Doset Hibi rings with an application to invariant theory

TL;DR

This paper introduces doset Hibi rings, a new class of algebraic structures, and applies them to study invariant rings, demonstrating their normality, Cohen-Macaulay property, and singularity characteristics.

Contribution

It defines doset Hibi rings and extends their application to analyze invariant rings of orthogonal groups, establishing their algebraic properties.

Findings

Rings of absolute orthogonal invariants are normal and Cohen-Macaulay.

These rings have rational singularities in characteristic zero.

Criteria for Gorenstein property are provided.

Abstract

We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigrouprings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is F-rational otherwise. We also state criteria of Gorenstein property of these rings.