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Ad ) to the point a1 ∧ · · · ∧ ad in ∧d V . Identifying ∧d V with K N , the i-th coordinate of a1 ∧ · · · ∧ ad is given by the determinant of the d-minor of the n×d matrix A = (aij ) (where the columns of the matrix A = (aij ) are the vectors a1 , . . , ad ), with row indices i1 , . . , id . Note that a1 ∧· · ·∧ad = 0 if and only if the matrix A belongs to the set Z of matrices of rank less than d. Moreover, it is easily seen that given (a1 , . . , ad ), (a1 , . . , there exists a matrix C ∈ GLd (K) such that A = AC), then a1 ∧ · · · ∧ ad = det(C)a1 ∧ · · · ∧ ad .

Let dim X(τ ) > 0. Let r ci Fi , ci ∈ k, (∗) i=1 be a linear relation of standard monomials Fi of degree m. Let Fi = pwi1 . . pwim . We divide the proof into the following two cases: Case 1: wi1 = τ, ∀i. First observe that X(τ ) being an integral scheme, pτ is a non-zero divisor in R(τ ); we cancel pτ and obtain a linear relation among monomials of degree m − 1. The required result follows by induction on m (note that when m = 1, the result follows by linear independence of {pi , i ∈ Id,n }). Case 2: wi1 < τ for some i.

Let λ be a dominant weight, and V (λ) be the irreducible G-module over C. , annulled by Xβ for all β ∈ R + ) in V (λ) of weight λ (for the T -action); we have that the weight λ in V (λ) has multiplicity one. 3 Schubert varieties 25 (for the T -action) of weight w(λ), and is unique up to scalars. Having fixed λ, we shall denote uλ (resp. uw,λ ) by just u (resp. uw ). Set Vw,Z (λ) = UZ+ (gC )uw . For any field K, let Vw,λ = Vw,Z (λ) ⊗ K, w ∈ W . Then VK (λ) := Vw0 ,λ (= Vw0 ,Z (λ) ⊗ K) is the Weyl module with highest weight λ, and for w ∈ W, Vw,λ is the Demazure module corresponding to w (and λ).

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