1994/11/21 by Peter Magyar, Magyar, Peter
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9411014
35pp, LaTeX
arxiv created 1994/11/21 · arxiv updated 2015/06/30
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 × \NN. For each D, we define the Schur module SD of GL(N). We introduce a projective variety \FFD and a line bundle \LLD, and describe the Schur module in terms of sections of \LLD. 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 Atiyah-Bott Fixed Point Theorem to establish a Weyl-type character formula of the form: \CharSD(x) = ∑t x\wt(t) \over ∏i,j (1-xi xj-1)^dij(t) , where t runs over certain standard tableaux of D. Our results are valid over fields of arbitrary characteristic.