lie algebra gl n with the highest weight (respectively, the dimension of the subspace V () V of the weight ). On the other side, as is also well known, the dimension of the weight subspace V () V admits a pure combinatorial description as the number of standard Young tableaux of shape and content : jk () \ Z n(n+1) 2 j = dimv () =jsty ( )j: (1:4) The equality (1.4) is a starting point of our investigations. We will try to translate the combinatorial information about the set of standard Young tableaux STY ( ) into the language of GT-patterns and vice versa.

10 1 Anatol N. Kirillov and Arkadiy D. Berenstein Now let us begin the construction of the main objects of this note: the group G n and its extension e Gn. At rst we dene the \elementary" transformations t j of the space of triangles X n. For this purpose, let us x a positive integer j 1 j < n, and introduce the sequences of numbers a 1 a j and b 1 b j. Given a triangle x =(x (n) x (1) ) 2 X n, we dene a 1 = x 1 j+1 a i = min(x i j+1 x i;1 j;1 ) 2 i j b j = x j+1 j+1 b i = max(x i j;1 x i+1 j+1 ) 1 i j ; 1: (1:5) Denition 1.1. The transformation t j : X n! X n is given by the following formulae Proposition 1.1. We have t j (x) =ex where ex ij = a i + b i ; x ij 1 i j (1:6) ex kl = x kl if l 6= j: a) t 2 j =1 1 j n ; 1 (1:7) b) t i t j = t j t i if ji ; jj 2 (1:8) c) (t j (x)) = (j j +1) (x) 1 j n ; 1 (1:9) where the action of transposition (j j +1) on the weight space R n is given by (j j +1)( 1 n )=( 1 j+1 j n ). Proof. The assertions a) and b) follow directly from the denition (1.6) of the transformation t j. As for c), let us remark, that b i + a i+1 = x i j;1 + x i+1 j+1 1 i j ; 1: So jex (j) j = X i (a i + b i ) ;jx (j) j = jx (j+1) j;jx (j) j + jx (j;1) j: Consequently, j+1 (ex) =jx (j+1) j;jex (j) j = jx (j) j;jx (j;1) j = j (x) and j (ex) =jex (j) j;jx (j;1) j = jx (j+1) j;jx (j) j = j+1 (x): Now let us consider the restriction of the action of t j on the Gelfand-Tsetlin polytopes. Fix vectors and in the space R n.

11 Proposition 1.2. We have for 1 j n ; 1 Combinatorics of the Gelfand-Tsetlin patterns 11 a) t j : K n! K n (1:1) b) t j : K! K t j : K ()! K ((j j +1) ): (1:11) Proof. It is sucient to show that the involutions t j conserve the inequalities (1.3). Consider \the elementary neighborhood" of x 2 K n : a c a c t j x ij ;! min(a b)+max(c d) ; x ij b d b d It is clear that (x := x ij ) min(a b) x max(c d). So a min(a b) min(a b) ; (x ; max(c d)) (min(a b) ; x)+max(c d) c: So all necessary inequalities are satised. We denote by G n the group generated by the involutions t 1 t n;1 : G n =<t 1 t n;1 >: (1:12) This group acts on the space of triangles X n and for any xedvector 2 R n + transforms the convex polytope K into itself and interchanges the polytopes K (). The group G n may be embedded in a bigger group e Gn. In order to construct such an extension, we dene a transformation T of the space X n in the following way: assume 2 R 1 and x 2 X n, then let us put T (x) =ex, where ex (j) = x (j), if 2 j n, and ex (1) = x 11 ;. We denote by e Gn the group generated by G n and T 2 R 1 : eg n =<t 1 t n;1 T 2 R 1 >: (1:13) The next proposition gives the description of some relations between the generators t i 1 i n ; 1, and T 2 R 1. Proposition 1.3. We have a) t 1 T = T ; t 1 or equivalently (t 1 T ) 2 =1 b) t i T = T t i 3 i n ; 1 c) (t 1 t 2 ) 6 =1 d) [t i t j ]= if ji ; jj 2 e) t 2 T t 2 T ;+ t 2 T ; = T t 2 T ;+ t 2 T ; t 2 : (1:14) f) (T t 2 t 1 T t 2 t 1 ) 3 =1 2 R 1 : Proof. The assertions a), b) and d) are evident.

