Veronese Algebras and Modules of Rings with Straightening Laws

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.