We give a geometrical description of the spin-embedding \$e_{sp}\$ of the symplectic dual polar space \$\Delta\cong DW(5,2^r)\$ by showing how the natural embedding of \$W(5,2^r)\$ into \$\PG(5,2^r)\$ is involved in the Grassman-embedding \$e_{gr}\$ of \$\Delta.\$ We prove that the map sending every quad of \$\Delta\$ to its nucleus realizes the natural embedding of \$W(5,2^r).\$ Taking the quotient of \$e_{gr}\$ over the space spanned by the nuclei of the quadrics corresponding to the quads of \$\Delta\$ gives an embedding isomorphic to \$e_{sp}.\$

Cardinali, I., Lunardon, (2008). A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 115, 1056-1064 [10.1016/j.jcta.2007.09.004].

### A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3

#### Abstract

We give a geometrical description of the spin-embedding \$e_{sp}\$ of the symplectic dual polar space \$\Delta\cong DW(5,2^r)\$ by showing how the natural embedding of \$W(5,2^r)\$ into \$\PG(5,2^r)\$ is involved in the Grassman-embedding \$e_{gr}\$ of \$\Delta.\$ We prove that the map sending every quad of \$\Delta\$ to its nucleus realizes the natural embedding of \$W(5,2^r).\$ Taking the quotient of \$e_{gr}\$ over the space spanned by the nuclei of the quadrics corresponding to the quads of \$\Delta\$ gives an embedding isomorphic to \$e_{sp}.\$
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Cardinali, I., Lunardon, (2008). A geometric description of the spin-embedding of symplectic dual polar spaces of rank 3. JOURNAL OF COMBINATORIAL THEORY. SERIES A, 115, 1056-1064 [10.1016/j.jcta.2007.09.004].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11365/18562`