Involutions of the Symmetric Group and Congruence B - Orbits ( Extended Abstract )
نویسندگان
چکیده
We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset. Résumé Nous étudions l’ensemble ordonné des classes de congruence de matrices symétriques ordonnées par containment de leurs fermetures. Nous donnons une description combinatoire de cet ensemble et calculons sa fonction rang. Nous étudions les relations entre cet ensemble et l’ensemble des involutions du groupe symérique ordonné selon l’ordre de Bruhat. Nous montrons qu’il existe une bijection parmi l’ensemble ordonné de classes de congruences de Borel des matrices anti-symétriques et l’ensemble des involutions du groupe symétrique. On termine en donnant deux formules pour la fonction rang pour ce dernier ensemble.
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