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Abstract
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The motivation for this article comes from the representation theory of general linear
groups
,
where
is
a
-adic
local field. In a natural way it leads to a partial order on the set of partitions of
. 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
and
satisfy
if there exist
permutations
and
of
such
that
and
have the same
parity and
for all
.
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.”
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Keywords
partitions, Zelevinsky operators, partial order,
representations of $p$-adic groups
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Mathematical Subject Classification 2010
Primary: 05A18, 06A07, 22E50
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Milestones
Received: 7 November 2018
Revised: 26 June 2019
Accepted: 30 June 2019
Published: 18 January 2020
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