论文标题
字符多项式和限制问题
Character Polynomials and the Restriction Problem
论文作者
论文摘要
字符多项式用于研究一般线性群对其置换矩阵亚组的多项式表示的限制。为计算字符多项式给出的类函数的内部产品获得了一个简单的公式。使用具有Eulerian因素化的生成函数计算对称和交替张量的字符多项式。这些用于计算具有双重性的Weyl模块的多项式。通过为Weyl模块的字符多项式和SPECHT模块多项式的多项式多项式使用，可以轻松计算稳定的限制系数。获得了Weyl模块中对称组不变的尺寸的生成函数。具有两个行，两个列和钩分区的分区，其Weyl模块在对称组下具有非零矢量不变。从严格的多项式函子类别到有限生成的FI模块类别的限制函数对限制问题进行了重新制定。
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.
