Zelevinsky operations for multisegments and a partial order on partitions

Peter Schneider and Ernst-Wilhelm Zink

Vol. 304 (2020), No. 1, 181–207

DOI: 10.2140/pjm.2020.304.181

Abstract

The motivation for this article comes from the representation theory of general linear groups ${GL}_n\left(F\right)$, where $F$ is a $p$-adic local field. In a natural way it leads to a partial order on the set of partitions of $n\ge 1$. By way of an example we showed in Schneider and Zink (2016) that this partial order is strictly contained in the well known dominance order. Our aim in the present paper is to describe it explicitly in terms of partitions alone, without any reference to representations. It is not too complicated to see that our partial order contains the refinement partial order. Our key result is that the additional relations are generated by the following “basic” relations: Two partitions $\mathcal{P}=\left({\ell }_1,\dots ,{\ell }_s\right)$ and ${\mathcal{P}}^{\prime }=\left({\ell }_1^{\prime },\dots ,{\ell }_s^{\prime }\right)$ satisfy $\mathcal{P}\le {\mathcal{P}}^{\prime }$ if there exist permutations $\sigma$ and $\pi$ of $\left(1,\dots ,s\right)$ such that ${\ell }_i^{\prime }$ and ${\ell }_{\pi \left(i\right)}^{\prime }$ have the same parity and ${\ell }_{\sigma \left(i\right)}=\left({\ell }_i^{\prime }+{\ell }_{\pi \left(i\right)}^{\prime }\right)∕2$ for all $i=1,\dots ,s$. The input from representation theory, coming from the work of Zelevinsky (1980), consists of certain operations on the set of so called multisegments. We prove our result by a careful analysis of the commutation relations between these “Zelevinsky operations.”

Keywords

partitions, Zelevinsky operators, partial order, representations of $p$-adic groups

Mathematical Subject Classification 2010

Primary: 05A18, 06A07, 22E50

Milestones

Received: 7 November 2018

Revised: 26 June 2019

Accepted: 30 June 2019

Published: 18 January 2020