A notion of inversion number associated to certain quiver flag varieties

Lucas Fresse¹

¹Institut Elie Cartan de Lorraine, Université de Lorraine, France

The electronic journal of combinatorics, Tome 25 (2018) no. 3

Résumé

We define an algebraic variety $X(d,A)$ consisting of matrices whose rows and columns are partial flags. This is a smooth, projective variety, and we describe it as an iterated bundle of Grassmannian varieties. Moreover, we show that $X(d,A)$ has a cell decomposition, in which the cells are parametrized by certain matrices of sets and their dimensions are given by a notion of inversion number. On the other hand, we consider the Spaltenstein variety of partial flags which are stabilized by a given nilpotent endomorphism. We partition this variety into locally closed subvarieties which are affine bundles over certain varieties called $Y_T$, parametrized by semistandard tableaux $T$. We show that the varieties $Y_T$ are in fact isomorphic to varieties of the form $X(d,A)$. We deduce that each variety $Y_T$ has a cell decomposition, in which the cells are parametrized by certain row-increasing tableaux obtained by permuting the entries in the columns of $T$ and their dimensions are given by the inversion number recently defined by P. Drube for such row-increasing tableaux.

Détail

DOI : 10.37236/7840

Classification : 14M15, 05A05, 05A19
Mots-clés : flag varieties, cell decomposition, Spaltenstein variety, semistandard Young tableaux, inversion number