Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Young Diagram Decompositions for Almost Symmetric Numerical Semigroups

Mehmet Yeşil

TL;DR

This work develops a unified combinatorial framework connecting almost symmetric numerical semigroups to Young diagrams, enabling new decomposition theorems via diagrammatic sums. By extending previous irreducible-case decompositions, it shows that almost symmetric semigroups with consecutive pseudo-Frobenius numbers (excluding the Frobenius number) admit a unique expression as a combination of a numerical semigroup, its dual, and an ordinary semigroup, realized through bonded, end-to-end, and conjoint diagram sums. The approach hinges on the numerical-set/Young-diagram correspondence, duality, and hook-length analyses to characterize when the sums yield numerical semigroups. The results provide explicit construction rules and verifiable conditions (in terms of conductors, generators, and gaps), with illustrative examples that highlight the combinatorial and algebraic synergy between the two representations. This has potential implications for structural classification and algorithmic generation of almost symmetric semigroups via the diagrammatic toolkit.

Abstract

This paper introduces new structural decompositions for almost symmetric numerical semigroups through the combinatorial lens of Young diagrams. To do that, we use the foundational correspondence between numerical sets and Young diagrams, which enables a visual and algorithmic approach to studying properties of numerical semigroups. Central to the paper, a decomposition theorem for almost symmetric numerical semigroups is proved, which reveals that such semigroups can be uniquely expressed as a combination of a numerical semigroup, its dual and an ordinary numerical semigroup.

Young Diagram Decompositions for Almost Symmetric Numerical Semigroups

TL;DR

This work develops a unified combinatorial framework connecting almost symmetric numerical semigroups to Young diagrams, enabling new decomposition theorems via diagrammatic sums. By extending previous irreducible-case decompositions, it shows that almost symmetric semigroups with consecutive pseudo-Frobenius numbers (excluding the Frobenius number) admit a unique expression as a combination of a numerical semigroup, its dual, and an ordinary semigroup, realized through bonded, end-to-end, and conjoint diagram sums. The approach hinges on the numerical-set/Young-diagram correspondence, duality, and hook-length analyses to characterize when the sums yield numerical semigroups. The results provide explicit construction rules and verifiable conditions (in terms of conductors, generators, and gaps), with illustrative examples that highlight the combinatorial and algebraic synergy between the two representations. This has potential implications for structural classification and algorithmic generation of almost symmetric semigroups via the diagrammatic toolkit.

Abstract

This paper introduces new structural decompositions for almost symmetric numerical semigroups through the combinatorial lens of Young diagrams. To do that, we use the foundational correspondence between numerical sets and Young diagrams, which enables a visual and algorithmic approach to studying properties of numerical semigroups. Central to the paper, a decomposition theorem for almost symmetric numerical semigroups is proved, which reveals that such semigroups can be uniquely expressed as a combination of a numerical semigroup, its dual and an ordinary numerical semigroup.
Paper Structure (11 sections, 12 theorems, 37 equations, 1 figure, 3 algorithms)

This paper contains 11 sections, 12 theorems, 37 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Numerical Sets and Semigroups
  3. Numerical Sets and Young Diagrams
  4. Operations on Young Diagrams
  5. Decompositions of Almost Symmetric Numerical Semigroups
  6. Case 1:
  7. Case 2:
  8. Case 1:
  9. Case 2:
  10. Case 1:
  11. Case 2:

Key Result

Proposition 3.2

N1 Let $R=\{0,r_{1},\dots,r_{n-1},r_{n},\rightarrow\}$ be a numerical set with the associated Young diagram $Y_{R}$. Then:

Figures (1)

  • Figure 1: Young diagram of $R\boxplus_{C}\{0,t+1,\rightarrow\}\boxplus_{B}R^{*}=R\boxplus_{C}\{0,t,\rightarrow\}\boxplus_{E}R^{*}$

Theorems & Definitions (31)

  • Example 2.1
  • Example 3.1
  • Proposition 3.2
  • Example 3.3
  • Definition 4.1
  • Example 4.2
  • Definition 4.3
  • Definition 4.4
  • Lemma 4.5
  • Remark 4.6
  • ...and 21 more