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A generalization of $q$-deformation of graphic arrangements to simplicial complexes

Tongyu Nian

TL;DR

The paper addresses the generalization of $q$-deformation of graphic arrangements to simplicial complexes and studies the resulting invariants. It defines the generalized $q$-deformation $\\mathcal{S}_\\Delta^q$ for a simplicial complex $\\Delta$ and establishes a $q$-deletion-contraction relation, connecting to the underlying chromatic polynomial via a limiting process. In the 1-dimensional case, it derives a closed form $\\chi(\\mathcal{S}_G^q,t)=(q-1)^\\ell \\chi\left(G,\frac{t-1}{q-1}\right)$ and extends the framework to graphic monomial arrangements $\\mathcal{M}(G,r)$ with $\\chi(\\mathcal{M}(G,r),t)=r^\\ell \\chi(G,\frac{t-1}{r})$, accompanied by explicit freeness bases and exponents transformations. The results unify $q$-deformations with freeness and supersolvability, extendable to fields with primitive $r$-roots of unity, and illuminate the interplay between combinatorial and algebraic properties of generalized hyperplane arrangements.

Abstract

The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and $q$-deformations of graphic arrangements. In this thesis, the author extends the definition of $q$-deformation to simplicial complexes, with the conjecture by Nian, Tsujie, Uchiumi and Yoshinaga. The author also investigates a special case called graphic monomial arrangement, including the characteristic polynomials and freeness with a further extension to fields with primitive roots.

A generalization of $q$-deformation of graphic arrangements to simplicial complexes

TL;DR

The paper addresses the generalization of -deformation of graphic arrangements to simplicial complexes and studies the resulting invariants. It defines the generalized -deformation for a simplicial complex and establishes a -deletion-contraction relation, connecting to the underlying chromatic polynomial via a limiting process. In the 1-dimensional case, it derives a closed form and extends the framework to graphic monomial arrangements with , accompanied by explicit freeness bases and exponents transformations. The results unify -deformations with freeness and supersolvability, extendable to fields with primitive -roots of unity, and illuminate the interplay between combinatorial and algebraic properties of generalized hyperplane arrangements.

Abstract

The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and -deformations of graphic arrangements. In this thesis, the author extends the definition of -deformation to simplicial complexes, with the conjecture by Nian, Tsujie, Uchiumi and Yoshinaga. The author also investigates a special case called graphic monomial arrangement, including the characteristic polynomials and freeness with a further extension to fields with primitive roots.
Paper Structure (6 sections, 15 theorems, 42 equations)

This paper contains 6 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Generalization to simplicial complexes
  4. One-dimensional case
  5. Graphic monomial arrangements
  6. Acknowledgement

Key Result

Theorem 2.1

If $\mathcal{A}\subset \mathbb{K}^\ell$ is free with exponents $(e_1,e_2,\cdots, e_\ell)$, then $\chi(\mathcal{A}, t)=\prod_{i=1}^\ell (t-e_i)$.

Theorems & Definitions (27)

  • Theorem 2.1: terao1981generalized
  • Theorem 2.2: Saito's criterion
  • Theorem 2.3: Terao terao1980arrangements
  • Definition 2.4
  • Definition 2.5
  • Conjecture 2.6: Nian-Tsujie-Uchiumi-Yoshinaga nian2024qdeformationchromaticpolynomialsgraphical
  • Theorem 2.7: Stanley (See edelman1994free-mz), Dirac dirac1961rigid-aadmsduh, Fulkerson-Gross fulkerson1965incidence-pjom, Nian-Tsujie-Uchiumi-Yoshinaga nian2024qdeformationchromaticpolynomialsgraphical
  • Definition 3.1
  • Conjecture 3.2
  • Proposition 3.3
  • ...and 17 more