论文标题
几何顶点分解，Gröbner基地和Frobenius分裂，用于常规的Hessenberg品种
Geometric vertex decomposition, Gröbner bases, and Frobenius splittings for regular nilpotent Hessenberg varieties
论文作者
论文摘要
我们通过研究定期的nilpotent nilpotent nilpotent hessenberg品种的特殊情况，并专注于$ \ mathrm {flags}（\ mathbb {c}^n）\ cong gl_n（c cong gl_n（c cont n）\ cont gl_n（c congl gl_n（c），我们通过研究定期nilpotent Hessenberg品种的特殊情况，对Hessenberg品种定义理想的Gröbner几何形状进行研究。最长的元素$ w_0 $ w_0 $ s_n $ of $ gl_n（\ mathbb {c}）$。我们的主要结果如下。令$ h $成为不可分解的Hessenberg功能。我们证明，$ w_0 $ - $ w_0 $ - bron的本地定义理想$ i_ {w_0，h} $的常规nilpotent hessenberg品种$ \ mathrm {hess}（\ sathsf {n}，h）$与$ h $相关的$ h $与适合选择的monomial corpsion monomial corpect ofgröbner基础。我们的gröbner基础由一个集合组成，$ \ {f^{w_0} _ {k，\ ell} \} $ $ i_ {w_0，h} $的生成器，由abe，dedieu，galetto，galetto和第二作者获得。我们还证明$ i_ {w_0，h} $在Klein和Rajchgot（基于Knutson，Miller和Yong的作品建立）上是几何顶点分解的。我们给出了上述结果的两个不同的证据。我们做出了这种非常规的博览会选择，因为我们的第一个证据引入并利用了具有独立关注的三角形完整交叉点的概念，而我们使用联络理论的第二个证明更可能是可以推广到$ w \ w \ neq w_0 $的一般$ w $ charts。 Finally, using our Gröbner analysis of the $f^{w_0}_{k,\ell}$ above and for $p>0$ any prime, we construct an explicit Frobenius splitting of the $w_0$-chart of $\mathrm{Flags}(\mathbb{C}^n)$ which simultaneously compatibly splits all the local defining ideals of $ i_ {w_0，h} $，AS $ h $范围在一组不可分解的Hessenberg函数上。最后的结果是以$ \ mathrm {flags}（\ Mathbb {c}^n）$而闻名的经典结果的本地Hessenberg类似物，以及Schubert和相反的Schubert品种的收藏中的$ \ Mathrm {flags}（\ Mathbb {C}^n）$。
We initiate a study of the Gröbner geometry of local defining ideals of Hessenberg varieties by studying the special case of regular nilpotent Hessenberg varieties in Lie type A, and focusing on the affine coordinate chart on $\mathrm{Flags}(\mathbb{C}^n) \cong GL_n(\mathbb{C})/B$ corresponding to the longest element $w_0$ of the Weyl group $S_n$ of $GL_n(\mathbb{C})$. Our main results are as follows. Let $h$ be an indecomposable Hessenberg function. We prove that the local defining ideal $I_{w_0,h}$ in the $w_0$-chart of the regular nilpotent Hessenberg variety $\mathrm{Hess}(\mathsf{N},h)$ associated to $h$ has a Gröbner basis with respect to a suitably chosen monomial order. Our Gröbner basis consists of a collection $\{f^{w_0}_{k,\ell}\}$ of generators of $I_{w_0,h}$ obtained by Abe, DeDieu, Galetto, and the second author. We also prove that $I_{w_0,h}$ is geometrically vertex decomposable in the sense of Klein and Rajchgot (building on work of Knutson, Miller, and Yong). We give two distinct proofs of the above results. We make this unconventional choice of exposition because our first proof introduces and utilizes a notion of a triangular complete intersection which is of independent interest, while our second proof using liaison theory is more likely to be generalizable to the general $w$-charts for $w \neq w_0$. Finally, using our Gröbner analysis of the $f^{w_0}_{k,\ell}$ above and for $p>0$ any prime, we construct an explicit Frobenius splitting of the $w_0$-chart of $\mathrm{Flags}(\mathbb{C}^n)$ which simultaneously compatibly splits all the local defining ideals of $I_{w_0,h}$, as $h$ ranges over the set of indecomposable Hessenberg functions. This last result is a local Hessenberg analogue of a classical result known for $\mathrm{Flags}(\mathbb{C}^n)$ and the collection of Schubert and opposite Schubert varieties in $\mathrm{Flags}(\mathbb{C}^n)$.
