有理对称函数的计算及其在不变量理论和PI-代数中的应用
Computing with rational symmetric functions and applications to invariant theory and PI-algebras
摘要：
设任意域中系数为d个变量的形式幂级数f是分解为一系列Schur函数的对称函数，并且设f是分母为（1 -单性）形式的二项式的积的有理函数。我们使用1903年Elliott在1916年MacMahon的Partition Analysis中进一步发展的经典组合方法来计算多重性的生成函数（即，系数）的Schur函数的表达式f。它是一个有理函数，分母的形式与f相似。我们应用该方法的几个问题的对称代数，以及在经典的不变理论，代数与多项式的身份，和非交换不变理论的问题。
英文摘要：
Let the formal power series f in d variables with coefficients in an arbitrary field be a symmetric function decomposed as a series of Schur functions, and let f be a rational function whose denominator is a product of binomials of the form (1 - monomial). We use a classical combinatorial method of Elliott of 1903 further developed in the Partition Analysis of MacMahon in 1916 to compute the generating function of the multiplicities (i.e., the coefficients) of the Schur functions in the expression of f. It is a rational function with denominator of a similar form as f. We apply the method to several problems on symmetric algebras, as well as problems in classical invariant theory, algebras with polynomial identities, and noncommutative invariant theory.
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