12 12 Anatol N. Kirillov and Arkadiy D. Berenstein c) It is sucient to show that (t 1 t 2 ) 3 =(t 2 t 1 ) 3 : X 3! X 3 : By direct computation we nd t 2 t 1 (x) =ex, where ex 12 = x 13 ; x 12 + max(x 23 2 (x)) ex 22 = x 33 ; x 22 + min(x 23 2 (x)) 2 (ex) = 3 (x) ex 11 = 2 (x) ex 13 = x 13 ex 23 = x 23 ex 33 = x 33 : Consequently, (t 2 t 1 ) 3 (x) = eex, where eex = x, if ( ) 6= (1 2) or (2,2), and eex 12 = x 13 ; x 12 ; max(x 23 3 (x)) + max(x 23 2 (x)) + max(x 23 1 (x)) eex 22 = x 33 ; x 22 ; min(x 23 3 (x)) + min(x 23 2 (x)) + min(x 23 1 (x)): (1:15) Similarly, we nd t 1 t 2 (x) =x, where x 12 = x 13 ; x 12 + max(x 23 1 (x)) x 22 = x 32 ; x 22 + min(x 23 1 (x)) x 11 = 3 (x) 3 (x) = 2 (x) x 3 = x 3 =1 2 3: Using these formulae, it is easy to see that for (t 1 t 2 ) 3 (x) we obtain the same expression (1.15). e) Let us dene t 2 := T t 2 T ; t 2 2 R 1. It is sucient to show that t 2t 2 = t 2t 2 : X 3! X 3 for any 2 R 1. Given x 2 X 3, the following relations hold x (t 2(x)) = x if ( ) 6= (1 2) or (2 2) x 12 (t 2(x)) = x 12 + max(x 23 x 11 + ) ; max(x 23 x 11 ) x 22 (t 2(x)) = x 22 + max(x 23 x 11 + ) ; max(x 23 x 11 ): Consequently, x 12 (t 2t 2(x)) = = x 12 + max(x 23 x 11 + )+max(x 23 x 11 + ) ; 2max(x 23 x 11 )=x 12 (t 2t 2(x)): By the same reasoning x 22 (t 2t 2(x)) = x 22 (t 2t 2):

13 Combinatorics of the Gelfand-Tsetlin patterns 13 f) It is sucient to prove (1.14,f) for n = 3. Let us consider a correspondence : X 3! X 3, given by : x 13 x 23 x 33 x 12 x 22 x 11 ;! x x 23 + x 33 x x 22 + x It is clear that is an automorphism of X 3. The following identities, which may be veried by a direct computation, reduce the proof of (1.14 f) to that of point c): ;1 t 1T = t 1 ;1 T ;t 2 t 1 T ; t 2 t 1 =(t 2 t 1 ) 2 : Note that c) is a particular case of f). One of our main results of this note, namely Theorem 1.1, gives the description of additional relations between the generators in the group e Gn. We don't know whether or not the set of relations given by (1.24) and (1.25) contains all relations, but assume that it is complete. In any case the validity of the relations (1.24) allows us to construct the section of the following exact sequence 1! Ker! G n * ) S n! 1 (t i )=(i i +1) and to nd a subgroup (not normal subgroup!), which is isomorphic to the symmetric group S n. Now let us pass to the construction of the involutions s i,which will generate the symmetric group and the transformations s () i. The transformations s () i will satisfy the colored braid relations ( Theorem 1.1 ). For this purpose consider the following elements in the groups G n and e Gn : p i = t i t i;1 t 1 v i = t i t i+1 t n;1 q i = p 1 p 2 p i u i = v n;1 v n;2 v n;i = t 1 t 2 t n;1 := p ;1 n;1 = v 1 s i = i;1 t 1 1;i s () i = i;1 T 1;i (1:16) where 2 i n ; 1 and put p 1 = q 1 = s 1 = t 1 v n =1. Proposition 1.4. We have a) q i = q i;1 p i = p ;1 i q i;1 u i = u i;1 v n;i = v ;1 n;i u i;1 consequently q 2 i =1 u 2 i =1 1 i n ; 1: b) s i = q i t 1 q i s () i = q i T ; q i : 1 i n ; 1: (1:17)

14 14 Anatol N. Kirillov and Arkadiy D. Berenstein c) q n;1 s i q n;1 = s n;i q n;1 s () i q n;1 = s () n;i 1 i n ; 1: d) (s i;1 s i ) 3 = q i;1 t 1 p i t 1 (t 2 t 1 ) 6 t 1 p ;1 i t 1 q i;1 2 i n ; 1: e) Crucial Lemma: Assume x 2 X n (x) =( 1 n ) then s i (x) =s ( i; i+1 ) i (x): (1:18) f) s i s () i = s (;) i s i 1 i n ; 1: g) Assume that s i t j = t j s i for all j <i; 1 then s i = t i;1 t i s i;1 t i t i;1 : (1:19) h) Formulas for the weights (s i (x)) = (i i +1)(x) (s () i (x)) = ( 1 i;1 i ; i+1 + i+2 n ) ((x)) = (n n ; 1 2 1)(x) (q n;1 (x)) = (x) :=w (x) where w 2 S n is the element of the maximal length in the symmetric group S n. Proof. a) We must prove that p i+1 q i p i+1 = q i : Let us use induction: p i+1 q i p i+1 = t i+1 p i q i;1 p i t i+1 p i = t i+1 q i;1 t i+1 p i = q i;1 p i = q i because [t i+1 q i;1 ]=. Consequently, Similarly we may prove that q 2 i = q i q i = q i;1 p ;1 i p i q i;1 = q 2 i;1 = = q 2 1 = t 2 1 =1: b) By induction, it is easy to see that v n;i u i;1 v n;i = u i;1 and u 2 i =1: So we have i;1 = q i u i+1 v ;1 1 v;1 2 t 1 2 i n ; 1 n;1 = q n;1 u n;1 v ;1 1 n = q n;1 u n;1 : (1:2) s i = i;1 t 1 ;(i;1) = q i (u i+1 v ;1 1 v;1 2 t 1v 2 v 1 u ;1 i+1 )q i = q i t 1 q i because [u i v ;1 1 v;1 2 t 1]=,if i 2. c) It is sucient to show that [q n;i q n;1 q i t 1 ]= [q n;i q n;1 q i T ]= 1 i n ; 1: (1:21) Let us assume that 2 i n ; 2. For i =1or i = n ; 1 the equalities (1.21) are clear. From a) by induction we nd q i = q j p j+1 p i = p ;1 i p ;1 j+1 q j if i j: (1:22)

