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Jean-Baptiste Gramain

Publications and source records attributed to Jean-Baptiste Gramain.

At least 19 recordsLinked to original sources
arXiv2026

Environment-Driven Emergence of Higher-Order Collective Behavior

Collective behavior is commonly attributed to direct interactions among system components. Using a minimal stochastic model, we show that higher-order collective structure can instead emerge from shared stochastic environments, even in the absence of interactions. Quantified via the O-information, environmental fluctuations induce both redundant and synergistic dependencies, with the latter occupying larger regions of the correlation space. We establish a no-go theorem showing that time-independent coupling between the system variables and a shared stochastic environment rules out synergistic higher-order behavior. Crucially, this constraint can be overcome dynamically: transitions between redundancy and synergy arise from time-dependent environmental coupling or from the nontrivial interplay between shared environments and direct interactions. Together, these results identify environmental mediation as a distinct mechanism of higher-order collective organization beyond the conventional interaction-centric paradigm.

physics.soc-ph↗
arXiv2025

Generalised hook lengths and Schur elements for Hecke algebras

We compare two generalisations of the notion of hook lengths for partitions. We apply this in the context of the modular representation theory of Ariki-Koike algebras. We show that the Schur element of a simple module is divisible by the Schur element of the associated (generalised) core. In the case of Hecke algebras of type $A$, we obtain an even stronger result: the Schur element of a simple module is equal to the product of the Schur element of its core and the Schur element of its quotient.

math.RT↗
arXiv2023

A discrete model for the growth and spread of the Scottish populations of red squirrels (Sciurus vulgaris) and grey squirrels (Sciurus carolinensis)

In this article, a model, discrete in space and time, is developed to describe the growth and spread of the Scottish populations of red squirrels (Sciurus vulgaris) and grey squirrel (Sciurus carolinensis). The initial state for the model is designed using a large dataset of records of sightings of individuals of both species reported by members of the public. Choices of parameters involved in the model and their values are informed by the analysis of this dataset for the period 2011-2016, and model predictions are compared to records for the years 2006-2019.

q-bio.PE↗
arXiv2020

On unitriangular basic sets for symmetric and alternating groups

We study the modular representation theory of the symmetric and alternating groups. One of the most natural ways to label the irreducible representations of a given group or algebra in the modular case is to show the unitriangularity of the decomposition matrices, that is, the existence of a unitriangular basic set. We study several ways to obtain such sets in the general case of a symmetric algebra. We apply our results to the symmetric groups and to their Hecke algebras and thus obtain new ways to label the simple modules for these objects. Finally, we show that these sets do not always exist in the case of the alternating groups by studying two explicit cases in characteristic 3.

math.RT↗
arXiv2020

Simultaneous core partitions with nontrivial common divisor

A tremendous amount of research has been done in the last two decades on $(s,t)$-core partitions when $s$ and $t$ are positive integers with no common divisor. Here we change perspective slightly and explore properties of $(s,t)$-core and $(\bar{s},\bar{t})$-core partitions for $s$ and $t$ with nontrivial common divisor $g$. We begin by revisiting work by D. Aukerman, D. Kane and L. Sze on $(s,t)$-core partitions for nontrivial $g$ before obtaining a generating function for the number of $(\bar{s},\bar{t})$-core partitions of $n$ under the same conditions. Our approach, using the $g$-core, $g$-quotient and bar-analogues, allows for new results on $t$-cores and self-conjugate $t$-cores that are {\it not} $g$-cores and $\bar{t}$-cores that are {\it not} $\bar{g}$-cores, thus strengthening positivity results of K. Ono and A. Granville, J. Baldwin et. al., and I. Kiming. We then detail a new bijection between self-conjugate $(s,t)$-core and $(\bar{s},\bar{t})$-core partitions for $s$ and $t$ odd with odd, nontrivial common divisor $g$. Here the core-quotient construction fits remarkably well with certain lattice-path labelings due to B. Ford, H. Mai, and L. Sze and C. Bessenrodt and J. Olsson. Along the way we give a new proof of a correspondence of J. Yang between self-conjugate $t$-core and $\bar{t}$-core partitions when $t$ is odd and positive. We end by noting $(s,t)$-core and $(\bar{s}, \bar{t})$-core partitions inherit Ramanujan-type congruences from those of $g$-core and $\bar{g}$-core partitions.

math.CO↗
arXiv2019

Restriction of characters to subgroups of wreath products and basic sets for the symmetric group

