论文标题
$ \ Mathbb f_q [[t]]^d $和螺旋移动操作员的晶格
Lattices in $\mathbb F_q[[T]]^d$ and spiral shifting operators
论文作者
论文摘要
我们研究了Hermite正常形式的类似物的代数和组合，该形式对$ \ Mathbb f_q [[t]]^d $的有限索引s子模型进行了分类。我们将两种正常形式都视为Gröbner基础理论的实例，在不同的单一顺序下，Hermite的正常形式对应于Lex顺序，而新的正常形式则是HLEX秩序。我们注意到，HLEX正常形式恢复了史密斯正常形式，这是Hermite正常形式所没有的功能。我们还确定了由HLEX正常形式引起的细胞分解的基础结构，该结构似乎具有独立感兴趣。值得注意的是，在$ \ Mathbb n^d $上收集了$ d $````螺旋移动操作员'''的统计数据，并以某种方式兼容，$ d $````螺旋移动operator''使用这些操作员，我们通过将所罗门和彼得格拉德斯基的结果以HLEX正常形式翻译而直接证明了一些新的组合身份。
We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of $\mathbb F_q[[T]]^d$. We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of $d$ ``spiral shifting operators'' on $\mathbb N^d$, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon and Petrogradsky in terms of the hlex normal form.
