INVERTING THE FROBENIUS MAP
INVERTING THE FROBENIUS MAP
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DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
J. Thibon;J. Thibon
中科院分区：
文献类型：
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作者：
J. Thibon;J. Thibon
著名的Frobenius特征映射是从对称群Sn的特征标空间到n次齐次对称函数空间的双射。本文证明了逆映射的一个公式。更精确地说，Sn的任意虚特征标x的值的母函数用x的Frobenius像的对称函数表示。我们还通过对Hecke代数特征标提供类似的公式，给出了这个结果的^-模拟，并给出了一些应用。§1 .对称群在本注记中，我们沿用Macdonald [M]的术语和符号。具体地，X = A（xi，x-2，.）将表示对称函数的代数（即，著名的Frobenius对应是对称群Sn（的集合共轭类）上的类函数空间与n次齐次对称函数空间之间的线性同构。具体地说，它将每个不可约特征标x发送给相应的舒尔函数s\。由于Sn的共轭类由n的划分来标记，因此对于任何类函数x o n $n，我们可以关联对称生成函数
The famous Frobenius characteristic map is a bijection from the space of characters of the symmetric group S„ to the space of homogeneous symmetric functions of degree n. In this note, a formula for the inverse map is proved. More precisely, the generating function for the values of an arbitrary virtual character x of Sn is expressed in terms of the symmetric function which is the Frobenius image of xWe also give a ^-analogue of this result by providing a similar formula for the Hecke algebra characters, and suggest some applications. §1 .The symmetric group In this note, we follow the terminology and notation of Macdonald [M]. In particular, Л = A(xi,x-2,...) will denote the algebra of symmetric functions (i.e., symmetric formal power series of bounded degree) in infinitely many variables The celebrated Frobenius correspondence is a linear isomorphism between the space of class functions on (the set conjugacy classes of) the symmetric group Sn and the space Л" of homogeneous symmetric functions of degree n. Specifically, it sends each irreducible character x to the corresponding Schur function s\. Since the conjugacy classes of S„ are labeled by partitions of n, with any class function x o n $n w e can associate a symmetric generating function