15 Combinatorics of the Gelfand-Tsetlin patterns 15 Consequently, q n;i q n;1 = p n;i+1 p n;1 : Now it is easy to see by induction that So Thus we have p n;i+k;1 = v n;i+k v ;1 k+1 p k 2 k i: p n;i+1 p n;1 =( q n;i q n;1 q i =( iy k=2 iy k=2 v n;i+k v ;1 k+1 ) t 1q i : v n;i+k v ;1 k+1 ) t 1: But the product Q i k=2 v n;i+kv ;1 k+1 does not contain the involutions t 1 and t 2 and therefore commutes with t 1. d) We use the identity (1.22). Let i ; j 1, then s i s j = q i t 1 q i q j t 1 q j = q j (p j+1 p i t 1 p ;1 i p ;1 j+1 t 1)q j : Now let us take j = i ; 1. Then (s j+1 s j ) 3 = q j (p j+1 t 1 p ;1 j+1 t 1) 3 q j = q j t 1 p j+1 t 1 (t 1 p ;1 j+1 t 1p j+1 ) 3 t 1 p ;1 j+1 t 1q j = = q j t 1 p j+1 t 1 (t 2 t 1 ) 6 t 1 p ;1 j+1 t 1q j since t 1 p ;1 j+1 t 1p j+1 = t 1 t 1 t j+1 t 1 t j+1 t 1 = t 2 t 1 t 2 t 1 : e) It is sucient to show (see (1.17) of Proposition 1.4) that if (x) =( 1 n ) then We note that t 1 (q i (x)) = T i+1 ; i (q i (x)): (q i (x)) = ( i+1 i 1 i+2 n ) 1 i n ; 1: Consequently, x 11 (t 1 (q i (x)) = i x 11 (T i+1 ; i (q i (x))) = i+1 ; ( i+1 ; i )= i : Crucial Lemma allows us to reduce any statment about s i to that about s i. For example, the relation (1.24,b) follows from that (1.25,b) if = i+1 ; i+2 and = i ; i+1.

16 16 Anatol N. Kirillov and Arkadiy D. Berenstein g) In fact, by (1.17) and part a), we have s i = q i t 1 q i = p ;1 i q i;1 t 1 q i;1 p i = p ;1 i s i;1 p i = = t 1 t i;2 (t i;1 t i s i;1 t i t i;1 )t i;2 t 1 : However, by assumption we have [es i t 1 ]==[es i t i;2 ]=,where es i is given by (1.19). So s i = es i = t i;1 t i s i;1 t i t i;1. f) Follows from Proposition 1.3, point a). Note that from (1.17) we obtain the formula for s i as a product of (i 2 + i ; 1) simple involutions t j, whereas formula (1.19) gives the expression for s i as a products of 4i ; 3 involutions. For example s 1 = t 1 q 1 = t 1 s 2 = t 1 t 2 t 1 t 2 t 1 q 2 = t 1 t 2 t 1 q 3 = t 1 t 2 t 1 t 3 t 2 t 1 s 3 = t 1 t 2 t 1 t 3 t 2 t 1 t 2 t 3 t 1 t 2 t 1 = t 2 t 3 t 1 t 2 t 1 t 2 t 1 t 3 t 2 : (1:23) Note that equality (1.23) for s 3 is equivalent to the following relation in the group G n, n 4 (see Corollary 1.1): (t 1 q 3 ) 4 =1. Now let us give a formulation of our main result of this Section. Theorem : The involutions s i 1 i n ; 1, satisfy the relations of the symmetric group S n, i.e. a) s 2 i =1 i =1 n; 1 b) (s i s i+1 ) 3 =1 i =1 n; 2 (1:24) c) s i s j = s j s i if 1 i j n ; 1 ji ; jj 2: 2 : The transformations s () i 2 R 1 we have satisfy the colored braid relations, i.e. for any a) s () i b) s () i c) s () i s () i = s (+) i in particular s (;) i =(s () i ) ;1 s (+) i+1 s () i = s () i+1 s(+) i s () i+1 1 i n ; 2 (1:25) s () j = s () j s () i if ji ; jj 2: Corollary 1.1 (of the Theorem 1.1) 1. Wehave the following relations between the generators t i in the group G n 1) t 2 i =1 t i t j = t j t i if ji ; jj 2 2) (t 1 t 2 ) 6 =1 3) (t 1 q i ) 4 =1 if 3 i n ; 1 where q i = p 1 p 2 p i = t 1 t 2 t {z} t 1 3t 2 t {z 1 } t it i;1 t 2 t {z 1 } :

