arXiv subjects

Jean-Philippe Labbé

Publications and source records attributed to Jean-Philippe Labbé.

At least 19 recordsLinked to original sources

arXiv2025

Solving one-body ensemble N-representability problems with spin

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $φ_i$ according to $0 \leq n_i \leq 2$. In this work, we first refine the underlying one-body $N$-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness $\boldsymbol w$ of the $N$-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope $Σ_{N,S}(\boldsymbol w) \subset [0,2]^d$. These constraints are independent of $M$ and the number $d$ of orbitals, while their dependence on $N, S$ is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

quant-ph↗

arXiv2023

Lineup polytopes of product of simplices

Consider a real point configuration $\mathbf{A}$ of size $n$ and an integer $r \leq n$. The vertices of the $r$-lineup polytope of $\mathbf{A}$ correspond to the possible orderings of the top $r$ points of the configuration obtained by maximizing a linear functional. The motivation behind the study of lineup polytopes comes from the representability problem in quantum chemistry. In that context, the relevant point configurations are the vertices of hypersimplices and the integer points contained in an inflated regular simplex. The central problem consists in providing an inequality representation of lineup polytopes as efficiently as possible. In this article, we adapt the developed techniques to the quantum information theory setup. The appropriate point configurations become the vertices of products of simplices. A particular case is that of lineup polytopes of cubes, which form a type $B$ analog of hypersimplices, where the symmetric group of type~$A$ naturally acts. To obtain the inequalities, we center our attention on the combinatorics and the symmetry of products of simplices to obtain an algorithmic solution. Along the way, we establish relationships between lineup polytopes of products of simplices with the Gale order, standard Young tableaux, and the Resonance arrangement.

math.CO↗

arXiv2022

Foundation of one-particle reduced density matrix functional theory for excited states

In [Phys. Rev. Lett. 127, 023001 (2021)] a reduced density matrix functional theory (RDMFT) has been proposed for calculating energies of selected eigenstates of interacting many-fermion systems. Here, we develop a solid foundation for this so-called $\boldsymbol{w}$-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights $\boldsymbol{w}$ in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case $\boldsymbol{w}=(1,0,\ldots)$. Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

quant-ph↗

arXiv2021

Lorentzian Coxeter systems and Boyd-Maxwell ball packings

In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: Given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd--Maxwell ball packings. Finally, we enumerate all the Coxeter systems that generate Boyd-Maxwell ball packings.

math.GR↗

arXiv2021

An effective solution to convex $1$-body $N$-representability

From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of $1$-fermion and $N$-fermion density matrices, therefore it coincides with the convex hull of the pure $N$-representable $1$-fermion density matrices. Consequently, the description of ground state physics through $1$-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the $1$-body $N$-representability problem to ensemble states with fixed spectrum $\mathbf{w}$, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the $1$-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of $\mathbf{w}$. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes.

quant-ph↗

arXiv2021

Combinatorial foundations for geometric realizations of subword complexes of Coxeter groups

Multitriangulations, and more generally subword complexes, yield a large family of simplicial complexes that are homeomorphic to spheres. Until now, all attempts to prove or disprove that they can be realized as convex polytopes faced major obstacles. In this article, we lay out the foundations of a framework -- built upon notions from algebraic combinatorics and discrete geometry -- that allows a deeper understanding of geometric realizations of subword complexes of Coxeter groups. Namely, we describe explicitly a family of chirotopes that encapsulate the necessary information to obtain geometric realizations of subword complexes. Further, we show that the space of geometric realizations of this family covers that of subword complexes, making this combinatorially defined family into a natural object to study. The family of chirotopes is described through certain parameter matrices. That is, given a finite Coxeter group, we present matrices where certain minors have prescribed signs. Parameter matrices are universal: The existence of these matrices combined with conditions in terms of Schur functions is equivalent to the realizability of all subword complexes of this Coxeter group as chirotopes. Finally, parameter matrices provide extensions of combinatorial identities; for instance, the Vandermonde determinant and the dual Cauchy identity are recovered through suitable choices of parameters.

math.CO↗

arXiv2020

Congruence Normality of Simplicial Hyperplane Arrangements via Oriented Matroids

A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Grünbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide additional structure to the catalogue of simplicial hyperplane arrangements by determining which arrangements always/sometimes/never lead to congruence normal lattices of regions. To this end, we use oriented matroids to recast shards as covectors to determine congruence normality of large hyperplane arrangements. As a consequence of this approach we derive in particular which lattices of regions of sporadic simplicial arrangements of rank 3 are always congruence normal. We also show that lattices of regions from finite Weyl groupoids of any rank are congruence normal.

math.CO↗

arXiv2019

Combinatorial Inscribability Obstructions for Higher-Dimensional Polytopes

For $3$-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every $f$-vector of $3$-polytopes, there exists an inscribable polytope with that $f$-vector. For higher-dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower-dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the $4$-dimensional cyclic polytopes with at least $8$ vertices---all of whose faces are inscribable---are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic $4$-polytopes with up to $7$ vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that $d$-dimensional cyclic polytopes with at least $d+4$ vertices are not circumscribable, and that no dual of a neighborly $4$-polytope with $8$ vertices, that is, no polytope with $f$-vector $(20,40,28,8)$, is inscribable.