In this paper, we give the decomposition into irreducible characters of the restriction to the wreath product $\mathbb{Z}_{p-1} \wr \mathfrak{S}_w$ of any irreducible character of $(\mathbb{Z}_p \rtimes \mathbb{Z}_{p-1}) \wr \mathfrak{S}_w$, where $p$ is any odd prime, $w \geq 0$ is an integer, and $\mathbb{Z}_p$ and $\mathbb{Z}_{p-1}$ denote the cyclic groups of order $p$ and $p-1$ respectively. This answers the question of how to decompose the restrictions to $p$-regular elements of irreducible characters of the symmetric group $\mathfrak{S}_n$ in the $\mathbb{Z}$-basis corresponding to the $p$-basic set of $\mathfrak{S}_n$ described by Brunat and Gramain in [1]. The result is given in terms of the Littlewood-Richardson coefficients for the symmetric group.

math.RT↗
arXiv2015

Perfect isometries and Murnaghan-Nakayama rules

This article is concerned with perfect isometries between blocks of finite groups. Generalizing a method of Enguehard to show that any two p-blocks of (possibly different) symmetric groups with the same weight are perfectly isometric, we prove analogues of this result for p-blocks of alternating groups (where the blocks must also have the same sign when p is odd), of double covers of alternating and symmetric groups (for p odd, and where we obtain crossover isometries when the blocks have opposite signs),of complex reflection groups G(d,1,n) (for d prime to p), of Weyl groups of type B and D (for p odd), and of certain wreath products. In order to do this, we need to generalize the theory of blocks, in a way which should be of independent interest.

math.RT↗
arXiv2011

On a conjecture of G. Malle and G. Navarro on nilpotent blocks

In a recent article, G. Malle and G. Navarro conjectured that the $p$-blocks of a finite group all of whose height 0 characters have the same degree are exactly the nilpotent blocks defined by M. Broué and L. Puig. In this paper, we check that this conjecture holds for spin-blocks of the covering group $2.\A_n$ of the alternating group $\A_n$, thereby solving a case excluded from the study of quasi-simple groups by Malle and Navarro.

math.RT↗
arXiv2011

On bar lengths in partitions

In this paper, we present, given a odd integer $d$, a decomposition of the multiset of bar lengths of a bar partition $λ$ as the union of two multisets, one consisting of the bar lengths in its $\bar{d}$-core partition $\bar{c}_d(λ)$ and the other consisting of modified bar lengths in its $\bar{d}$-quotient partition. In particular, we obtain that the multiset of bar lengths in $\bar{c}_d(λ)$ is a sub-multiset of the multiset of bar lengths in $λ$. Also we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of the symmetric group. The proof involves a recent similar result for partitions, proved in [1].

math.CO↗
arXiv2011

Generalized hook lengths in symbols and partitions

In this paper, we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G. Malle and G. Navarro in [3].

math.CO↗
arXiv2010

On core and bar-core partitions

If $s$ and $t$ are relatively prime J. Olsson proved in 2008 that the $s$-core of a $t$-core partition is again a $t$-core partition, and that the $s$-bar-core of a $t$-bar-core partition is again a $t$-bar-core partition. Here generalized results are proved for partitions and bar-partitions when the restriction that $s$ and $t$ be relatively prime is removed.

math.CO↗
arXiv2009

A 2-basic set of the alternating group

In this note, we construct a 2-basic set of the alternating group \A_n. To do this, we construct a 2-basic set of the symmetric group \sym_n with an additional property, such that its restriction to \A_n is a 2-basic set. We adapt here a method developed in \cite{BrGr} for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Külshammer, Olsson and Robinson in \cite{KOR}.

math.RT↗
arXiv2009

Defect of characters of the symmetric group

Following the work of B. Kuelshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the \ell-defect of characters of the symmetric group S_n, where \ell > 1 is an arbitrary integer. We prove that the \ell -defect is given by an analogue of the hook-length formula, and use it to prove, when n < \ell^2, an \ell-version of the McKay Conjecture in S_n .

math.RT↗
arXiv2008

A basic set for the alternating group

This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric group $\sym_n$ has a $p$-basic set with some additional properties, allowing us to deduce a $p$-basic set for $\A_n$. Our main tool is the generalized perfect isometries introduced by Külshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition number of $\A_n$.

math.RT↗
arXiv2007

Generalized blocks of unipotent characters in the finite general linear group

In a paper of 2003, B. Külshammer, J. B. Olsson and G. R. Robinson defined $\ell$-blocks for the symmetric groups, where $\ell >1$ is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this paper a definition of $d$-sections in the finite general linear group, and construct $d$-blocks of unipotent characters, where $d \geq 1$ is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer's Second Main Theorem.

math.RT↗