Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts
Splitting the Square of a Schur Function into its Symmetric and Antisymmetric Parts
复制标题
将 Schur 函数的平方分为对称部分和反对称部分
DOI：
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
B. Leclerc
中科院分区：
文献类型：
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作者：
Christophe Carré;B. Leclerc
我们提出了两个舒尔函数积的一种新的组合描述。在舒尔函数SI的平方的特殊情况下，它允许以一种非常自然的方式区分平方的对称和反对称部分。换句话说，它同时描述了体积体S2（SI）和Λ2（SI）在舒尔函数的基础上的膨胀。更一般地说，我们对多重性cIJK = SISJ， skk的组合解释导致了这些多重性有趣的q-类似物cIJK(q)。我们使用的组合对象是多米诺表，即由1 × 2的矩形盒子组成的表，其中装满了沿行弱递增和沿列严格递增的整数。许多作者已经考虑过标准的多米诺骨牌表[33]，[6],[34],[8],[1]，但是，据我们所知，用Yamanouchi多米诺骨牌表来表达Littlewood-Richardson系数是新的，以及第7节中描述的双注入，以及第8节中定义的多米诺骨牌表的对角线类的概念。这种构造导致了一个新的对称函数族（h函数）的定义，其相关性质将在第9节中总结。
We propose a new combinatorial description of the product of two Schur functions. In the particular case of the square of a Schur function SI, it allows to discriminate in a very natural way between the symmetric and antisymmetric parts of the square. In other words, it describes at the same time the expansion on the basis of Schur functions of the plethysms S2(SI) and Λ2(SI). More generally our combinatorial interpretation of the multiplicities cIJK = SISJ, SKleads to interesting q-analogues cIJK(q) of these multiplicities. The combinatorial objects that we use are domino tableaux, namely tableaux made up of 1 × 2 rectangular boxes filled with integers weakly increasing along the rows and strictly increasing along the columns. Standard domino tableaux have already been considered by many authors [33], [6], [34], [8], [1], but, to the best of our knowledge, the expression of the Littlewood-Richardson coefficients in terms of Yamanouchi domino tableaux is new, as well as the bijection described in Section 7, and the notion of the diagonal class of a domino tableau, defined in Section 8. This construction leads to the definition of a new family of symmetric functions (H-functions), whose relevant properties are summarized in Section 9.