arXiv subjects

Nicolas M. Thiéry

Publications and source records attributed to Nicolas M. Thiéry.

At least 19 recordsLinked to original sources

arXiv2024

Controlling the C3 super class linearization algorithm

C3 is an algorithm used by several widely used programming languages such as Python to support multiple inheritance in object oriented programming (OOP): for each class, C3 computes recursively a linear extension of the poset of all its super classes (the Method Resolution Order, MRO) from user-provided local information (an ordering of the direct super classes). This algorithm can fail if the local information is not consistent. For large hierarchies of classes, as encountered when modeling hierarchies of concepts from abstract algebra in the SageMath computational system, maintaining consistent local information by hand does not scale and leads to unpredictable C3 failures. This paper reports on the authors' work to analyze and circumvent this maintenance nightmare. First, we discovered through extensive computer exploration that there exists posets admitting no consistent local information; we exhibit the smallest one which has 10 elements. Then, we provide and analyze an algorithm that, given a poset and a linear extension, automatically builds local information for C3 in such a way that guarantees that it will never fail, at the price of a slight relaxation of the hypotheses. This algorithm has been used in production in SageMath since 2013.

math.CO↗

arXiv2021

Using Jupyter for reproducible scientific workflows

Literate computing has emerged as an important tool for computational studies and open science, with growing folklore of best practices. In this work, we report two case studies - one in computational magnetism and another in computational mathematics - where domain-specific software was exposed to the Jupyter environment. This enables high-level control of simulations and computation, interactive exploration of computational results, batch processing on HPC resources, and reproducible workflow documentation in Jupyter notebooks. In the first study, Ubermag drives existing computational micromagnetics software through a domain-specific language embedded in Python. In the second study, a dedicated Jupyter kernel interfaces with the GAP system for computational discrete algebra and its dedicated programming language. In light of these case studies, we discuss the benefits of this approach, including progress toward more reproducible and reusable research results and outputs, notably through the use of infrastructure such as JupyterHub and Binder.

cs.MS↗

arXiv2020

Classification of P-oligomorphic groups, conjectures of Cameron and Macpherson

Let G be a group of permutations of a denumerable set E. The profile of G is the function phi which counts, for each n, the number phi(n) of orbits of G acting on the n-subsets of E. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever the profile phi(n) is bounded by a polynomial -- we say that G is P-oligomorphic --, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked whether the orbit algebra of G -- a graded commutative algebra invented by Cameron and whose Hilbert function is phi -- is finitely generated. In this paper we establish a classification of (closed) P-oligomorphic permutation groups in terms of finite permutation groups with decorated blocks. It follows from the classification that the orbit algebra of any P-oligomorphic group is isomorphic to (a straightforward quotient of) the invariant ring of some finite permutation group. This answers positively both Cameron's conjecture and Macpherson's question. The orbit algebra is in fact Cohen-Macaulay; therefore the generating series of phi is a rational fraction whose numerator has positive coefficients, while the denominator admits a combinatorial description. In addition, the classification provides a finite data structure for encoding closed P-oligomorphic groups. This paves the way for computing with them and enumerating them as well as for proofs by structural induction. Finally, the relative simplicity of the classification gives hopes to extend the study to, e.g., the class of (closed) permutations groups with sub-exponential profile. The proof exploits classical notions from group theory -- notably block systems and their lattice properties --, commutative algebra, and invariant theory.

math.CO↗

arXiv2018

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number, possibly infinite, $φ_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.

math.CO↗

arXiv2018

The orbit algebra of a permutation group with polynomial profile is Cohen-Macaulay

Let $G$ be a group of permutations of a denumerable set $E$. The profile of $G$ is the function $ϕ_G$ which counts, for each $n$, the (possibly infinite) number $ϕ_G(n)$ of orbits of $G$ acting on the $n$-subsets of $E$. Counting functions arising this way, and their associated generating series, form a rich yet apparently strongly constrained class. In particular, Cameron conjectured in the late seventies that, whenever $ϕ_G(n)$ is bounded by a polynomial, it is asymptotically equivalent to a polynomial. In 1985, Macpherson further asked if the orbit algebra of $G$ - a graded commutative algebra invented by Cameron and whose Hilbert function is $ϕ_G$ - is finitely generated. In this paper, we announce a proof of a stronger statement: the orbit algebra is Cohen-Macaulay. The generating series of the profile is a rational fraction whose numerator has positive coefficients and denominator admits a combinatorial description. The proof uses classical techniques from group actions, commutative algebra, and invariant theory; it steps towards a classification of ages of permutation groups with profile bounded by a polynomial.

math.CO↗

arXiv2016

Braid moves in commutation classes of the symmetric group

We prove that the expected number of braid moves in the commutation class of the reduced word $(s_1 s_2 \cdots s_{n-1})(s_1 s_2 \cdots s_{n-2}) \cdots (s_1 s_2)(s_1)$ for the long element in the symmetric group $\mathfrak{S}_n$ is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot's theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.

math.CO↗

arXiv2016

Interoperability in the OpenDreamKit Project: The Math-in-the-Middle Approach

OpenDreamKit --- "Open Digital Research Environment Toolkit for the Advancement of Mathematics" --- is an H2020 EU Research Infrastructure project that aims at supporting, over the period 2015--2019, the ecosystem of open-source mathematical software systems. From that, OpenDreamKit will deliver a flexible toolkit enabling research groups to set up Virtual Research Environments, customised to meet the varied needs of research projects in pure mathematics and applications. An important step in the OpenDreamKit endeavor is to foster the interoperability between a variety of systems, ranging from computer algebra systems over mathematical databases to front-ends. This is the mission of the integration work package (WP6). We report on experiments and future plans with the \emph{Math-in-the-Middle} approach. This information architecture consists in a central mathematical ontology that documents the domain and fixes a joint vocabulary, combined with specifications of the functionalities of the various systems. Interaction between systems can then be enriched by pivoting off this information architecture.

