A graphical calculus for 2-block Spaltenstein varieties
Let \(V = {\mathbb C}^n\) be an \(n\)-dimensional complex vector space. For a sequence of integers \(\{i_1, i_2, \dots, i_m\}\) with \(0 < i_1 < i_2 < \cdots < i_m = n\), a partial flag of \(V\) of type \((i_1,i_2,\dots, i_m)\) is a sequence of subspaces \(F_{i_1} \subset F_{i_2} \subset \cdots \subset F_{i_m}\) with \(\text{dim}F_{i_j} = i_j\). A complete (or full) flag is one of type \((1,2,\dots, n)\). Consider a nilpotent endomorphism \(N : V \to V\). The variety of all complete flags that are invariant under \(N\) is known as the Springer fiber of \(N\). More generally, the variety of \(N\)-invariant partial flags of some type \((i_1,i_2,\dots,i_m)\) is known as a Spaltenstein variety of type \((i_1,i_2,\dots,i_m)\). The nilpotent endomorphism \(N\) is determined by the sizes of its Jordan blocks and hence can be associated with a partition \(\lambda\) of \(n\). Spaltenstein showed that the irreducible components of the Springer fiber are in one-to-one correspondence with standard Young tableaux of shape \(\lambda\). More generally, the irreducible components of a Spaltenstein variety (of a given type) can also be (bijectively) associated with a set of tableaux determined by the type.NEWLINENEWLINEThe goal of this paper is to study the geometry of the irreducible (and so-called ``generalized irreducible'') components of a Spaltenstein variety in the case when \(N\) has two Jordan blocks by generalizing known results for the special case of the Springer fiber. The author first gives an explicit description of these irreducible components by generalizing the description given for Springer fibers by \textit{F. Fung} [Adv. Math. 178, No. 2, 244--276 (2003; Zbl 1035.20004)]. Following \textit{C. Stroppel} and \textit{B. Webster} [Comment. Math. Helv. 87, No. 2, 477--520 (2012; Zbl 1241.14009)], associated to a row-strict tableaux of a given type, the author defines the notion of a ``generalized irreducible component'' of a Spaltenstein variety (of that given type). The collection of generalized irreducible components contains the honest irreducible components. Two key tools are used in the paper: the ``dependence graph'' associated to a tableaux and the ``circle diagram'' associated to a pair of tableaux. Dependence graphs extend the notion of cup diagrams used in the work of Fung and Stroppel-Webster while circle diagrams (which are built out of two extended cup diagrams) were introduced for Springer fibers by \textit{C. Stroppel} [Compos. Math. 145, No. 4, 954--992 (2009; Zbl 1187.17004)].NEWLINENEWLINEThe first main result is a graphical description of generalized irreducible components in terms of the associated dependence graph. This is also extended to the intersection of a pair of such components. Circle diagrams can be used to determine when the intersection of two generalized irreducible components is empty. By showing that generalized irreducible components and (non-empty) intersections of pairs of such can be identified with iterated fiber bundles, the author computes the cohomology of such spaces. Finally, it is shown that the direct sum over all pairs of the cohomology groups of pair-wise intersections can be given an algebra structure via colored cobordisms.
scientific article; zbMATH DE number 6031024
scientific article