arXiv subjects
Publications and source records attributed to Jean-Gabriel Luque.
We introduce partial $r$-Bell polynomials in three combinatorial Hopf algebras. We prove a factorization formula for the generating functions which is a consequence of the Zassenhauss formula.
Partial multivariate Bell polynomials have been defined by E.T. Bell in 1934. These polynomials have numerous applications in Combinatorics, Analysis, Algebra, Probabilities, etc. Many of the formulae on Bell polynomials involve combinatorial objects (set partitions, set partitions in lists, permutations, etc.). So it seems natural to investigate analogous formulae in some combinatorial Hopf algebras with bases indexed by these objects. The algebra of symmetric functions is the most famous example of a combinatorial Hopf algebra. In a first time, we show that most of the results on Bell polynomials can be written in terms of symmetric functions and transformations of alphabets. Then, we show that these results are clearer when stated in other Hopf algebras (this means that the combinatorial objects appear explicitly in the formulae). We investigate also the connexion with the Fa{à} di Bruno Hopf algebra and the Lagrange-B{ü}rmann formula.
We generalize the construction of multitildes in the aim to provide multitilde operators for regular languages. We show that the underliying algebraic structure involves the action of some operads. An operad is an algebraic structure that mimics the composition of the functions. The involved operads are described in terms of combinatorial objects. These operads are obtained from more primitive objects, namely precompositions, whose algebraic counter-parts are investigated. One of these operads acts faithfully on languages in the sense that two different operators act in two different ways.
In the aim to understand the generalization of Stirling numbers occurring in the bosonic normal ordering problem, several combinatorial models have been proposed. In particular, Blasiak \emph{et al.} defined combinatorial objects allowing to interpret the number of $S_{\bf{r,s}}(k)$ appearing in the identity $(a^†)^{r_n}a^{s_n}\cdots(a^†)^{r_1}a^{s_1}=(a^†)^α\displaystyle\sum S_{\bf{r,s}}(k)(a^†)^k a^k$, where $α$ is assumed to be non-negative. These objects are used to define a combinatorial Hopf algebra which specializes to the enveloping algebra of the Heisenberg Lie algebra. Here, we propose a new variant of this construction which admits a realization with variables. This means that we construct our algebra from a free algebra $\mathbb{C}\langle A \rangle$ using quotient and shifted product. The combinatorial objects (B-diagrams) are slightly different from those proposed by Blasiak \emph{et al.}, but give also a combinatorial interpretation of the generalized Stirling numbers together with a combinatorial Hopf algebra related to Heisenberg Lie algebra. The main difference comes from the fact that the B-diagrams have the same number of inputs and outputs. After studying the combinatorics and the enumeration of B-diagrams, we propose two constructions of algebras called Fusion algebra $\mathcal{F}$, defined using formal variable and another algebra $\mathcal{B}$ constructed directly from the B-diagrams. We show the connection between these two algebras and that $\mathcal{B}$ can be endowed with a Hopf structure. We recognize two already known combinatorial Hopf subalgebras of $\mathcal{B}$ : $\mathrm{WSym}$ the algebra of word symmetric functions indexed by set partitions and $\mathrm{BWSym}$ the algebra of biword symmetric functions indexed by set partitions into lists.
In this paper we study the state complexity of catenation combined with symmetric difference. First, an upper bound is computed using some combinatoric tools. Then, this bound is shown to be tight by giving a witness for it. Moreover, we relate this work with the study of state complexity for two other combinations: catenation with union and catenation with intersection. And we extract a unified approach which allows to obtain the state complexity of any combination involving catenation and a binary boolean operation.
An intriguing correspondence between four-qubit systems and simple singularity of type $D_4$ is established. We first consider an algebraic variety $X$ of separable states within the projective Hilbert space $\mathbb{P}(\mathcal{H})=\mathbb{P}^{15}$. Then, cutting $X$ with a specific hyperplane $H$, we prove that the $X$-hypersurface, defined from the section $X\cap H\subset X$, has an isolated singularity of type $D_4$; it is also shown that this is the "worst-possible" isolated singularity one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of $X$, which is nothing but the Cayley hyperdeterminant of type $2\times 2\times 2\times 2$, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the $D_4$-singularity.
