GROUPS GENERATED BY INVOLUTIONS GELFAND TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX

We construct families of piecewise linear representations cpl representations of the symmetric group Sn and the a ne Weyl group e Sn of type A n acting on the space of triangles Xn We nd a nontrivial family of local cpl invariants for the action of the symmetric group Sn on the space Xn and construct one global invariant w r t the action of the a ne Weyl group e Sn so called cocharge 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 de ne and study a new class of representations of the sym metric group Sn namely the continuous piecewise linear representations cpl representations of Sn in the space of triangles Xn By de nition a triangle x Xn is a triangular array of real numbers x xij xij R i j n More precisely following GZ GZ BZ we consider the Gelfand Tsetlin cone Kn consisting of all triangles x Xn such that xij xi j i j n xij xi j i j n xij i j n This is a nondegenerate convex polyhedral cone in the space Xn R n n having n generators see Remark It is well known e g GZ that the integral Anatol N Kirillov and Arkadiy D Berenstein points set Kn Z of the cone Kn is in a one to one correspondence with the set STY n of standard Young tableaux having all entries not exceeding n The cpl action of the symmetric group Sn on the space Xn given in our paper is such that it conserves the Gelfand Tsetlin cone Kn 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 Sch utzenberger LS LS see Theorem Our main observation is that a great many of combinatorial constructions on the set of standard Young tableaux e g the Sch utzenberger involution Sch EG Ki the dual Sch utzenberger involution Sch a promotion transformation Sch EG the action of the symmetric group LS LS the crystal graph structure on the set STY n Ka Ka the construction of cocharge LS Ki may be transfered to the Gelfand Tsetlin cone Kn and even to the whole space of triangles Xn Our constructions are based on a consideration of elementary transforma tions tj j n and T R De nition Assume that x Xn then tj x e x T x ee x where e xik xik if k j e xij min xi j xi j max xi j xi j xij and we presuppose that x j xj j j n ee xik xik if i k ee x x We denote by Gn t tn the group generated by ti i n The transformations ti i n satisfy the following relations see Corollary i t i titj tji if jj ij ii t t ii t qi if i n where qi t t t z t t t z titi t z The restriction of the involutions ti to the set of standard Young tableaux STY of a given shape and content admits a simple combinatorial in terpretation 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 A We assume that the relations are the de ning relations for the group Gn By any way the group Gn seems to be very interesting It is easy to see that the order of the group G is equal to But if n then Gn is in nite and for any N there exist an epimorphism of the group G on the symmetric group SN see comments after Corollary The group Gn admits an extension e Gn by means of R e Gn t tn T R Combinatorics of the Gelfand Tsetlin patterns We have the following relations between the generators in the group e Gn i T T T ii t T tjT tj T j n R iii titj tjti if jj ij iv T t t T t t for any R v T t T t T t t T t T t T vi t T qjt T qj j n vii T qjT qj qjT qjT j n R where transformation qj is de ned in The proofs of the relations i v are based on direct computations The main di culties arise in the proof of the vi In order to understand better the relations and let us consider the following elements in the group e Gn si qit q i i n s i qiT q i i n The main result of Section is Theorem which is equivalent to the relations i vi and asserts that The involutions si i n satisfy the relations of the symmetric group Sn i e a s i i n b sisi i n c sisj sjsi if ji jj The transformations s i i n R satisfy the colored braid relations i e for any R we have a s i s i s i i n b s i s i s i s i s i s i i n c s i s j s j s i if ji jj The relation vi is equivalent to the statement that the transformations s i and s j commute if ji jj In oder to prove the last statement we use another expression for s i i n as a product of the Lusztig involutions Lu We recall the corresponding de nitions because we use not exactly the same involutions that contained in Lu but their analogs for the space of triangles Xn Anatol N Kirillov and Arkadiy D Berenstein De nition For each triple of integers ijk i j k n let us de ne a transformations Rijk Xn Xn in the following manner Rijk x e x x Xn where e xi j xi j xi k xi k min xj k xj k xi j xi j e xj j xj j xi k xi k min xj k xj k xi j xi j e x x if i j or j j Let us denote by Ln the group generated by all Rijk with i j k n We have the following relations between the generators in the group Ln i Rijk ii Rijk Ri j k Ri j k Rijk if j ijk i j k j iii RijkRijlRiklRjkl iv RijlRikl if i j k l n We assume that the relations are the de ning ones for the group Ln To go further let us de ne the transformations T i acting on the space Xn T i e x x Xn where e xii xii e x x if i j The statement that s i and s j commute if ji jj follows from the following expression for s i i n Theorem s i Ri iRi i R i T i R i Ri iRi i where Rjk Rjkk j k n The proof of identity is based on induction and on the recurence formula for s i see In order to obtain the corresponding results for the involutions si see a we use Crucial Lemma Assume x Xn x n then si x s i i i x where x is a weight of the triangle x Xn i e i x jx i j jx i j i n x The relations between generators of the group e Gn allow us to construct many interesting subgroups in e Gn For example Let R be xed then the elements s s s s sn s n are the standard generators of the symmetric group Sn Combinatorics of the Gelfand Tsetlin patterns Let n be the real numbers and e si e s i i sis i i for i n Let us put e s e s  e sn e sn e s s e s e sn Then e s es esn are the standard generators of the a ne Weyl group of the type A n It is important that the cocharge cn x of a triangle x Xn Section is an invariant w r t the action of any involutions e s i i i n Let us summarize the content of the Section of our paper We construct a family of the cpl representations of the symmetric group nd a family of the cpl representations of the a ne Weyl group of the type A n see item after Corollary 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 Our next step is to give a combinatorial inter pretation 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 sn on the space of triangles Xn In Section we prove see Theorem that the following cpl functions on the space of triangles Xn are sj invariants j x min x j x j xj j xj j ij x min xi j xi j xi j xi j