A free Lie algebra as a module over the full linear group
A free Lie algebra as a module over the full linear group
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作为完整线性群上的模块的自由李代数
DOI：
10.1070/sm1996v187n02abeh000109
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
V M Zhuravlev
中科院分区：
文献类型：
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作者：
V M Zhuravlev
在本文中，我们考虑特征为零的域上的树李代数。该代数是完整线性群上的模，齐次元素的空间在此作用下保持不变。我们研究齐次空间分解为不可约分量并计算它们的重数。计算这些多重性的一种方法涉及它们与对应于相同长度的独立循环的乘积的元素的共轭类上的对称群的不可约特征的值的联系。在第二部分中，我们给出了计算此类字符值的明确公式。该公式类似于对称群不可约模维数的钩子公式。在计算重数的第二种方法中，我们利用维特公式来计算自由李代数的多齐次分量的维数。本文的其余部分讨论自由双元李代数的希尔伯特级数与该代数中不可约模重数的生成级数之间的关系。
In this paper we consider a tree Lie algebra over a field of characteristic zero. This algebra is a module over the full linear group, and the spaces of homogeneous elements are invariant under this action. We study the decomposition of the homogeneous spaces into irreducible components and calculate their multiplicities. One method for calculating these multiplicities involves their connection with the values of the irreducible characters of the symmetric group on conjugacy classes of elements corresponding to a product of independent cycles of the same length. In the second section we give an explicit formula for calculating such character values. This formula is analogous to the hook formula for the dimension of the irreducible modules of the symmetric group. In the second method for calculating multiplicities we make use of Witt's formula for the dimensions of the polyhomogeneous components of a free Lie algebra. The rest of this paper deal with relations between the Hilbert series of a free two-generator Lie algebra and the generating series of the multiplicities of the irreducible modules in this algebra.