math.MG↗

arXiv2018

Area difference bounds for dissections of a square into an odd number of triangles

Monsky's theorem from 1970 states that a square cannot be dissected into an odd number of triangles of the same area, but it does not give a lower bound for the area differences that must occur. We extend Monsky's theorem to "constrained framed maps"; based on this we can apply a gap theorem from semi-algebraic geometry to a polynomial area difference measure and thus get a lower bound for the area differences that decreases doubly-exponentially with the number of triangles. On the other hand, we obtain the first superpolynomial upper bounds for this problem, derived from an explicit construction that uses the Thue-Morse sequence.

math.MG↗

arXiv2018

Cambrian acyclic domains: counting $c$-singletons

We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos--Reiner for a particular acyclic domain called Fishburn's alternating scheme, we provide explicit formulae for the size of any Cambrian acyclic domain and characterize the Cambrian acyclic domains of minimum or maximum size.

math.CO↗

arXiv2017

Bounds for entries of $γ$-vectors of flag homology spheres

We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Δ$, we show that its $γ$-vector $γ^Δ=(1,γ_1,γ_2,\ldots)$ satisfies: \begin{align*} γ_j=0,\text{ for all } j>γ_1, \quad γ_2\leq\binom{γ_1}{2}, \quad γ_{γ_1}\in\{0,1\}, \quad \text{ and }γ_{γ_1-1}\in\{0,1,2,γ_1\}, \end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Δ$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $γ$-vectors of flag homology spheres (of any dimension); these are the first examples of this kind. In addition, we prove a flag analog of Perles' 1970 theorem on $k$-skeleta of polytopes with "few" vertices, specifically: the number of combinatorial types of $k$-skeleta of flag homology spheres with $γ_1\leq b$, of any given dimension, is bounded independently of the dimension.

math.CO↗

arXiv2016

On inversion sets and the weak order in Coxeter groups

In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and their position relative to the imaginary cone. Finally, we discuss inversion sets of infinite reduced words and the notions of biconvex and biclosed sets of positive roots.

math.GR↗

arXiv2015

A Perron theorem for matrices with negative entries and applications to Coxeter groups

Handelman (J. Operator Theory, 1981) proved that if the spectral radius of a matrix $A$ is a simple root of the characteristic polynomial and is strictly greater than the modulus of any other root, then $A$ is conjugate to a matrix $Z$ some power of which is positive. In this article, we provide an explicit conjugate matrix $Z$, and prove that the spectral radius of $A$ is a simple and dominant eigenvalue of $A$ if and only if $Z$ is eventually positive. For $n\times n$ real matrices with each row-sum equal to $1$, this criterion can be declined into checking that each entry of some power is strictly larger than the average of the entries of the same column minus $\frac{1}{n}$. We apply the criterion to elements of irreducible infinite nonaffine Coxeter groups to provide evidences for the dominance of the spectral radius, which is still unknown.

math.GR↗

arXiv2015

Cluster Algebras of Type $D_4$, Tropical Planes, and the Positive Tropical Grassmannian

We show that the number of combinatorial types of clusters of type $D_4$ modulo reflection-rotation is exactly equal to the number of combinatorial types of tropical planes in $\mathbb{TP}^5$. This follows from a result of Sturmfels and Speyer which classifies these tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian $\operatorname{Gr}(3,6)$. Speyer and Williams show that the positive part $\operatorname{Gr}^+(3,6)$ of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type $D_4$. We provide a structural bijection between the rays of $\operatorname{Gr}^+(3,6)$ and the almost positive roots of type $D_4$ which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type $D_4$ to describe the equivalence of "positive" tropical planes in $\mathbb{TP}^5$, giving a combinatorial model which characterizes the combinatorial types of tropical planes using automorphisms of pseudotriangulations of the octogon.

math.CO↗

arXiv2015

Hirsch polytopes with exponentially long combinatorial segments

In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes. We use this definition to investigate further properties of combinatorial segments. Besides recovering the two stated bounds, we show a refined bound for banner complexes, and study the behavior of the maximal length of combinatorial segments with respect to two usual operations, namely join and one-point suspension. Finally, we show the limitations of combinatorial segments by constructing pure normal simplicial complexes in which all combinatorial segments between two particular facets achieve the length $Ω(n2^{d})$. This includes vertex-decomposable---therefore Hirsch---polytopes.

math.CO↗

arXiv2014

Fan realizations of subword complexes and multi-associahedra via Gale duality

We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $Δ_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $Δ_{9,2}$ and $Δ_{11,3}$.

math.CO↗

arXiv2014

Limit Directions for Lorentzian Coxeter Systems

Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space, limit directions are limits of injective sequences in the orbit of some base point. Limit roots are limit directions that can be obtained starting from simple roots. In this article, we study the limit directions arising from any point when the representation space is a Lorentz space. In particular, we characterize the light-like limit directions using eigenvectors of infinite-order elements. This provides a spectral perspective on limit roots, allowing for efficient computations. Moreover, we describe the space-like limit directions in terms of the projective Coxeter arrangement.

math.GR↗

arXiv2013

Asymptotical behaviour of roots of infinite Coxeter groups

Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E, and then we exhibit a countable subset of E, formed by limit points for the dihedral reflection subgroups of W. We explain that this subset is built from the intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E.

math.GR↗