Combinatorial Mathematics by Holton D.A. (ed.)

By Holton D.A. (ed.)

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9 we know that this intersection rank condition may be written in terms of the vanishing of a set of homogeneous polynomials in the Plücker coordinates of Vi and Wj . On the other hand, observe that the product Drap(k E )m × Drap(k E )n is first embedded in a product of grassmannians, and secondly in a product of projective spaces, by taking the product of the corresponding Plücker embeddings. We conclude that Σ(M ) is a k-variety (cf. 10). 13 A more detailed description of the k-variety structure of Σ(M ) (resp.

By the following definitions we introduce the main objects of interest in this chapter. 10 Let (M, D) ∈ Relpos(E) × Drap(k E ) (resp. (M, D) ∈ Relpos(E) × Drap(E)). Define Σ(M ) ⊂ Drap(k E ) × Drap(k E ) as the fiber over M of the quotient mapping, and Σ(M, D) ⊂ Drap(k E ) as the fiber over D of the mapping Σ(M ) −→ Drap(k E ) induced by the second projection. Write Σ(M, D) = Σ(M, k D ). We call Σ(M ) the Universal Schubert cell of type M , and Σ(M, D) the Schubert cell defined by the flag D and the type M .

9 and the following commutative diagram Drapn (E × Drapm (E)/SE N(l+1)×(λ+1) → l,λ∈N ↓ Drapn (k ) × Drapm (k E )/Gl(k E ). 2) E Remark that the oblique arrow is the mapping (D, D ) → M (D, D ), that the horizontal arrow is injective and that the down arrow is surjective. This achieves the proof. 2 Schubert cells and Schubert varieties In what follows we identify the set Relpos(E) of types of relative position of flags of E with its image in the set of matrices with integral coefficients N(l+1)×λ+1) .

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