GROUPS GENERATED BY INVOLUTIONS GELFAND TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX
GROUPS GENERATED BY INVOLUTIONS GELFAND TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX
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由对合生成的组 GELFAND TSETLIN 模式和年轻 TableAux 的组合
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发表时间：
2011
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通讯作者：
N. O’Connell
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作者：
N. O’Connell
cpl我们构建家庭的分段线性表示表示新形式的对称组Sn和ne e组Sn型n作用于空间三角形Xn我们nd cpl局部不变量的非平凡家庭的作用空间的对称组Sn Xn和构建一个全局不变的w r t e组的新形式ne Sn的作用我们称为cocharge nd的连续模拟Kostka Foulkes多项式和晶体图我们给一个代数版本的一些组合转换组标准的年轻的场景在本文中,我们介绍de ne和研究的一个新阶层的代表符号度量组Sn即连续分段线性表示cpl Sn的表征空间的三角形Xn de定义一个三角形Xn是实数的三角形数组x xij xij R i j n后更精确地广州广州BZ我们考虑Gelfand Tsetlin锥Kn x Xn这样组成的三角形xij xi j我j n xij xi j n xij我n空间里这是一个非简并凸多面锥Xn R n n有发电机看到备注众所周知e g广州积分Anatol n Kirillov Arkadiy D Berenstein点集的Kn Z锥Kn的一一对应设置猪圈n标准年轻的舞台造型让所有条目不超过n的cpl作用空间的对称组Sn Xn给出我们的论文是这样保存Gelfand Tsetlin锥Kn在猪圈n这样的行动恰逢对称群年轻组标准舞台造型由Lascoux和M P原理图utzenberger LS LS看到定理我们主要观察是一个伟大的许多年轻组合结构的一系列标准舞台造型e g同步信道utzenberger对合原理图如Ki对偶原理图utzenberger对合同步信道晋升转换原理图如对称群的行动LS LS晶体图结构在猪圈n咔咔cocharge LS Ki的建设可能会转移到Gelfand Tsetlin锥Kn甚至整个空间的三角形Xn我们建筑规划设计是基于考虑小学transforma tj j n和T R De定义假设x Xn tj e x T ee x, e xik xik如果k j e xij分钟xi jξj马克斯xiξxij我们假定x j xj j n ee xik xik如果我k ee x x表示Gn T tn集团由ti我n ti的转换生成n满足以下看到推论的关系我t titj tji如果jj ij ii t t二气如果我n气t t t t t t z伶猴z对合ti的限制标准的年轻舞台造型猪圈给定形状和内容的承认一个简单的组合工作,这些限制是普通不苟为了证明舒尔函数使用对称e g BK SW Sa和部分我们假设关系的德宁关系组Gn通过任何方式组Gn似乎很有趣很容易看到订单的的组G是相等的,但如果n然后Gn是在夜间和对任何n有存在一个满射的的群G上的对称组SN看到评论后必然的集团Gn承认一个扩展e Gn的手段的R e Gn t tn t R组合的的Gelfand Tsetlin模式我们有了以下关系之间的发电机在的组e Gn我t t t 2 t t tjT tj t j n R三世titj tjti如果jj ij iv t t t t t t任何R v t t t t t t t t t tt t vi t t qjt第七t qj j n t qjt qj qjt qjt j n R变换qj de ned的证明我v的关系是基于直接计算的主要di部分出现在证明六世为了更好地理解的关系,让我们考虑以下元素e组Gn si qit问我我n s qit问我我n节的主要结果是定理相当于我vi和断言的关系退化如果我n满足对称的关系组Sn e b s我n茜茜我n c sisj sjsi如果霁jj转换年代我n R满足彩色编织关系我任何R e s我我我我n b年代年代我我我年代年代我n c s j s年代我如果霁jj vi的关系相当于声明,我和年代的转换j通勤如果霁jj在奥得河证明最后声明中我们使用另一个表达式的年代我n作为一个产品Lusztig对合陆我们回忆起相应的定义,因为我们不使用陆中包含完全相同的退化,但他们的类似物的空间三角形Xn Anatol N Kirillov Arkadiy D Berenstein De定义为每个三个整数ijk我j k N让我们德不转换Rijk Xn Xn以下列方式Rijk e x x Xn其中eξj xiξkξk分钟xj k xj kξj xi e xj j xjξkξk分钟xj k xj kξjξj e x如果我j或让我们表示Ln生成的集团所有与我j k Rijk N我们有以下的关系发电机组中Ln我Rijk ii Rijk Ri j k Ri j k Rijk如果ijk我第四j k三世RijkRijlRiklRjkl RijlRikl j k l n我们假设的关系是德宁的集团Ln进一步让我们德ne转换T我表演空间Xn T e x x Xn在第十二e第十二e x如果我j声明,我和年代通勤如果霁jj遵循从以下表达式我n定理年代我Ri iRi我我T R Ri iRi, Rjk Rjkk j k n为基础的身份证明感应和recurence公式年代我看到退化为了获得相应的结果如果看到我们使用至关重要引理假设Xn x n然后如果我我x, x是一个三角形的重量Xn我e x jx我j jx j n e组的关系发电机Gn允许我们构建许多有趣的子组例如让R e Gn xed然后年代年代sn年代n元素的标准发电机对称组sn的组合Gelfand Tsetlin模式让n实数和e si e s我我姐姐为我n让我们把e e年代 e sn e sn e s s e e sn然后e年代es esn的标准发电机n ne韦尔组的类型是很重要的,三角形的cocharge cn x x Xn节是一个不变的w r t任何退化的行动e s n我我让我们总结我们的论文的部分内容我们构建一个家庭cpl的对称群的表示和产线的一个家庭新形式表示的一个ne组n后看到项推论和构造类型的cpl表示彩色编织关系我们开发了一个几何证明的技术分段线性函数之间的一些非琐碎的身份看到定理我们的下一步是给一个组合的枫桥夜泊转换正在考虑和研究连续分段线性不变量r w t s sn生成的对称群的作用空间在第一部分中，我们证明了下列cpl函数在三角形Xn空间上是sj不变量j x min xj xj j xj j j j j j j j j j j j j j j的定理
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