Seidel and Pieri products in cominuscule quantum K-theory

We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space

Characterization and classification of minuscule Kac–Moody representations built from colored posets

Colored minuscule and d-complete partially ordered sets encode information that can be used to construct many objects that arise in semisimple and Kac–Moody Lie theory. The colored minuscule posets can be used to construct the minuscule representations of the semisimple Lie algebras. R.M. Green extended this picture with his full heap colored posets, showing they are sufficient to construct beautiful “doubly infinite” representations of affine Kac–Moody algebras. We reformulate and build upon his work. We obtain necessary poset coloring conditions for the construction of representations of Kac–Moody algebras that uniformly incorporate the minuscule representations of semisimple Lie algebras and Green's representations of the affine algebras. While doing so, we delineate which defining relations for the algebras correspond to which coloring properties for the posets and obtain representations of the Borel subalgebras as well. This leads to the development of new “frontier census” coloring properties and new uniform definitions of four kinds of colored posets: Finite and infinite minuscule and d-complete posets. This is the first definition of an infinite colored d-complete poset. Building upon work of R.A. Proctor, J.R. Stembridge, R.M. Green, and Z.S. McGregor-Dorsey, we classify these posets and their associated Dynkin diagrams. This in turn classifies all minuscule representations that can be built from colored posets.Doctor of Philosoph

Minuscule Heaps over Dynkin diagrams of type Ã

A minuscule heap is a partially ordered set, together with a labeling of its elements by the nodes of a Dynkin diagram, satisfying certain conditions derived by J. Stembridge. This paper classi es the minuscule heaps over the Dynkin diagram of type ~ A

Combinatoire autour des groupes de permutations généralisées

Ce mémoire constitue un travail de synthèse de mes travaux dans le domaine de la combinatoire énumérative et algébrique autour des groupes de permutations généralisées : ce terme désignera ici les groupes de Coxeter classiques et certains groupes de réflexions complexes. La plupart des résultats présentés dans ce mémoire peuvent être classifiés en deux catégories. Il s'agit soit de résultats algébriques dont nous donnons des descriptions combinatoires explicites, soit de résultats énumératifs qui ont une signification dans un contexte algébrique particulier. Pour faire cela, on s'appuie souvent sur des fonctions particulières à valeurs entières positives qui sont usuellement appelées statistiques de permutations