Springer fibers and Schubert points

Abstract

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

An equivariant basis for the cohomology of Springer fibers

“…Remark 3.4. Note that the definition above is closely related to the notion of a Springer dimension pair considered by the first author and Tymoczko in [32]. In that paper, the convention is that the row-strict tableaux have increasing entries (from left to right), while our convention is that the entries are decreasing (from left to right).…”

“…This change in conventions is routine; to convert from one to the other, apply the permutation w 0 such that w 0 (i) = n − i + 1 for all i. A Springer inversion from this paper corresponds to a unique Springer dimension pair as defined in [32] (up to transformation under w 0 ). If (i, j) is a Springer inversion then (n − i + 1, n − j + 1) is a Springer dimension pair, where j denotes the smallest element in row j such that i < j .…”

“…We now describe the set W (α) ⊆ W . This subset is analogous to the set of Schubert points defined by the first author and Tymoczko in [32], although our conventions differ, as discussed in Remark 3.4 above. To any Υ ∈ RSCT n (α) we define u Υ to be the unique permutation (as defined in Lemma 5.7) such that the inversion vector of Υ equals the Lehmer code of u Υ .…”

“…Springer showed that the symmetric group acts on the cohomology of each Springer fiber, the top-dimensional cohomology is an irreducible representation, and each irreducible symmetric group representation can be obtained in this way [35,36]. As a consequence, Springer fibers frequently arise in geometric representation theory and algebraic combinatorics; see [14,15,18,20,32,34] for just a few examples.…”

The geometry and combinatorics of Springer fibers

“…For instance we have the following: It turns out that these permutations w T index a set of Schubert varieties whose union has the same Betti numbers as a Springer fiber. More precisely we have the following [34].…”