We investigate the enumeration of two families of polycubes, namely pyramids and espaliers, in connection with the multi-indexed Dirichlet convolution
We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.
The clustering properties of Jack polynomials are relevant in the theoretical study of the fractional Hall states. In this context, some factorization properties have been conjectured for the $(q,t)$-deformed problem involving Macdonald polynomials. The present paper is devoted to the proof of this formula. To this aim we use four families of Jack/Macdonald polynomials: symmetric homogeneous, nonsymmetric homogeneous, shifted symmetric and shifted nonsymmetric.
We give noncommutative versions of the Redfield-Pólya theorem in WSym, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras.
Multitildes are regular operators that were introduced by Caron et al. in order to increase the number of Glushkov automata. In this paper, we study the family of the multitilde operators from an algebraic point of view using the notion of operad. This leads to a combinatorial description of already known results as well as new results on compositions, actions and enumerations.
The aim of the paper is to propose geometric descriptions of multipartite entangled states using algebraic geometry. In the context of this paper, geometric means each stratum of the Hilbert space, corresponding to an entangled state, is an open subset of an algebraic variety built by classical geometric constructions (tangent lines, secant lines) from the set of separable states. In this setting we describe well-known classifications of multipartite entanglement such as $2\times 2\times(n+1)$, for $n\geq 1$, quantum systems and a new example with the $2\times 3\times 3$ quantum system. Our description completes the approach of Miyake and makes stronger connections with recent work of algebraic geometers. Moreover for the quantum systems detailed in this paper we propose an algorithm, based on the classical theory of invariants, to decide to which subvariety of the Hilbert space a given state belongs.
Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators with some additional properties concerning the leading monomial. These polynomials were introduced by Griffeth in the general setting of the complex reflections groups $G(r,p,N)$ and studied by one of the authors (C. Dunkl) in the specialization $r=p=1$ (i.e. for the symmetric group). By adapting a construction due to Lascoux, we describe an algorithm allowing us to compute explicitly the Jack polynomials following a Yang-Baxter graph. We recover some properties already studied by C. Dunkl and restate them in terms of graphs together with additional new results. In particular, we investigate normalization, symmetrization and antisymmetrization, polynomials with minimal degree, restriction etc. We give also a shifted version of the construction and we discuss vanishing properties of the associated polynomials.
We investigate the asymptotic behavior of the Selberg-like integral $$ \frac1{N!}\int_{[0,1]^N}x_1^p\prod_{i<j}(x_i-x_j)^2\prod_ix_i^{a-1}(1-x_i)^{b-1}dx_i$$, as $N\to\infty$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.
Nonlinear statistics (i.e. statistics of permanents) on the eigenvalues of invariant random matrix models are considered for the three Dyson's symmetry classes $β=1,2,4$. General formulas in terms of hyperdeterminants are found for $β=2$. For specific cases and all $β$s, more computationally efficient results are obtained, based on symmetric functions expansions. As an application, we consider the case of quantum transport in chaotic cavities extending results from [D.V. Savin, H.-J. Sommers and W. Wieczorek, {\it Phys. Rev. B} {\bf 77}, 125332 (2008)].
The decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the Vandermonde determinants in the Schur basis. Such a computation is notoriously difficult and the coefficients of the expansion have not yet been interpreted. In our paper, we give an expression of these coefficients in terms of hyperdeterminants of sparse tensors. We use this result to construct an algorithm allowing to compute one coefficient of the development without computing the others. Thanks to a program in {\tt C}, we performed the calculation for the square of the Vandermonde up to an alphabet of eleven lettres.
We investigate the homogeneous symmetric Macdonald polynomials $P_λ(\X;q,t)$ for the specialization $t=q^k$. We show an identity relying the polynomials $P_λ(\X;q,q^k)$ and $P_λ(\frac{1-q}{1-q^k}\X;q,q^k)$. As a consequence, we describe an operator whose eigenvalues characterize the polynomials $P_λ(\X;q,q^k)$.
We prove that a $q$-deformation $\Disc k\X q$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\Disc k\X q$ on different basis of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis.