Intersections of Schubert Varieties

✍ Scribed by S.B. Mulay

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 214 KB
• Volume: 186
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0390

✦ Synopsis

Let k be a field and let FL n, k denote the variety of full flags on an n-dimensional vector space over k. This variety can also be identified with Ž . the quotient GrB where G s GL n, k , and B is the subgroup consisting Ž w x. of all upper-triangular matrices. It is well known that e.g., see 7 Ž . FL n, k can be decomposed, in a canonical way, into n! affine cells. A Ž . Schubert subvariety of FL n, k is, by definition, the zariski-closure of w x such an affine cell. We employ the notation of 7 in the following discussion. Let W denote the affine cell corresponding to the permutation g S in the canonical decomposition @W of the flag variety, and let X n denote the associated Schubert subvariety. Let W be the ''opposite big cell,'' i.e., the quotient TrB where T is the subgroup of unipotent