Standard Monomial Theory
نویسندگان
چکیده
We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Zi associated to G = GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain standard tableaux and generalizes the Standard Monomial basis for Schubert varieties. In this paper, we prove the results announced in [LkMg] for the case of Type An−1 (the groups GL(n) and SL(n)). That is, we construct an explicit basis for certain “generalized Demazure modules”, natural finite-dimensional representations of the group B of upper triangular matrices. These modules can be constructed in an elementary way as flagged Schur modules [Mg1,Mg2,RS1,RS2]. They include as special cases almost all natural examples of B-modules, and their characters include most of the known generalizations of Schur polynomials. We view these representations geometrically via Borel-Weil theory as the space of global sections of a line bundle over a Bott-Samelson variety. Thus, our theory also describes the coordinate ring of this variety. Notations: G = GL(n,F) or SL(n,F), where F is an algebraically closed field of arbitrary characteristic or F = Z ; B is the Borel subgroup consisting of upper triangular matrices; T is the maximal torus consisting of diagonal matrices;W is the symmetric group Sn generated by the adjacent transpositions (simple reflections) si = (i, i + 1); Pi ⊃ B is the minimal parabolic subgroup of G associated to si, namely Pi = { (xij) ∈ G | xij = 0 if i > j and (i, j) 6=(i+1, i) }. For any word i = (i1, . . . , il), with letters 1 ≤ ij ≤ n − 1, the Bott-Samelson variety is the quotient space Zi = Pi1 × Pi2 × · · · × Pil / B l , where B acts by (p1, p2, . . . , pl) · (b1, b2, . . . , bl) = (p1b1, b −1 1 p2b2, . . . , b −1 l−1plbl).
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تاریخ انتشار 2008