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Andrews--Gordon type identities with parity restrictions through particle motion

Jehanne Dousse, Jihyeug Jang

Abstract

In this paper, we use the particle motion bijection introduced by Warnaar and developed by the two authors, Jouhet and Konan, to study q-series and partition identities of the Andrews--Gordon type with parity restrictions. These restrictions are of the type ``even (resp. odd) parts appear an even number of times". We prove $q$-series identities where a multisum equals a sum of products, which generalise identities of Andrews and Kim--Yee in a similar way that Stanton's identities generalised the Andrews--Gordon identities. As a consequence of our results, we obtain a simple proof of a recent identity of Chern--Li--Stanton--Xue--Yee related to Ariki--Koike algebras.

Andrews--Gordon type identities with parity restrictions through particle motion

Abstract

In this paper, we use the particle motion bijection introduced by Warnaar and developed by the two authors, Jouhet and Konan, to study q-series and partition identities of the Andrews--Gordon type with parity restrictions. These restrictions are of the type ``even (resp. odd) parts appear an even number of times". We prove -series identities where a multisum equals a sum of products, which generalise identities of Andrews and Kim--Yee in a similar way that Stanton's identities generalised the Andrews--Gordon identities. As a consequence of our results, we obtain a simple proof of a recent identity of Chern--Li--Stanton--Xue--Yee related to Ariki--Koike algebras.
Paper Structure (6 sections, 37 theorems, 129 equations, 1 figure)

This paper contains 6 sections, 37 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Particle motion and its properties
  3. Proof of Theorem \ref{['thm:main']} and Corollary \ref{['cor:odd']}
  4. Proof of Theorem \ref{['thm:even1']}
  5. Proof of \ref{['thm:even2']} and Corollary \ref{['cor:even']}
  6. Open problems

Key Result

Theorem 1.2

For integers $k \geq 1$ and $0 \leq r \leq k$, the number of partitions $\lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$ of $n$ such that $\lambda_i - \lambda_{i+k} \geq 2$ for all $i$, and where the part $1$ appears at most $(k-r)$ times, is equal to the number of partitions of $n$ into pa

Figures (1)

  • Figure 1: Illustration of applying $5$ particle motions starting from index $1$ in the frequency sequence $f = (\dots, 0, f_1=4, f_2=0,2,1,3,1,0,\dots)$. The symbol $\Rightarrow$ indicates a particle motion, and $\rightarrow$ indicates a focus shift.

Theorems & Definitions (67)

  • Example 1.1
  • Theorem 1.2: Gordon's identities Go
  • Theorem 1.3: Andrews--Gordon identities AndrewsGordon
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6: Andrews Andrews2010
  • Theorem 1.7: Andrews Andrews2010
  • Theorem 1.8: Kim--Yee KY2013, sum side by Kurşungöz Kursungoz2010
  • Theorem 1.9: Kim--Yee KY2013, sum side by Kurşungöz Kursungoz2010
  • Theorem 1.10: Unification of Theorems \ref{['thm:1']}--\ref{['thm:4']}
  • ...and 57 more