Séminaire Lotharingien de Combinatoire, 91B.65 (2024), 12 pp.
Pierre Bonnet and Charlotte Hardouin
A Galois Structure on The Orbit
of Large Steps Walks in The Quadrant
Abstract.
The enumeration of weighted walks in the quarter plane reduces to studying
a functional equation with two catalytic variables. When the steps
of the walk are small,
Bousquet-Mélou and Mishna defined a group called the group of
the walk which turned out to be crucial in the classification of
the small steps models.
In particular, its action on the catalytic variables provides a
convenient set of changes
of variables in the functional equation. This particular set
called the orbit has been generalized to models with
arbitrary large steps
by Bostan, Bousquet-Mélou and Melczer (BBMM).
However, the orbit had till now no underlying group.
In this article, we endow the orbit with the action of a Galois
group, which extends
the notion of the group of the walk to models with large steps. As
an application, we
look into a general strategy to prove the algebraicity of models
with small backwards
steps, which uses the
fundamental objects that are invariants and
decoupling. The group action
on the orbit allows us to develop a Galoisian approach to these two notions. Up
to the knowledge of the finiteness of the orbit, this gives
systematic procedures to test their existence and construct them. Our constructions
lead to the first proofs of algebraicity of weighted models with large steps,
proving in particular a conjecture of BBMM, and allowing to find
new algebraic models with large steps.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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