17 Combinatorics of the Gelfand-Tsetlin patterns Besides the relations 1) - 3) of Corollary 1.1, part 1, we have the following relations between the generators t i, and T in the group e Gn 1) T T = T + 2) t 1 T = T ; t 1 t i T = T t i if i 3 3) (T t 2 t 1 T t 2 t 1 ) 3 =1 for any 2 R 1 4) T t 2 T ;+ t 2 T ; t 2 = t 2 T t 2 T ;+ t 2 T ; 5) T q j T q j = q j T q j T 3 j n ; 1 6) (t 1 T q j t 1 T q j ) 2 =1 3 j n ; 1: Proof. 1 The assertions 1) and 2) are proved in Proposition 1.3. On the other hand, if i ; j 2, then s j s i = j;1 (t 1 s i;j+1 ) ;(j;1) and (s j s i ) 2 =1i (t 1 s i;j+1 ) 2 =1. But if i 3, then (t 1 s i ) 2 =(t 1 q i ) 4 : 2. The properties 1) - 4) follow from Proposition 1.3. The identity 5) follows from the fact that the transformations s () i and s () j commute, if ji ; jj 2 (see Theorem 1.1). The last identity 6) follows from an observation that the elements s i s () i and s j s () j commute, if ji ; jj 2, which is a consequence of Corollary 1.4. Corollary 1.2 (of the Theorem 1.1). Let us dene Then s s 1 s n;1 type A (1) n;1, and (s() s := s () = s n;1 s n;2 s 2 s 1 T s 2 s n;1 : are the standard generators of the ane Weyl group of the (x)) = s() ((x)) = ( n + 2 n;1 1 ; ). Proof. It is sucient to show that (s s 1 ) 3 =1and (s s n;1 ) 3 =1. We have the following chains of the equivalent statements i) (s s 1 ) 3 =1() (s 2 s 1 T s 2 s 1 ) 3 =1() (T s 1 s 2 ) 3 =1. The last relation is equivalent to the following one (T t 2 t 1 t 2 t 1 ) 3 = 1, which isa particular case of (1.14,f) when =. ii) (s s n;1 ) 3 =1() [(s 2 s 3 s n;2 s n;1 s n;2 s 3 s 2 )s 1 T ] 3 =1. The last relation is equivalent to one of the form (s 2 s 1 T ) 3 = 1, which is also a particular case of (1.14,f). It is also clear, that s 2 = 1 and s s j = s j s, if 2 j n ; 2. Corollary 1.3. The elements s i s () i 1 i n 2 R 1 satisfy the following relations 1) (s i s () i ) 2 =1 1 i n ; 1 2) (s i s () i s i+1 s () i+1 )3 =1 1 i n ; 2 2 R 1 3) s i s () i s j s () j = s j s () j s i s () i if ji ; jj 2 2 R 1 : Proof. We must prove 2) only when i =1,i.e. (s 1 T s 2 s () 2 )3 =1. But this relation is exactly (1.14,f). The statement 3) follows from Corollary 1.4.

18 18 Anatol N. Kirillov and Arkadiy D. Berenstein Using the same arguments as in the proof of Corollary 1.2, we mayprove that if es := s ( ) = s n;1 s () n;1 s n;2s () n;2 s 2s () 2 s 1s () 1 T s 2 s () 2 s n;1 s () n;1 then es s 1 s () s 1 n;1s () n;1 of the type A (1) and n;1 are the standard generators of the ane Weyl group (s ( ) (x)) = ( n + n + 2 n;1 1 ; n ; ): Let us give some comments. i) Part 1 of Theorem 1.1 follows from part 2 and the equality (1.18). ii) The assertion b) of part 2 follows from Proposition 1.3, part e). In fact, formula (1.25, b) is equivalent to the following one T s (+) 2 T = s () 2 T + s () 2 which in turn coincides with (1.14). iii) Assertion c) of part 2 is equivalent to the following identity T s () j;i+1 = s() j;i+1 T if j ; i 2: (1:26) In fact, using (1.16) we nd that if j i, then s () i s () j = i;1 (T s () j;i+1 )1;i : iv) We assume that relations between the generators t i,1 i n ; 1, (respectively between t i and T ) as described in Corollary1.1, are the dening relations for the group G n (respectively for the group e Gn ). v) Let us say a few words about the group G n for n =3and n =4. G 3 = f t 1 t 2 j t 2 1 = t 2 2 =(t 1 t 2 ) 6 =1g G 4 = f t 1 t 2 t 3 j t 2 1 = t 2 2 = t 2 3 =(t 1 t 3 ) 2 =(t 1 t 2 ) 6 =(t 1 t 2 t 1 t 3 t 2 ) 4 =1g The group G 3 is nite of the order 12, contains the normal subgroup N :=< 1 (t 1 t 2 ) 2 > of the order 3 with the factor group G=N = Z 2 Z 2. The group G 4 seems to be very interesting. For example, for arbitrary n the symmetric group S n is a factor of the group G 4. Namely, let us dene an epimorphism := n : G 4! S n by setting (t 1 )=s 1 (t 2 )=s 2 s 4 s 6 (t 3 )=s 3 s 5 s 7 where s i =(i i +1)2 S n 1 i n ; 1, is a simple transpositions. In fact, one can show that is agree with the relations in the group G 4 and (t 1 ) (t 2 ) and (t 3 ) are really generate the symmetric group S n. Consequently, the group G n is innite, if n 4. The main diculties in the demonstration of Theorem 1.1 appear to be to prove that the generators s () i and s () i commute, if ji ; jj 2, or equivalently, to verify identity (1.26). Our strategy is to prove more precise result about the action of transformations s () i on the triangles. Before stating the theorems about the structure of the mapping s () i, let us dene some additional operators acting on the space of triangles X n.

