Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A proof of the Göllnitz-Gordon-Andrews identities via commutative algebra

Rupam Barman, Alapan Ghosh, Gurinder Singh

TL;DR

The paper addresses proving the Göllnitz-Gordon-Andrews partition identities by translating them into an algebraic framework. It builds graded algebras and homogeneous ideals so that generating functions for partition sets coincide with Hilbert-Poincaré series, and then derives recursions that match known partition recurrences. By establishing the equality $\mathcal{C}_{(r-1)J+\ell}=HP_i^{2J+1}$ via a limiting argument in the $q$-adic topology, it provides a general commutative-algebra proof of the identities (with the classical case $J=0$). This approach extends the identities to a broader family, highlighting a deep link between combinatorial partition theory and graded-algebra invariants with potential broader impact in algebraic combinatorics.

Abstract

The Göllnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. Göllnitz and B. Gordon. In this article, we present a commutative algebra proof of the Göllnitz-Gordon-Andrews identities. More generally, we establish a family of identities, the special cases of which are the Göllnitz-Gordon-Andrews identities. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincaré series of suitably constructed graded algebras.

A proof of the Göllnitz-Gordon-Andrews identities via commutative algebra

TL;DR

The paper addresses proving the Göllnitz-Gordon-Andrews partition identities by translating them into an algebraic framework. It builds graded algebras and homogeneous ideals so that generating functions for partition sets coincide with Hilbert-Poincaré series, and then derives recursions that match known partition recurrences. By establishing the equality via a limiting argument in the -adic topology, it provides a general commutative-algebra proof of the identities (with the classical case ). This approach extends the identities to a broader family, highlighting a deep link between combinatorial partition theory and graded-algebra invariants with potential broader impact in algebraic combinatorics.

Abstract

The Göllnitz-Gordon-Andrews identities generalize the partition identities discovered independently by H. Göllnitz and B. Gordon. In this article, we present a commutative algebra proof of the Göllnitz-Gordon-Andrews identities. More generally, we establish a family of identities, the special cases of which are the Göllnitz-Gordon-Andrews identities. In the proof, we relate the generating functions associated with these identities to the Hilbert-Poincaré series of suitably constructed graded algebras.
Paper Structure (3 sections, 7 theorems, 57 equations)

This paper contains 3 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Proof of Theorem \ref{['Thm2']}

Key Result

Theorem 1.1

Let $r$ and $i$ be positive integers with $1\leq i\leq r$. Then

Theorems & Definitions (18)

  • Theorem 1.1: Göllnitz-Gordon-Andrews identities
  • Theorem 1.2
  • Definition 2.1: Graded ring
  • Definition 2.2: Homogeneous ideal
  • Definition 2.3: Graded $\mathbb{F}$-algebra
  • Definition 2.4: Weight of a polynomial
  • Example 2.5: Gradation by weight
  • Definition 2.6: Hilbert-Poincaré series
  • Lemma 3.1
  • proof
  • ...and 8 more