cs.MS↗

arXiv2015

Spectral gap for random-to-random shuffling on linear extensions

In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling.

math.PR↗

arXiv2014

Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile

The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number $φ_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra introduced by P. J. Cameron. In this paper we give a closer look at this association, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that $φ_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $φ_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.

math.CO↗

arXiv2013

The biHecke monoid of a finite Coxeter group and its representations

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w in W a combinatorial module T_w whose support is the interval [1,w]_R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

math.CO↗

arXiv2011

An evaluation approach to computing invariants rings of permutation groups

Using evaluation at appropriately chosen points, we propose a Gröbner basis free approach for calculating the secondary invariants of a finite permutation group. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. This is confirmed by extensive benchmarks using a Sage implementation.

math.CO↗

arXiv2011

On the representation theory of finite J-trivial monoids

In 1979, Norton showed that the representation theory of the 0-Hecke algebra admits a rich combinatorial description. Her constructions rely heavily on some triangularity property of the product, but do not use explicitly that the 0-Hecke algebra is a monoid algebra. The thesis of this paper is that considering the general setting of monoids admitting such a triangularity, namely J-trivial monoids, sheds further light on the topic. This is a step to use representation theory to automatically extract combinatorial structures from (monoid) algebras, often in the form of posets and lattices, both from a theoretical and computational point of view, and with an implementation in Sage. Motivated by ongoing work on related monoids associated to Coxeter systems, and building on well-known results in the semi-group community (such as the description of the simple modules or the radical), we describe how most of the data associated to the representation theory (Cartan matrix, quiver) of the algebra of any J-trivial monoid M can be expressed combinatorially by counting appropriate elements in M itself. As a consequence, this data does not depend on the ground field and can be calculated in O(n^2), if not O(nm), where n=|M| and m is the number of generators. Along the way, we construct a triangular decomposition of the identity into orthogonal idempotents, using the usual Möbius inversion formula in the semi-simple quotient (a lattice), followed by an algorithmic lifting step. Applying our results to the 0-Hecke algebra (in all finite types), we recover previously known results and additionally provide an explicit labeling of the edges of the quiver. We further explore special classes of J-trivial monoids, and in particular monoids of order preserving regressive functions on a poset, generalizing known results on the monoids of nondecreasing parking functions.

math.RT↗

arXiv2010

Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe.

math.QA↗

arXiv2010

Deformed diagonal harmonic polynomials for complex reflection groups

We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version.

math.CO↗

arXiv2009

Algèbre combinatoire et effective: des graphes aux algèbres de Kac, via l'exploration informatique

This manuscript synthesizes almost fifteen years of research in algebraic combinatorics, in order to highlight, theme by theme, its perspectives. In part one, building on my thesis work, I use tools from commutative algebra, and in particular from invariant theory, to study isomorphism problems in combinatorics. I first consider algebras of graph invariants in relation with Ulam's reconstruction conjecture, and then, more generally, the age algebras of relational structures. This raises in return structural and algorithmic problems in the invariant theory of permutation groups. In part two, the leitmotiv is the quest for simple yet rich combinatorial models to describe algebraic structures and their representations. This includes the Hecke group algebras of Coxeter groups which I introduced and which relate to the affine Hecke algebras, but also some finite dimensional Kac algebras in relation with inclusions of factors, and the rational Steenrod algebras. Beside being concrete and constructive, such combinatorial models shed light on certain algebraic phenomena and can lead to elegant and elementary proofs. My favorite tool is computer exploration, and the algorithmic and effective aspects play a major role in this manuscript. In particular, I describe the international open source project *-Combinat which I founded back in 2000, and whose mission is to provide an extensible toolbox for computer exploration in algebraic combinatorics and to foster code sharing among researchers in this area. I present specific challenges that the development of this project raised, and the original algorithmic, design, and development model solutions I was led to develop.

math.CO↗

arXiv2009

The biHecke monoid of a finite Coxeter group

The usual combinatorial model for the 0-Hecke algebra of the symmetric group is to consider the algebra (or monoid) generated by the bubble sort operators. This construction generalizes to any finite Coxeter group W. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it has |W| simple and projective modules. In order to construct a combinatorial model for the simple modules, we introduce for each w in W a combinatorial module whose support is the interval [1,w] in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra. This involves the introduction of a w-analogue of the combinatorics of descents of W and a generalization to finite Coxeter groups of blocks of permutation matrices.

math.CO↗

arXiv2008

Hopf Algebras of Graphs

We define graded Hopf algebras with bases labeled by various types of graphs and hypergraphs, provided with natural embeddings into an algebra of polynomials in infinitely many variables. These algebras are graded by the number of edges and can be considered as generalizations of symmetric or quasi-symmetric functions.

math.CO↗

arXiv2008

Deformation of symmetric functions and the rational Steenrod algebra

In 1999, Reg Wood conjectured that the quotient of Q[x_1,...,x_n] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group S_n. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a non-commutative q-deformation of the ring of symmetric functions, which specializes at q=1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood's conjecture to q formal and to any q complex not of the form -a/b, with a in {1,...,n} and b a positive natural number.

math.CO↗