On the Grassmann modules for the unitary groups
نویسنده
چکیده
Let V be 2n-dimensional vector space over a field K equipped with a nondegenerate skew-ψ-Hermitian form f of Witt index n ≥ 1, let K0 ⊆ K be the fix field of ψ and let G denote the group of isometries of (V, f). For every k ∈ {1, . . . , 2n}, there exist natural representations of the groups G ∼= U(2n,K/K0) and H = G∩SL(V ) ∼= SU(2n,K/K0) on the k-th exterior power of V . With the aid of linear algebra, we prove some properties of these representations. We also discuss some applications to projective embeddings and hyperplanes of Hermitian dual polar spaces.
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