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Shuffle bialgebras

María Ronco (2011)

Annales de l’institut Fourier

The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

Simple Graphs as Simplicial Complexes: the Mycielskian of a Graph

Piotr Rudnicki, Lorna Stewart (2012)

Formalized Mathematics

Harary [10, p. 7] claims that Veblen [20, p. 2] first suggested to formalize simple graphs using simplicial complexes. We have developed basic terminology for simple graphs as at most 1-dimensional complexes. We formalize this new setting and then reprove Mycielski’s [12] construction resulting in a triangle-free graph with arbitrarily large chromatic number. A different formalization of similar material is in [15].

Singularités génériques des variétés de Schubert covexillaires

Aurélie Cortez (2001)

Annales de l’institut Fourier

On montre que les composantes irréductibles du lieu singulier d’une variété de Schubert dans G L n / B , associée à une permutation covexillaire, sont paramétrées par certains des points coessentiels du graphe de la permutation. On donne une description explicite de ces composantes et l’on décrit la singularité le long de chacune d’entre elles.

Skew-symmetric cluster algebras of finite mutation type

Anna Felikson, Michael Shapiro, Pavel Tumarkin (2012)

Journal of the European Mathematical Society

In the famous paper [FZ2] Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by cluster algebras of finite mutation type which have finitely many exchange matrices (but are allowed to have infinitely many cluster variables). In this paper we classify all cluster algebras of finite mutation type with skew-symmetric exchange matrices....

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