Noncommutative Symmetrical Functions
Noncommutative Symmetrical Functions
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DOI：
10.1006/aima.1995.1032
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发表时间：
1995-05
影响因子：
1.7
通讯作者：
I. Gelfand;D. Krob;A. Lascoux;B. Leclerc;V. Retakh;J. Thibon
中科院分区：
文献类型：
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作者：
I. Gelfand;D. Krob;A. Lascoux;B. Leclerc;V. Retakh;J. Thibon
本文在拟行列式概念的基础上给出了对称函数的一个非交换理论。我们开始与一个正式的理论，对应的情况下，对称函数的无穷多个独立的变量。这使我们能够赋予所得到的代数与一个霍普夫结构，这导致了一个新的方法计算下降代数。它还对一些经典结构给出了统一的重新解释。其次，我们研究对称多项式的非交换类似物。根据所考虑的特定类型的应用，人们会得出不同的结构。例如，当一个中心变量具有非交换系数的多项式被分解为线性因子的乘积时，这些因子的根与展开多项式的根不同。因此，根据一个人是否有兴趣在建设一个多项式与给定的根或在扩大的产品的线性因素，人们必须考虑两个不同的专门化的形式对称函数。第三种类型出现时，人们寻找一个非交换的推广有关的概念，特征多项式的矩阵的应用。这种构造可以应用于，例如，由泛包络代数的生成元形成的非交换矩阵。
This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. It also gives unified reinterpretation of a number of classical constructions. Next, we study the noncommutative analogs of symmetric polynomials. One arrives at different constructions, according to the particular kind of application under consideration. For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. This construction can be applied, for instance, to the noncommutative matrices formed by the generators of the universal enveloping algebraor of