Let $V$ be a $2n$-dimensional vector space over a field $\mathbb{F}$ equipped with a non-degenerate alternating form $\xi.$ Let $\mathcal{G}_n$ be the $n$-grassmannian of $\PG(V)$ and $\Delta_n$ the dual of the polar space $\Delta$ associated to $\xi$. Then $\mathcal{G}_n$ and $\Delta_n$ are naturally embedded in the vector space $W_n=\wedge^nV$ and $V_n\subseteq W_n$ respectively, where dim($W_n$)$={ 2n \choose n}$ and dim($V_n$)$={ 2n \choose n}-{ 2n \choose {n-2}}.$ The spaces $W_n$ and $V_n$ can be regarded as modules for the symplectic group $Sp(2n, \mathbb{F}).$ If char($\mathbb{F}$)$\not= 2$, we will define two forms $\alpha$ and $\beta$ of $W_n$ which coincide on $V_n$ and we will investigate the relation between these two forms and the collineation of $W_n$ naturally induced by $\xi$. We will obtain a description of the module $W_n$ in terms of the two subspaces of $W_n$ where the linear functionals induced by $\alpha$ and $\beta$ are equal and respectively opposite.\\

Cardinali, I., Pasini, A. (2012). Two forms related to the symplectic dual polar space in odd characteristic. DESIGNS, CODES AND CRYPTOGRAPHY, 64(1-2), 47-60 [10.1007/s10623-011-9545-6].

### Two forms related to the symplectic dual polar space in odd characteristic

#### Abstract

Let $V$ be a $2n$-dimensional vector space over a field $\mathbb{F}$ equipped with a non-degenerate alternating form $\xi.$ Let $\mathcal{G}_n$ be the $n$-grassmannian of $\PG(V)$ and $\Delta_n$ the dual of the polar space $\Delta$ associated to $\xi$. Then $\mathcal{G}_n$ and $\Delta_n$ are naturally embedded in the vector space $W_n=\wedge^nV$ and $V_n\subseteq W_n$ respectively, where dim($W_n$)$={ 2n \choose n}$ and dim($V_n$)$={ 2n \choose n}-{ 2n \choose {n-2}}.$ The spaces $W_n$ and $V_n$ can be regarded as modules for the symplectic group $Sp(2n, \mathbb{F}).$ If char($\mathbb{F}$)$\not= 2$, we will define two forms $\alpha$ and $\beta$ of $W_n$ which coincide on $V_n$ and we will investigate the relation between these two forms and the collineation of $W_n$ naturally induced by $\xi$. We will obtain a description of the module $W_n$ in terms of the two subspaces of $W_n$ where the linear functionals induced by $\alpha$ and $\beta$ are equal and respectively opposite.\\
##### Scheda breve Scheda completa Scheda completa (DC)
2012
Cardinali, I., Pasini, A. (2012). Two forms related to the symplectic dual polar space in odd characteristic. DESIGNS, CODES AND CRYPTOGRAPHY, 64(1-2), 47-60 [10.1007/s10623-011-9545-6].
File in questo prodotto:
File
forms-odd-char.pdf

non disponibili

Tipologia: Post-print
Licenza: NON PUBBLICO - Accesso privato/ristretto
Dimensione 387.06 kB
Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11365/3162