A BorelWeil Theorem for Schur Modules
Abstract
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 $\NN \times \NN$. For each $D$, we define the Schur module $S_D$ of $GL(N)$. We introduce a projective variety $\FF_D$ and a line bundle $\LL_D$, and describe the Schur module in terms of sections of $\LL_D$. For diagrams with the ``northeast'' property, $$(i_1,j_1),\ (i_2, j_2) \in D \to (\min(i_1,i_2),\max(j_1,j_2)) \in D ,$$ which includes the skew diagrams, we resolve the singularities of $\FD$ and show analogs of Bott's and Kempf's vanishing theorems. Finally, we apply the AtiyahBott Fixed Point Theorem to establish a Weyltype character formula of the form: $$ {\Char}_{S_D}(x) = \sum_t {x^{\wt(t)} \over \prod_{i,j} (1x_i x_j^{1})^{d_{ij}(t)}} \ ,$$ where $t$ runs over certain standard tableaux of $D$. Our results are valid over fields of arbitrary characteristic.
 Publication:

arXiv eprints
 Pub Date:
 November 1994
 arXiv:
 arXiv:alggeom/9411014
 Bibcode:
 1994alg.geom.11014M
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 35pp, LaTeX