Springer Fibers of Hook Type and Schubert Points

Felemu Olasupo Jon

Abstract: This paper is a survey article aimed at extending the work of Tymoczko and Precup in [4] and compute the Betti numbers of Springer varieties Spr? in type A. Specifically, we focus on the Springer varieties associated to nilpotent operators in Jordan form of type ? = n-k, 1k and ? = ?1, ?1-1, 1k, (n ? 5, ?1 ? 3, k = n-(2?1-1)) and show that the Betti numbers of Spr? equals the Betti numbers of a particular union of Schubert varieties by setting the redundant permutation to zero. Furthermore, we consider a special partition of generic shape ? = n k, 1k, n 6, k = 3 and give a closed formula which is useful in computing the number of redundant permutations and also a closed formula that helps in determining the number of standard tableaux (of the shape ? = n-k, 1k, k = 3, n ? 6) that generate the redundant permutations.

Keywords: Flag variety, Springer variety, Standard tableau, Row-strict tableau Betti numbers