arXiv subjects

Abdelmalek Abdesselam

Publications and source records attributed to Abdelmalek Abdesselam.

At least 19 recordsLinked to original sources

arXiv2026

A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups

There is a well understood way of generating random coverings of a fixed manifold by sampling homomorphisms from the fundamental group of this manifold into the symmetric group. We prove a central limit theorem for the number of connected components of these random coverings when the fundamental group is nilpotent. This provides a nonabelian generalization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least two. Our result relies on the work of du Sautoy and Grunewald on the subgroup growth zeta functions of nilpotent groups, and on Delange's generalization of the Wiener-Ikehara Tauberian theorem.

math.PR↗

arXiv2026

A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

We prove a central limit theorem (CLT) for the number of joint orbits of random tuples of commuting permutations. In the uniform sampling case this generalizes the classic CLT of Goncharov for the number of cycles of a single random permutation. We also consider the case where tuples are weighted by a factor other than one, per joint orbit. We view this as an analogue of the Ewens measure, for tuples of commuting permutations, where our CLT generalizes the CLT by Hansen. Our proof uses saddle point analysis, in a context related to the Hardy-Ramanujan asymptotics and the theorem of Meinardus, but concerns a multiple pole situation. The proof is written in a self-contained manner, and hopefully in a manner accessible to a wider audience. We also indicate several open directions of further study related to probability, combinatorics, number theory, an elusive theory of random commuting matrices, and perhaps also geometric group theory.

math.PR↗

arXiv2025

A combinatorial formula for the coefficients of multidimensional resultants

The classical multidimensional resultant can be defined as the, suitably normalized, generator of a projective elimination ideal in the ring of universal coefficients. This is the approach via the so-called inertia forms or Trägheitsformen. Using clever substitutions, Mertens and Hurwitz gave a criterion, for recognizing such inertia forms, which amounts to a linear system for their numerical coefficients. In this article we explicitly solve this linear system. We do so by identifying a subset of the available equations which forms a unitriangular system. The key notion we use is that of transversal, i.e., a selection of a monomial term in each of the homogeneous polynomials at hand. We need two such transversals which are disjoint and extremal, in the sense that they relate to extremizers of a, possibly new, determinantal inequality for differences of two substochastic matrices. Thanks to this notion of extremal pair of transversals, we derive an explicit formula for the coefficients of general multidimensional resultants, as a sum of terms made of a sign times a product of multinomial coefficients, thereby explicitly showing they are integer-valued. As an application of our formula, we recover Sombra's bound on the height of resultants, in the classical case.

math.AC↗

arXiv2025

Proof of a conjecture by Starr and log-concavity for random commuting permutations

We prove a conjecture by Shannon Starr regarding the asymptotics for the number of tuples of commuting permutations with given number of joint orbits. These numbers generalize unsigned Stirling numbers of the first kind which count how many single permutations have a given number of cycles. In the case of pairs of permutations, these numbers are related to D'Arcais polynomials and the Nekrasov-Okounkov formula. As a consequence of the above asymptotics, we confirm a log-concavity conjecture in the regime of typical values for the number of joint orbits. As a result of possible indepentent interest in applied mathematics and mathematical physics, we also provide detailed asymptotics, using Mellin transform techniques, for certain multiple series or multivariate Ramanujan sums which are related to ordinary generating functions of Dirichlet convolutions of power laws. Besides these multiple sums asymptotics, our proofs use bivariate saddle point analysis related to the Meinardus theorem in the delicate case of multiple poles for the associated Dirichlet series.

math.CO↗

arXiv2024

Structure constants for simple Lie algebras from principal $\mathfrak{sl}_2$-triple

For a simple complex Lie algebra $\mathfrak{g}$, fixing a principal $\mathfrak{sl}_2$-triple and highest weight vectors induces a basis of $\mathfrak{g}$ as vector space. For $\mathfrak{sl}_n$, we describe how to compute the Lie bracket in this basis using transvectants. This generalizes a well-known rule for $\mathfrak{sl}_2$ using Poisson brackets and degree 2 monomials in two variables. Our proof method uses a graphical calculus for classical invariant theory. Other Lie algebra types are discussed.

