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Partition identities associated with $A_r$-Surface singularities

Pooneh Afsharijoo, Pedro D. González Pérez, Hussein Mourtada

TL;DR

This work connects $A_r$-surface singularities to combinatorial partition identities via arc Hilbert-Poincaré series. It introduces 3-colored partitions and a family $\mathcal{F}_r$, proving that the counts $F_r(n)$ are independent of $r$ for all $r\ge2$ and equal to the number of partitions in which the part $1$ can appear in 3 colors while other parts appear in 2 colors; the generating-series is given by $\sum_{n\ge0} F_r(n) q^n=\frac{\mathbf{H}^2}{(q)_1}$ with $\mathbf{H}=\prod_{j\ge1}(1-q^j)^{-1}$. The proof synthesizes arc-space geometry, conjectural descriptions of initial ideals for $A_{r-1}$ singularities, and intricate generating-series manipulations anchored in Durfee-rectangle frameworks and Gordon-type identities, yielding a unifying infinite family of partition identities tied to $A_r$-type singularities.

Abstract

We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and Andrews-Gordon identities, as well as in their recent extensions. The identities established in this paper are associated with the $A_r$ surface singularities via the arc HP-series, which provides a measure of singularities of algebraic varieties defined using arc spaces.

Partition identities associated with $A_r$-Surface singularities

TL;DR

This work connects -surface singularities to combinatorial partition identities via arc Hilbert-Poincaré series. It introduces 3-colored partitions and a family , proving that the counts are independent of for all and equal to the number of partitions in which the part can appear in 3 colors while other parts appear in 2 colors; the generating-series is given by with . The proof synthesizes arc-space geometry, conjectural descriptions of initial ideals for singularities, and intricate generating-series manipulations anchored in Durfee-rectangle frameworks and Gordon-type identities, yielding a unifying infinite family of partition identities tied to -type singularities.

Abstract

We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and Andrews-Gordon identities, as well as in their recent extensions. The identities established in this paper are associated with the surface singularities via the arc HP-series, which provides a measure of singularities of algebraic varieties defined using arc spaces.
Paper Structure (4 sections, 14 theorems, 83 equations, 1 figure)

This paper contains 4 sections, 14 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A possible initial ideal of the space of arc at an $A_{r-1}$ singularity
  3. Useful generating series
  4. Hilbert-Poincaré series of the algebra ${\mathcal{S}}/J_r$

Key Result

Theorem A

(Gordon's identities). Given integers $r\geq 2$ and $1\leq i \leq r,$ let $\mathcal{B}_{r,i}(n)$ denote the set of partitions of $n$ of the form $(\lambda_1,\dots, \lambda_s)$, where $\lambda_{j}-\lambda_{j+r-1} \geq 2$ and at most $i-1$ parts equal to $1$ and denote its cardinal by $B_{r,i}(n)$. Le

Figures (1)

  • Figure 1: The Young diagram of the partition $\lambda=(7,7,6,4,3,3,2)$. Durfee rectangles and Durfee squares in the figure show that $\lambda \in \mathcal{D}_{4,2}(32)$.

Theorems & Definitions (30)

  • Theorem A
  • Theorem B
  • Conjecture 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 20 more