Two forms related to the symplectic dual polar space in odd characteristic
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 \(W_n:=\bigwedge^n\,V\) and \(V_n\subseteq\,W_n\), respectively. Suppose that a hyperbolic basis \((e_1,\dots,e_{2n})\) of \(V\) is chosen so that \(\xi(e_i,e_j)=\xi(e_{i+n},e_{j+n})=0\), \(\xi(e_{i},e_{j+n})=\delta_{ij}\), and \(\xi(e_{i+n},e_{j})=-\delta_{ij}\) for any \(1\leq\,i,j\leq\,n\). Using coordinates with respect to the standard basis \((e_J)_{J\in{I\choose{n}}}\) of \(W_n\) where \(I=\{1,2,\dots,2n\}\) and \(I\choose{n}\) is the family of \(n\)-subsets of \(I\), \(J=\{j_1,\dots,j_n\}\), \(1\leq\,j_1<\dots<j_n\leq\,2n\) and \(e_J=e_{j_1}\wedge\,e_{j_2}\wedge\dots\wedge\,e_{j_n}\), the authors define two forms \(\alpha\) and \(\beta\) of \(W_n\) in case of char\(({\mathbb F})\not=2\) as follows: Assume that \(x:=(x_J)_{J\in{I\choose{n}}}\) and \(y:=(y_J)_{J\in{I\choose{n}}}\), then \[ \alpha(x,y):=\sum_{J\in{I\choose{n}}}(-1)^{\varepsilon(J)n}x_J.y_{\rho(J)} \] with \(\varepsilon(J)=|J\cap\{n+1,\dots,2n\}|\), \(\rho(J)=\{\rho(j) : j\in\,J\}\), and \(\rho(j)=j+n\), if \(j\leq\,n\), and \(\rho(j)=j-n\), if \(j>n\), and \[ \beta(x,y):=\sum_{J\in{I\choose{n}}}(-1)^{\theta(J)}x_J.y_{I\setminus\,J} \] with \(\theta(J)=n(n+1)/2+\sum_{j\in\,J}j\). The forms \(\alpha\) and \(\beta\) are symmetric, if \(n\) is even, and alternating, if \(n\) is odd; they are non-degenerate on \(W_n\) and coincide on \(V_n\). The forms \(\alpha\) and \(\beta\) are preserved by the group \(Sp(2n,{\mathbb F})\). Restricted to \(V_n\) the forms \(\alpha\) and \(\beta\) can be degenerate and their radical is the largest invariant subspace of \(V_n\) for the symplectic group, called the \textit{radical of \(V_n\)}, where \(V_n\) is regarded as the Weyl module for \(Sp(2n,{\mathbb F})\) with respect to the \(n\)-th fundamental dominant weight. The authors obtain a description of the module \(W_n\) in terms of two subspaces of \(W_n\) where the linear functionals induced by \(\alpha\) and \(\beta\) are equal and respectively opposite.
scientific article; zbMATH DE number 6037525