Symmetric and Pseudo-Symmetric Numerical Semigroups via Young Diagrams and Their Semigroup Rings

TL;DR

This paper explores the structure of symmetric and pseudo-symmetric numerical semigroups using Young diagrams, introducing new operations and decompositions, and characterizes when their semigroup rings have Gorenstein or Kunz subrings.

Contribution

It introduces novel operations on Young diagrams for these semigroups and provides exact conditions for the existence of Gorenstein and Kunz subrings in their semigroup rings.

Findings

New decompositions of symmetric and pseudo-symmetric semigroups.

Characterization of semigroup rings with Gorenstein subrings.

Criteria for semigroup rings to have Kunz subrings.

Abstract

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into an over semigroup and its dual. It is also given exactly for what kind of numerical semigroup $S$, the semigroup ring $\Bbbk[\![S]\!]$ has at least one Gorenstein subring and has at least one Kunz subring.