19 Combinatorics of the Gelfand-Tsetlin patterns 19 Denition 1.2. For each triple of integers (ijk) 1 i<j<k n, let us give a transformation R ijk : X n! X n by the following formulae R ijk (x) =ex x 2 X n where ex ij = x ij + x ik ; x i k;1 ; min(x jk ; x j k;1 x ij ; x i j;1 ) ex jj = x jj ; x ik + x i k;1 + min(x jk ; x j k;1 x ij ; x i j;1 ) (1:27) ex = x if ( ) 6= (i j) or (j j): Proposition 1.5. The operators R ijk satisfy the following relations: a) (R ijk ) 2 =1 b) R ijk R i j k = R i j k R ijk if j(ijk) \ (i j k )j 6= 2 c) (R ijk R ijl R ikl R jkl ) 2 =1 d) (R ijl R ikl ) 3 =1 e) (R ijk (x)) = (x) x 2 X n : Proof. The assertions a) b) and e) are almost evident, and c) d) may be checked by direct computation. Remark 1.1. i) The assertions a) ; c) are essentially due to G. Lusztig [Lu2]. ii) It seems plausible that relations a) ; d) are the dening ones for the group L n, generated by all R ijk with 1 i<j<kn. To go further, let us dene the transformations T (i) and ' i acting on the space X n by the formulae: T (i) (x) =ex ' i (x) =ex x 2 X n where ex ii = x ii ; ex ii = x ii + i+1 (x) ; i (x) both ex and ex are equal to x i ( ) 6= (i i): It is clear from the denitions that if x 2 X n = (x) = ( 1 n ), then ' i (x) =T i ; i+1 (x), and (T (i) (x)) = ( 1 i ; i+1 + n ) (' i (x)) = (i i +1) (x): Now we are ready to state our result about the structure of the transformations s i and s () i. Theorem 1.2. The following equalities are fullled s i = R i;1 i R i;2 i R 1 i ' i R 1 i R i;2 i R i;1 i s () i (1:28a) = R i;1 i R i;2 i R 1 i T (i) R 1 i R i;2 i R i;1 i (1:28b) where 1 i n ; 1 and R jk := R jkk+1 1 j <k n ; 1. The proof of Theorem 1.2 will be given in Appendix.

20 2 Anatol N. Kirillov and Arkadiy D. Berenstein Corollary 1.4. Let x 2 X n x =(x (n) x (1) ). Then 1 : s i (x) = (x (n) x (i+1) ex (i) x (i;1) x (1) ) 1 i n ; 1 and a vector ex (i) 2 R i depends only on the components of the vectors x (i;1) x (i) and x (i+1). 2 : s () i (x) = (x (n) x (i+1) ex (i) x (i;1) x (1) ) and a vector ex (i) depends only on and the components of vectors x (i;1) x (i) and x (i+1). In particular, from Corollary 1.2, part 1, follows that s i t j = t j s i if 1 j <i;1 and consequently, s i s j = s j s i, if i ; j 2. This proves the part 1 of Theorem 1.1 and also the recurrence relation (1.19) for s i. Similarly, from Corollary 1.2, part 2, it follows that T s () i = s () i T for any 2 R and 3 i n ; 1. This proves part 2 of Theorem 1.1 (see (1.26)). As another application of Theorem 1.2 we will deduce a recurrence relation for the transformations s () i, but before doing so it is necessary to introduce some additional notations. Denition 1.3. Let us give a mapping [i i +1]:X n! X n 1 i n ; 1 by [i i +1](x) =ex x 2 X n where ex ki = x k i+1 + x k i;1 ; x ki if k<i ex ii = x i i+1 + x i+1 i+1 ; x ii ex ik = x i+1 k and ex i+1 k = x i k if k>i and for all remaining elements ex = x. Lemma 1.1. The mappings [i i +1] 1 i n ; 1, satisfy the relations of the symmetric group S n, i.e. a) [i i +1] 2 =1 b) [i i +1] [i +1 i+2] [i i +1]=[i +1 i+2] [i i +1] [i +1 i+2] c) [i i +1] [j j +1]=[j j +1] [i i +1] if ji ; jj 2 d) ([i i + 1](x)) = (i i +1)(x) x 2 X n : It is not dicult to show that the group generated by all [i i +1] 1 i n ; 1, is really isomorphic to the symmetric group S n. Proposition 1.6. Let us take i 2. Then s i = R i;1 i [i ; 1 i][i i +1]s i;1 [i i +1][i ; 1 i]r i;1 i s () i = R i;1 i [i ; 1 i][i i +1]s () i;1 [i i +1][i ; 1 i]r i;1 i: (1:29) Proof. It is easy to see, that R k i =[i ; 1 i][i i +1]R k i;1 [i i + 1][i ; 1 i] T (i) =[i ; 1 i][i i +1]T (i) [i i +1][i ; 1 i] (1:3)

