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Christophe Tollu

Publications and source records attributed to Christophe Tollu.

13 recordsLinked to original sources
arXiv2024

Elementary methods for splitting representations of Rook monoids: a gentle introduction to groupoids

We show that the algebra of the coloured rook monoid $R_n^{(r)}$, {\em i.e.} the monoid of $n \times n$ matrices with at most one non-zero entry (an $r$-th root of unity) in each column and row, is the algebra of a finite groupoid, thus is endowed with a $C^*$-algebra structure. This new perspective uncovers the representation theory of these monoid algebras by making manifest their decomposition in irreducible modules.

cs.DM↗
arXiv2018

(Pure) transcendence bases in $ϕ$-deformed shuffle bialgebras

Computations with integro-differential operators are often carried out in an associative algebra with unit, and they are essentially non-commutative computations. By adjoining a cocommutative co-product, one can have those operators perform act on a bialgebra isomorphic to an enveloping algebra. That gives an adequate framework for a computer-algebra implementation via monoidal factorization, (pure) transcendence bases and Poincaré--Birkhoff--Witt bases. In this paper, we systematically study these deformations, obtaining necessary and sufficient conditions for the operators to exist, and we give the most general cocommutative deformations of the shuffle co-product and an effective construction of pairs of bases in duality. The paper ends by the combinatorial setting of local systems of coordinates on the group of group-like series.

cs.SC↗
arXiv2015

The mechanics of shuffle products and their siblings

We carry on the investigation initiated in [15] : we describe new shuffle products coming from some special functions and group them, along with other products encountered in the literature, in a class of products, which we name $φ$-shuffle products. Our paper is dedicated to a study of the latter class, from a combinatorial standpoint. We consider first how to extend Radford's theorem to the products in that class, then how to construct their bi-algebras. As some conditions are necessary do carry that out, we study them closely and simplify them so that they can be seen directly from the definition of the product. We eventually test these conditions on the products mentioned above.

math.CO↗
arXiv2014

Combinatorics of $ϕ$-deformed stuffle Hopf algebras

In order to extend the Schützenberger's factorization to general perturbations, the combinatorial aspects of the Hopf algebra of the $ϕ$-deformed stuffle product is developed systematically in a parallel way with those of the shuffle product.

math.CO↗
arXiv2010

Combinatorial Deformations of Algebras: Twisting and Perturbations

The framework used to prove the multiplicative law deformation of the algebra of Feynman-Bender diagrams is a \textit{twisted shifted dual law} (in fact, twice). We give here a clear interpretation of its two parameters. The crossing parameter is a deformation of the tensor structure whereas the superposition parameters is a perturbation of the shuffle coproduct of Hoffman type which, in turn, can be interpreted as the diagonal restriction of a superproduct. Here, we systematically detail these constructions.

cs.SC↗
arXiv2010

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.

math.CO↗
arXiv2010

Partial monoids: associativity and confluence

A partial monoid $P$ is a set with a partial multiplication $\times$ (and total identity $1_P$) which satisfies some associativity axiom. The partial monoid $P$ may be embedded in a free monoid $P^*$ and the product $\star$ is simulated by a string rewriting system on $P^*$ that consists in evaluating the concatenation of two letters as a product in $P$, when it is defined, and a letter $1_P$ as the empty word $ε$. In this paper we study the profound relations between confluence for such a system and associativity of the multiplication. Moreover we develop a reduction strategy to ensure confluence and which allows us to define a multiplication on normal forms associative up to a given congruence of $P^*$. Finally we show that this operation is associative if, and only if, the rewriting system under consideration is confluent.

cs.DM↗
arXiv2009

Sweedler's duals and Schützenberger's calculus

We describe the problem of Sweedler's duals for bialgebras as essentially characterizing the domain of the transpose of the multiplication. This domain is the set of what could be called ``representative linear forms'' which are the elements of the algebraic dual which are also representative on the multiplicative semigroup of the algebra. When the algebra is free, this notion is indeed equivalent to that of rational functions of automata theory. For the sake of applications, the range of coefficients has been considerably broadened, i.e. extended to semirings, so that the results could be specialized to the boolean and multiplicity cases. This requires some caution (use of ``positive formulas'', iteration replacing inversion, stable submodules replacing finite-rank families for instance). For the theory and its applications has been created a rational calculus which can, in return, be applied to harness Sweedler's duals. A new theorem of rational closure and application to Hopf algebras of use in Physics and Combinatorics is provided. The concrete use of this ``calculus'' is eventually illustrated on an example.

math.CO↗
arXiv2008

Hyperdeterminantal computation for the Laughlin wave function

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.

math.CO↗
arXiv2007

Hopf algebras of diagrams

We investigate several Hopf algebras of diagrams related to Quantum Field Theory of Partitions and whose product comes from the Hopf algebras WSym or WQSym respectively built on integer set partitions and set compositions. Bases of these algebras are indexed either by bipartite graphs (labelled or unlabbeled) or by packed matrices (with integer or set coefficients). Realizations on biword are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture.

math.CO↗