Geometric Vertex Decomposition, Gröbner Bases, and Frobenius Splittings for Regular Nilpotent Hessenberg Varieties

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 Flags(Cn)≅GLn(C)/B$${\textrm{Flags}}(\mathbb {C}^n) \cong GL_n(\mathbb {C})/B$$ corresponding to the longest element w0$$w_0$$ of the Weyl group Sn$$S_n$$ of GLn(C)$$GL_n(\mathbb {C})$$. Our main results are as follows. …