Let \$V\$ be the Weyl module of dimension \${ 2n \choose n}-{2n \choose n-2}\$ for the symplectic group \$\Sp(2n,\F)\$ whose highest weight is the \$n\$-th fundamental dominant weight. The module \$V\$ affords the grassmann embedding of the symplectic dual polar space \$DW(2n-1,\F)\$, therefore \$V\$ is also called the grassmann-module for the symplectic group. We consider the smallest case for \$Char(\F)\$ odd for which \$V\$ is reducible, namely \$n = 4\$ and \$Char(\F)=3\$. In this case the unique factor \$R\$ of \$V\$ has vector dimension \$1\$. Here we provide a geometric description for \$R\$ and study some relations between \$R\$ and other objects associated with the grassmann embedding.

Cardinali, I. (2010). On the Grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3. DISCRETE MATHEMATICS, 310, 3219-3227 [10.1016/j.disc.2009.10.017].

### On the Grassmann module of symplectic dual polar spaces of rank 4 in characteristic 3

#### Abstract

Let \$V\$ be the Weyl module of dimension \${ 2n \choose n}-{2n \choose n-2}\$ for the symplectic group \$\Sp(2n,\F)\$ whose highest weight is the \$n\$-th fundamental dominant weight. The module \$V\$ affords the grassmann embedding of the symplectic dual polar space \$DW(2n-1,\F)\$, therefore \$V\$ is also called the grassmann-module for the symplectic group. We consider the smallest case for \$Char(\F)\$ odd for which \$V\$ is reducible, namely \$n = 4\$ and \$Char(\F)=3\$. In this case the unique factor \$R\$ of \$V\$ has vector dimension \$1\$. Here we provide a geometric description for \$R\$ and study some relations between \$R\$ and other objects associated with the grassmann embedding.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11365/18563`