21 Combinatorics of the Gelfand-Tsetlin patterns 21 where 1 k < i n ; 1. Identity (1.29) follows by induction from (1.19), (1.3) and Theorem 1.2, parts 1 and 2. The recurrence relations (1.19) and (1.29) are used as an induction base in a rst proof of Theorem 1.2. However, it is possible to solve these recurrence relations in an explicit form and consequently to obtain a second proof of Theorem 1.2. Before describing the solutions of (1.19) and (1.29), let us give the appropriate denitions. It is convenient to use the notations (a) + = max(a ) (a) ; = min(a ) a 2 R: At rst we dene the sequence of piece-wise linear functions Q k n(a 1 a n ), 1 k n, inductively, inthe following way: Q 1 1(a 1 ):=;a 1 Q 1 n(a 1 a n ):=(Q 1 n;1(a 1 a n )) ; +(Q 1 n;1(a 2 a n )) + (1:31) Q k n(a 1 a n ):=Q 1 n(a k a n a 1 a 2 a k;1 ) 1 k n: Secondly, we dene the linear functionals ' ij := ' (n) ij : X n! R 1 i j n ; 1 on the space of triangles X n by the formulae: ' ij (x) :=x i;1 j + x ij ; x i;1 j;1 ; x i j+1 if 2 i j n ; 1 (1:32) ' 11 (x) := ' 1 j (x) :=x 1 j + x j j ; x 1 j+1 ; x j+1 j+1 if 1 <j n ; 1: Theorem 1.3. Let us x a positive integer k, 2 k n ; 1, and a triangle x 2 X n. Assume that Then s k (x) =ex s () k (x) =ex: ex ik = x ik + Q i k(' 2k (x) ' kk (x) ' 1k (x)) (1:33) ex ik = x ik + Q i k(' 2k (x) ' kk (x) ' 1k (x)+ k+1 (x) ; k (x)+): (1:34) The proof of Theorem 1.3 will be given elsewere.we shall give the exact formulae for s k, if k 3, in Section 3. Remark 1.2. We know (see (1.2)) that n = q n;1 u n;1. Assume additionally that (q n;1 u n;1 ) h = 1 h may be equal to 1. Consider the subgroup n G n generated by the elements s i = i;1 t 1 1;i 1 i<nh. Then from Corollary 1.1 it follows that a) s 2 i =1 1 i<nh b) (s i s i+1 ) 3 =1 1 i nh ; 2 c) s i s j = s j s i if 2 ji ; jj n ; 2:

22 22 Anatol N. Kirillov and Arkadiy D. Berenstein Proof. From the denition it is easy to see that s i s j = ;i (t 1 s j;i+1 ) i,ifj i. So (s i s i+1 ) 3 = 1 i (t 1 s 2 ) 3 = 1: But t 1 s 2 = (t 2 t 1 ) 2 and by Proposition 1.3 part c) we know that (t 1 t 2 ) 6 = 1. Similarly (s i s j ) 2 = 1 i (t 1 s j;i+1 ) 2 = 1 and under the assumption 2 ji ; jj n ; 2 the assertion c) follows from Theorem 1.1 part 1 ). In particular, for any 1 a (n ; 1)h the involutions s a s a+n;2 generate a subgroup of G n which is isomorphic to the symmetric group S n. Remark 1.3. The involutions q i 1 i n;1, also give a system of generators for the group G n, because we have t 1 = q 1 t i = q i;1 q i q i;1 q i;2 if i 2 (q := 1): (1:35) The proof of (1.35) follows from Proposition 1.4, point a). In fact, we have q i;1 q i q i;1 q i;2 = q i;1 q i;1 p i p ;1 i;1 q i;2q i;2 = p i p ;1 i;1 = t i if i 2: The relations between the generators q i 1 i n ; 1, follow from Corollary 1.1 and have the following form 1) q 2 i =1 1 i n ; 1 2) (q 1 q 2 ) 6 =1 (q 1 q i ) 4 =1 if 3 i n ; 1 3) [ q i+1 q i q i;1 q i ]= if 3 i n ; 2 (1:36) 4) [ q i q i+1 q i q i;1 q j q j+1 q j q j;1 ]= if ji ; jj 2: In Section 2 we will show that in the case when 2 Z n + is a partition, the restriction of the involution q n;1 on the set K Z ' STY ( n) coincides with the Schutzenberger involution S (see e.g. [Sch1],[Sch2],[EG],[Ki1]). The similarly combinatorial interpretation admits the involutions q i 1 i n ; 1: So our construction gives the extension of the Schutzenberger involutions q i on the space of triangles X n and describes the group generated by q 1 q n;1 i.e. the relations between q 1 q n;1 : The next steps are Theorems 1.1 and 2.3 (see Section 2) which give us the connection between the natural action of the symmetric group S n =< s 1 s n;1 > on the set STY ( n),which was introduced and studied by A. Lascoux and M.-P. Schutzenberger [LS2],[LS3], and the Schutzenberger involutions s i = q i q 1 q i : (1:37) The equality (1.37), restricted on the set STY ( n), is a pure combinatorial assertion and may be deduced directly from the properties of the plactic monoid (e.g. [LS2]) and the Robinson-Schensted correspondence. Details will appear elsewhere.

