A Borel-Weil Construction for Schur Modules
نویسنده
چکیده
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 SD of GL(N). We introduce a projective variety FD and a line bundle LD, and describe the Schur module in terms of sections of LD. For diagrams with the “northeast” property, (i1, j1), (i2, j2) ∈ D ⇒ (min(i1, i2),max(j1, j2)) ∈ 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 Weyl-type character formula of the form:
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