# On the Grassmann modules for the unitary groups

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### Additional excerpts

...In De Bruyn and Van Maldeghem [13], we showed that with the pair T = (O,K), there is associated a polar space PT ....

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...Following the terminology of Cardinali, De Bruyn and Pasini [5], the embedding e/R is precisely the minimal full polarized embedding of ∆T (up to isomorphism)....

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...In Propositions 4.9 till 4.13 of De Bruyn and Van Maldeghem [13] a number of automorphisms of PT are described....

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...The following proposition was proved in De Bruyn and Van Maldeghem [13]....

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...By Cooperstein [6, 7] and De Bruyn [9, 10], e′′(p) := 〈v̄1 ∧ v̄2 ∧ v̄3〉 is a point of PG(W ) and the map e′′ from the point set of ∆ to the point set of PG(W ) defines a full embedding of ∆ into PG(W ), called the Grassmann embedding of ∆....

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### Additional excerpts

...Then gr(Hk) = er(Hk) = ( 2n k ) ; see [1, 3, 14, 16, 18]....

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##### References

1,115 citations

### "On the Grassmann modules for the un..." refers background in this paper

...Let Π be a polar space (Tits [12], Veldkamp [13]) of rank n ≥ 2....

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851 citations

### "On the Grassmann modules for the un..." refers background in this paper

...admits subquadrangles isomorphic to Q(4, q), see Payne and Thas [10]....

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...The generalized quadrangle Q−(5, q) admits subquadrangles isomorphic to Q(4, q), see Payne and Thas [10]....

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156 citations

### "On the Grassmann modules for the un..." refers background in this paper

...With Π there is associated a point-line geometry ∆ which is called a dual polar space, see Cameron [3]....

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68 citations