23 Combinatorics of the Gelfand-Tsetlin patterns 23 Remark 1.4. In a particular case that the a weight 2 R n + has the form =(a n ) i.e. 1 = 2 = = n = a, the group G n conserves the convex polytope K () for any highest weight 2 R n + = ( 1 2 n ). It is easy to see that a restriction of t 1 to the polytope K () becomes the identical map and consequently for all 1 i n ; 1 we have s i j K () = Id K () : The following example shows that (t 3 t 2 ) 6 x 6= x (in fact, for this example (t 3 t 2 ) 24 x = x) and it is not clear whether or not it is possible to nd the subgroup in G n which is isomorphic to the symmetric group S N (for some N) after the restriction of the action of G n on the polytope K (). We nish this section by introducing some additional transformations of the space X n and by considering of the stable behavior of the involutions s i. First, consider the map I: X n! X n which is dened in the following way: let x 2 X n, then I(x) =ex, where Proposition 1.7. We have ex ij = x 1n ; x j;i+1 j 1 i j n: (1:38) 1) i (I(x)) = x 1n ; i (x): 2) The map I commutes with the actions of s i and q i, i.e. Is i = s i I and q i I = Iq i 1 i n ; 1 3) I T = T ; I I s () i = s (;) i I 1 i n ; 1: Proof. Using Denition 1.1 for the involution t j we nd that for x 2 X n, x ij (t j (Ix)) = x 1n ; x j;i+1 j (t j (x)): Consequently, It j = t j I,i.e.the actions of t j and I commute. Secondly, for a partition we consider the restrictions of the maps s (1) i Gelfand-Tsetlin polytope K : to the e i := pr s (1) i f i := pr s (;1) i 1 i n ; 1 (1:39) where pr : X n! K is the projection map.

24 24 Anatol N. Kirillov and Arkadiy D. Berenstein Proposition 1.8. We have s i f i s i = e i 1 i n: Proof. The assertion follows from denitions (1.16), Proposition 1.4 point f) and (1.39). Nowwe consider the stable behavior of the involutions s i and q i. Given the space of triangles X n;1, we dene the following embeddings ' : X n;1,! X n =1 2: given x =(x (n;1) x (1) ) 2 X n;1, then ' 1 (x) =(x (n) x (n;1) x (1) ) (1:4a) where x (n) =(x 1 n;1 x n;1 n;1 ) and ' 2 (x) =(ex (n) ex (1) ) (1:4b) where ex (1) = ex (i+1) =(x 1 i x ii ) 1 i n ; 1: Proposition 1.9. We have 1) s i (' 1 (x)) = s i (x) if 1 i n ; 2 s n;1 (' 1 (x)) = (x 1 n;1 x n;2 n;2 ): (1:41) 2) t i (' 2 (x)) = ' 2 (t i;1 (x)) q i (' 2 (x)) = ' 2 (q i;1 (x)) s i (' 2 (x)) = ' 2 (s i;1 (x)) where 1 i n ; 1: (1:42) Remark 1.5. All our main results (including those dealing with a cocharge c (see Section 3)) after small modication are still valid for the space of truncated triangles X n m m n [BZ1]. By denition the space X n m consists of all sequences x =(x (n) x (m) ) where x (j) = (x 1 j x j j ) 2 R j. In what follows, we always consider the space X n m as the subspace in X n under the constraint x i j = x i j+1 1 i j m ; 1: (1:43) Given a truncated triangle x 2 X n m, it is convenient to use notations (x) :=x (n), (x) :=x (m) and to dene a weight := (x) of a truncated triangle x 2 X n m as a vector = ( m+1 n ) 2 R m such that j = jx (j) j;jx (j;1) j m < j n. For given vectors 2 R n 2 R m and 2 R n;m we consider the following subspaces of the space X n m X n m = fx 2 X n m j (x) =g X n n m = fx 2 X n m j (x) =g (1:44) X n n m() =fx 2 X n n m j (x) =g:

