Geometric vertex decomposition, Gr\"obner bases, and Frobenius splittings for regular nilpotent Hessenberg varieties

We initiate a study of the Gr\"obner 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 Gr\"obner basis with respect to a suitably chosen monomial order. Our Gr\"obner 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 Gr\"obner 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 $\mathrm{Flags}(\mathbb{C}^n)$.