math.RT↗

arXiv2024

Bessenrodt--Ono inequalities for $\ell$-tuples of pairwise commuting permutations

Let $S_n$ denote the symmetric group. We consider \begin{equation*} N_{\ell}(n) := \frac{\left\vert Hom\left( \mathbb{Z}^{\ell},S_n\right) \right\vert}{n!} \end{equation*} which also counts the number of $\ell$-tuples $π=\left( π_1, \ldots, π_{\ell}\right) \in S_n^{\ell}$ with $π_i π_j = π_j π_i$ for $1 \leq i,j \leq \ell$ scaled by $n!$. A recursion formula, generating function, and Euler product have been discovered by Dey, Wohlfahrt, Bryman and Fulman, and White. Let $a,b, \ell \geq 2$. It is known by Bringman, Franke, and Heim, that the Bessenrodt--Ono inequality \begin{equation*} Δ_{a,b}^{\ell}:= N_{\ell}(a) \, N_{\ell}(b) - N_{\ell}(a+b) >0 \end{equation*} is valid for $a,b \gg 1$ and by Bessenrodt and Ono that it is valid for $\ell =2$ and $a+b >9$. In this paper we prove that for each pair $(a,b)$ the sign of $\{Δ_{a,b}^{\ell} \}_{\ell}$ is getting stable. In each case we provide an explicit bound. The numbers $N_{\ell}\left( n\right) $ had been identified by Bryan and Fulman as the $n$-th orbifold characteristics, generalizing work by Macdonald and Hirzebruch--Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products, where $N_2(n)=p(n) $ represents the partition function.

math.CO↗

arXiv2024

A bijection for tuples of commuting permutations and a log-concavity conjecture

Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$ which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $\ell+1\rightarrow \ell$ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $A(\ell,n,k)$ with respect to $k$. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula.

math.CO↗

arXiv2024

On a mod $3$ property of $\ell $-tuples of pairwise commuting permutations

Let $S_n$ denote the symmetric group of permutations acting on $n$ elements. We investigate the double sequence $\{N_{\ell}(n)\}$ counting the number of $\ell$ tuples of elements of the symmetric group $S_n$, where the components commute, normalized by the order of $S_n$. Our focus lies on exploring log-concavity with respect to $n$: $$ N_{\ell}(n)^2 - N_{\ell}(n-1) \,\, N_{\ell}(n+1) \geq 0.$$ We establish that this depends on $n \pmod{3}$ for sufficiently large $\ell$. These numbers are studied by Bryan and Fulman as the $n$th orbifold characteristics, generalizing work of Macdonald and Hirzebruch--Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, $N_2(n)$ represents the partition numbers $p(n)$, while $N_{3}(n)$ represents the number of non-equivalent $n$-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since $ \vert S_n \vert \,\, N_{\ell}(n) = \left\vert Hom \left( \mathbb{Z}^{\ell},S_n\right) \right\vert $.

math.CO↗

arXiv2024

Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations

Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the $A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a well known property of unsigned Stirling numbers of the first kind. As the $p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length polynomials. In this article, we prove the $p=\infty$ case of our conjecture. We start from an expression for the $A(p,n,k)$ which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The last step essentially amounts to the log-concavity in $k$ of a generalized Turán number, namely, the maximum product of $k$ positive integers whose sum is $n$.