25 Combinatorics of the Gelfand-Tsetlin patterns 25 By denition, a truncated triangle x 2 X n m is called a truncated Gelfand-Tsetlin pattern i x belongs to the cone K n. We denote by K n m the cone of all truncated GT-patterns and its intersections with subspaces (1.44) correspondently by Kn m K n and K n (). It is clear that K n and K n () are the convex, compact polytopes in the space R c n m +, where c n m = 1 (n ; m)(n + m + 1). It is wellknown (see Section 2 or [BZ1 ]), that if and be the partitions, jnj = p 2 and is a composition of the same integer p, then the number of an integral points in the convex polytope K n (respectively in K n ()) is equal to the dimension of the representation V n of the Lie algebra gl n;m (see Section 2)(respectively the dimension of the subspace V ( n ) V n of the weight ). On the other hand, as is also well known, the dimension of the weight subspace V ( n ) V n admits a pure combinatorial description as the number of ( skew) standard Young tableaux of the shape and content jk n Z ()j := jkn () \ Z c n m j = dimv ( n ) =jsty ( n )j: (1:45) Using the constraint (1.43), it is possible to dene the action of the symmetric group S n;m on the space of truncated triangles X n m by setting i = s m+i 1 i n ; m ; 1: (1:46) The fact that the involutions i really generate the symmetric group S n;m follows from Theorem 1.1 and Corollary 1.2. Exercise 1.1. Fix real number and integers i j s.t. 1 i j n. Let us dene a transformation T (i j) : X n! X n in the following way T (i j) (x) :=ex x 2 X n where ex ij = x ij ; and ex kl = x kl, if (k l) 6= (i j). Show that (i) (t j T (i j) ) 2 =1 (ii) t k T (i j) = T (i j) t k if jk ; jj 2 (iii) t k T (i j) t k T (i j) ; t kt (i j) ; = T (i j) which is a generalization of (1.14e). Exercise 1.2. Fix real number q and integer j transformation e tj : X n! X n : e tj := t j [q](x) =ex where ex i k = x i k if k 6= j t k T (i j) ;+ t kt (i j) ; t k if jk ; jj =1 2 R 1 j n. Let us dene a ex i j = min(x i j+1 qx i;1 j;1 )+max(x i+1 j+1 qx i j;1 ) ; qx i j : Here we presuppose that x j := +1 x j j;1 := ;1 1 j n ; 1. Show that (i) e t 2 j =(1; q)e tj + q Id Xn e ti e tj = e tj e ii if ji ; jj 2 (ii) (e tj (x)) = ( 1 j;1 j+1 +(1; q) j q j j+1 n ):

26 26 Anatol N. Kirillov and Arkadiy D. Berenstein where (x) :=( 1 n ) is the weight of triangle x. Thus we obtain a representation of the Hecke algebra H n (q) onthe space of weights R n. Problems. (i) To nd the dening relations between the transformations e tj. (ii) Is it possible to extend this representation of the Hecke algebra H n (q) to the whole space of triangles X n? (iii) Is it possible to construct a cpl-representation of the braid group B n on the space of triangles X n? x2 Combinatorial description of the basic transformations. In this section we give a combinatorial description of the restrictions of the transformations constructed in the previous section to the set of standard Young tableaux of a given shape and content. We will denote this last set by STY ( ). Also we exploit the notations STY ( n ) for the set of skew standard Young tableaux of a skew shape n and content, and STY ( n n) for the set of all skew standard Young tableaux of the shape n with all entries not exceeding n. We will assume in the sequel that l() n l() m l() n for some xed positive integers m n. Atrstwe remind the well-known bijection ( e.g.[gz1],[gz2],[bz1]) between the set STY ( n ) and the set of integral points in the convex polytope K n (). So, take T 2 STY ( n ). As it is well known ( e.g. [Ma] ), one may consider the tableaux T as a sequence of Young diagrams = (m) m+1 (n) = (2:1) such that all skew diagrams (i) n (i;1) m < i n are a horizontal strip. Let us dene the triangle x = x(t ) = (x (n) x (m) ), where x (i) is the shape of the diagram (i). Then we have T 2 STY ( n ) i x(t ) 2 K n Z (): Let us construct the inverse map K n Z ()! STY ( n ). Given a point (), we are lling the shape n by the numbers 1 naccording to the x 2 K n Z following rule: in the i-th row ( n ) i of the skew diagram n we write exactly x k i+m ; x k i+m+1 numbers equal to k, starting from k =1. Let us consider an explanatory example. Assume 1 4 T := =(5 5 4) =(3 1) =( ): Then we have the following sequence of shapes for (2.1): =(3 1) (4 1 1) (4 3 1) (4 4 2) (5 4 4) (5 5 4) = :