Borel–Weil Theorem for Configuration Varieties and Schur Modules
✍ Scribed by Peter Magyar
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 449 KB
- Volume
- 134
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
We present a generalization of the classical Schur modules of GL(n) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagram D is an arbitrary finite subset of N_N. For each D, we define the Schur module S D of GL(n). We introduce a projective variety F D and a line bundle L D , and describe the Schur module in terms of sections of L D . For diagrams with the ``northeast'' property
which includes the skew diagrams, we resolve the singularities of F D and show analogues of Bott's and Kempf's vanishing theorems. Finally, we apply the Atiyah Bott Fixed Point Theorem to establish a Weyl-type character formula of the form:
where t runs over certain standard tableaux of D. Our results are valid over fields of arbitrary characteristic.