math.CO↗

arXiv2022

Non-Abelian correlation inequalities and stable determinantal polynomials

We consider the correlations of invariant observables for the $O(N)$ and $\mathbb{C}\mathbb{P}^{N-1}$ models at zero coupling, namely, with respect to the natural group-invariant measure. In the limit where one takes a large power of the integrand, we show that these correlations become inverse powers of the Kirchhoff polynomial. The latter therefore provide a simplified toy model for the investigation of inequalities between products of correlations. Properties such as ferromagnetic behavior for spin model correlations correspond, in this asymptotic limit, to log-ultramodularity which is a consequence of the Rayleigh property of the Kirchhoff polynomial. In addition to the above rigorous asymptotics, the main result of this article is a general theorem which shows that inverse half-integer powers of certain determinantal stable polynomials, such as the Kirchhoff polynomials, satisfy generalizations of the GKS 2 inequalities and the Ginibre inequalities. We conclude with some open problems, e.g., the question of whether the last statement holds for powers which are not half-integers. This leads to a Hirota-bilinear analogue of the complete monotonicity property recently investigated by Scott and Sokal.

math-ph↗

arXiv2021

A local injective proof of log-concavity for increasing spanning forests

We give an explicit combinatorial proof of a weighted version of strong log-concavity for the generating polynomial of increasing spanning forests of a finite simple graph equipped with a total ordering of the vertices. In contrast to similar proofs in the literature, our injection is local in the sense that it proceeds by moving a single edge from one forest to the other. In the particular case of the complete graph, this gives a new combinatorial proof of log-concavity of unsigned Stirling numbers of the first kind where a pair of permutations is transformed into a new pair by breaking a single cycle in the first permutation and gluing two cycles in the second permutation, while all the other spectator cycles are left untouched.

math.CO↗

arXiv2019

A Second-Quantized Kolmogorov-Chentsov Theorem via the Operator Product Expansion

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field.

math.PR↗

arXiv2019

An algebraic independence result related to a conjecture of Dixmier on binary form invariants

In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier's collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof à la Racah, namely, only using the Chu-Vandermonde Theorem, for Dixon's Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan's 1868 method in classical invariant theory.

math.RT↗

arXiv2019

Quadratic Involutions on Binary Forms

There is a classical geometric construction which uses a binary quadratic form to define an involution on the space of binary d-ics. We give a complete characterization of a general class of such involutions which are definable using compound transvectant formulae. We also study the associated varieties of forms which are preserved by such involutions. Along the way we prove a recoupling formula for transvectants, which is used to deduce a system of equations satisfied by the coefficients in these involutions.

math.AG↗

arXiv2018

Towards three-dimensional conformal probability

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and second-quantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Second-quantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. We formulate this program in both the Archimedean and $p$-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where $p$-adic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a $p$-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author's talk at the 6th International Conference on $p$-adic Mathematical Physics and its Applications, Mexico 2017.

math.PR↗

arXiv2018

The weakly dependent strong law of large numbers revisited

We give a short, self-contained, and elementary proof of the strong law of large numbers under a power law decay hypothesis for joint second moments. The result is related to the classical one by Lyons. However, we also provide a rate of convergence. Our proof does not use maximal inequalities and is instead inspired by the method of multiscale large versus small field decompositions in constructive quantum field theory.

math.PR↗

arXiv2016

On the reconstruction problem for Pascal lines

Given a sextuple of distinct points $A, B, C, D, E, F$ on a conic, arranged into an array $\left[\begin{array}{ccc} A & B & C F & E & D \end{array}\right]$, Pascal's theorem says that the points $AE \cap BF, BD \cap CE, AD \cap CF$ are collinear. The line containing them is called the Pascal of the array, and one gets altogether sixty such lines by permuting the points. In this paper we prove that the initial sextuple can be explicitly reconstructed from four specifically chosen Pascals. The reconstruction formulae are encoded by some transvectant identities which are proved using the graphical calculus for binary forms.

math.AG↗

arXiv2015

16,051 formulas for Ottaviani's invariant of cubic threefolds

We provide explicit combinatorial formulas for Ottaviani's degree 15 invariant which detects cubics in 5 variables that are sums of 7 cubes. Our approach is based on the chromatic properties of certain graphs and relies on computer searches and calculations.

math.AG↗