46 research outputs found
Stable Noncrossing Matchings
Given a set of men represented by points lying on a line, and
women represented by points lying on another parallel line, with each
person having a list that ranks some people of opposite gender as his/her
acceptable partners in strict order of preference. In this problem, we want to
match people of opposite genders to satisfy people's preferences as well as
making the edges not crossing one another geometrically. A noncrossing blocking
pair w.r.t. a matching is a pair of a man and a woman such that
they are not matched with each other but prefer each other to their own
partners in , and the segment does not cross any edge in . A
weakly stable noncrossing matching (WSNM) is a noncrossing matching that does
not admit any noncrossing blocking pair. In this paper, we prove the existence
of a WSNM in any instance by developing an algorithm to find one in a
given instance.Comment: This paper has appeared at IWOCA 201
Combinatorics of symplectic invariant tensors
International audienceAn important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group . Our formulation is completely explicit and provides a very precise link to -noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon.Un problème important de la théorie des invariantes est de décrire le sous espace d’une puissance tensorielle d’une représentation invariant à l’action du groupe. Suivant la classique de Weyl, le théorème fondamental premier pour la représentation standard du groupe sympléctique dit que tous les invariants peuvent être exprimés entre un nombre fini d’entre eux. Par ailleurs, un théorème fondamental second détermine les relations entre ces invariants basiques.Ici, nous présentons une preuve transparente d’un théorème fondamental second pour la représentation standard du groupe sympléctique . Notre formulation est complètement explicite et elle fournit un lien très précis avec les couplages parfaits -noncroissants, plus précis qu’un dénombrement de la dimension. Comme corollaire nous exhibons un phénomène de crible cyclique