A Borel-Weil 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 Atiyah-Bott Fixed Point Theorem to establish a Weyl-type character formula of the form: $$ {\Char}_{S_D}(x) = \sum_t {x^{\wt(t)} \over \prod_{i,j} (1-x_i x_j^{-1})^{d_{ij}(t)}} \ ,$$ where $t$ runs over certain standard tableaux of $D$. Our results are valid over fields of arbitrary characteristic.

35pp, LaTeX