2010/04/27 by Lucas Fresse, Fresse, Lucas · 1 citation
Mathematics · #05E10 #14M15 (primary) #20G05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.CO #math.RT #msc:05E10 #msc:14M15 #msc:20G05
paper · pdf · doi:10.48550/arxiv.1004.4839
33 pages
arxiv created 2010/04/27 · arxiv updated 2010/04/28
Let u\inEnd(ℂn) be nilpotent. The variety of u-stable complete flags is called the Springer fiber over u. Its irreducible components are parameterized by a set of standard Young tableaux. The Richardson (resp. Bala-Carter) components of Springer fibers correspond to the Richardson (resp. Bala-Carter) elements of the symmetric group, through Robinson-Schensted correspondence. Every Richardson component is isomorphic to a product of standard flag varieties. On the contrary, the Bala-Carter components are very susceptible to be singular. First, we characterize the singular Bala-Carter components in terms of two minimal forbidden configurations. Next, we introduce two new families of components, wider than the families of Bala-Carter components and Richardson components, and both in duality via the tableau transposition. The components in the first family are characterized by the fact that they have a dense orbit of special type under the action of the stabilizer of u, whereas all components in the second family are iterated fiber bundles over projective spaces.