Veronese Algebras and Modules of Rings with Straightening Laws

TL;DR

This paper investigates whether Veronese rings of algebras with straightening laws (ASLs) retain the ASL structure, providing positive results in specific cases and constructing supporting posets in low dimensions.

Contribution

It demonstrates that certain Veronese rings and modules of rings with straightening laws can be endowed with ASL structures, expanding understanding of their algebraic properties.

Findings

Veronese rings of Hibi rings have an ASL structure in some cases

Veronese modules of polynomial rings support an ASL structure

Constructs posets supporting ASL structures in dimensions up to three

Abstract

Do the Veronese rings of an algebra with straightening laws (ASL) still have an ASL structure? We give positive answers to this question in some particular cases, namely for the second Veronese algebra of Hibi rings and of discrete ASLs. We also prove that the Veronese modules of the polynomial ring have a structure of module with straightening laws. In dimension at most three we present a poset construction that has the required combinatorial properties to support such a structure.