论文标题
$ \ mathrm {gl} _n \ mathbb {f} _q $中的循环类型因法化
Cycle type factorizations in $\mathrm{GL}_n \mathbb{F}_q$
论文作者
论文摘要
Huang，Lewis，Morales，Reiner和Stanton的最新工作表明，$ \ Mathrm {Gl} _N \ Mathbb {F} _Q $的常规椭圆元素与对称组的$ n $ cyccles类似。 1981年，史丹利（Stanley）列举了置换量为$ n $ cycles的产品。我们研究了$ \ mathrm {gl} _n \ mathbb {f} _q $中的类似问题。更准确地说，我们为$ \ mathrm {gl} _n \ Mathbb {f} _q $定义了一个周期类型的概念，并寻求枚举固定数量的常规椭圆元素的元素，其产品具有给定的周期类型。在某些特殊情况下，我们通过为必要的字符值引入简化公式，使用由于Frobenius引起的标准字符理论技术提供明确的公式。我们还针对大的$ Q $解决了计算常规椭圆元素随机元组具有给定周期类型的可能性的问题。最后，我们以列举公式和一些开放性问题的多项式性结论。
Recent work by Huang, Lewis, Morales, Reiner, and Stanton suggests that the regular elliptic elements of $\mathrm{GL}_n \mathbb{F}_q$ are somehow analogous to the $n$-cycles of the symmetric group. In 1981, Stanley enumerated the factorizations of permutations into products of $n$-cycles. We study the analogous problem in $\mathrm{GL}_n \mathbb{F}_q$ of enumerating factorizations into products of regular elliptic elements. More precisely, we define a notion of cycle type for $\mathrm{GL}_n \mathbb{F}_q$ and seek to enumerate the tuples of a fixed number of regular elliptic elements whose product has a given cycle type. In some special cases, we provide explicit formulas, using a standard character-theoretic technique due to Frobenius by introducing simplified formulas for the necessary character values. We also address, for large $q$, the problem of computing the probability that the product of a random tuple of regular elliptic elements has a given cycle type. We conclude with some results about the polynomiality of our enumerative formulas and some open problems.
