Borel–Weil Theorem for Configuration Varieties and Schur Modules

Abstract

We present a generalization of the classical Schur modules ofGL(n) exhibiting the same interplay among algebra, geometry, and combinatorics. A generalized Young diagramDis an arbitrary finite subset ofN×N. For eachD, we define the Schur moduleSDofGL(n). We introduce a projective variety FDand 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 FDand 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:charSD(x)=∑txwt(t)∑i, j(1−xix−1j)dij(t),wheretruns over certain standard tableaux ofD. Our results are valid over fields of arbitrary characteristic.