A notion of inversion number associated to certain quiver flag varieties
Summary: 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 \textit{P. Drube} [Electron. J. Comb. 23, No. 1, Research Paper P1.43, 22 p. (2016; Zbl 1333.05035)] for such row-increasing tableaux.
scientific article; zbMATH DE number 6933723