arXiv 1408.35922014-08math.RTmath.CO

经典表示论的组合方法

A combinatorial approach to classical representation theory

Martin Rubey, Bruce W. Westbury

摘要：

不变量理论的一个基本问题是描述多线性函数在群G作用下在表示V上的自同态代数。根据Weyl的经典，第一个主要（后来：基本）定理的不变理论提供了一个有限的生成集，这个代数，而第二个主要定理描述了线性关系之间的那些基本不变式。我们使用图解的方法进行外尔的程序更进一步，提供明确的基础的子空间的第r张量幂的权力V不变的作用下的G，这是另外保存的行动的长周期的对称群r字母。我们研究的表示基本上是那些发生在Weyl的书：定义表示的辛群，定义表示的对称群被认为是线性表示，和伴随表示的线性群。特别是，我们提出了一个透明的，组合证明的第二个基本定理的定义表示的辛群Sp（2n）。我们的公式是明确的，并提供了一个非常精确的链接（n+1）-非交叉完美匹配，超越了维度计数。将我们的论证扩展到这些表示的k次对称幂，所涉及的组合对象原来是（n+1）-非交叉k-正则图。作为推论，我们得到这些对象的循环筛分现象和自然旋转行动的实例。在一般情况下，我们推导出分支规则对应的图代数的表示在一个统一的方式。我们还计算了限制在对称群作用下的图代数的模的Frobenius特征，并且当n比r足够大时，得到了V的r次张量幂的同构分解.

英文摘要：

A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental) theorem of invariant theory provides a finite spanning set for this algebra, whereas a a second main theorem describes the linear relations between those basic invariants. We use diagrammatic methods to carry Weyl's programme a step further, providing explicit bases for the subspace of the r-th tensor powers of V invariant under the action of G, that are additionally preserved by the action of the long cycle of the symmetric group on r letters. The representations we study are essentially those that occur in Weyl's book: the defining representations of the symplectic groups, the defining representations of the symmetric groups considered as linear representations, and the adjoint representations of the linear groups. In particular, we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic groups Sp(2n). Our formulation is explicit and provides a very precise link to (n+1)-noncrossing perfect matchings, going beyond a dimension count. Extending our argument to the k-th symmetric powers of these representations, the combinatorial objects involved turn out to be (n+1)-noncrossing k-regular graphs. As corollaries we obtain instances of the cyclic sieving phenomenon for these objects and the natural rotation action. In general, we derive branching rules for the diagram algebras corresponding to the representations in a uniform way. We also compute the Frobenius characteristics of modules of the diagram algebras restricted to the action of the symmetric group and obtain the isotypic decomposition of the r-th tensor power of V when n is large enough in comparison to r.

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英文摘要：

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