Grassmann and Weyl embeddings of orthogonal Grassmannians (Q2435035)

Let \(V\) be a \((2n+1)\)-dimensional vector space over a field \(\mathbb F\), endowed with a non-singular quadratic form \(q\) of Witt index \(n\). Let \(\Delta\) be the building of type \(\mathsf B_n\) whose elements are the subspaces of \(V\) which are totally singular for \(q\), and for \(k \leq n\) denote \(\Delta_k \subset \Delta\) the \(k\)-Grassmannian of \(\Delta\), whose points are the totally singular \(k\)-subspaces of \(V\). Then \(\Delta_k\) is a point-line geometry. The paper under review studies two classes of embeddings of \(\Delta_k\) into projective spaces, called the Grassmann embeddings and the Weyl embeddings. The Grassmann embedding is the natural embedding \(\varepsilon_k : \Delta_k \rightarrow \mathrm{PG}(W_k)\), where \(W_k = \bigwedge^k V\) and where \(\mathrm{PG}(W_k)\) denotes the associated projective space. To define the Weyl embedding, notice that \(\Delta_k\) has a natural action of the orthogonal group \(\mathrm{SO}(2n+1,\mathbb F)\). Given \(k < n\), denote \(\lambda_k\) the \(k^{th}\) fundamental weight of \(\mathrm{SO}(2n+1,\mathbb F)\), and set \(\lambda_n\) twice the last fundamental weight. Then \(\Delta_k\) is identified with the orbit of a highest weight line in the projective space of the Weyl module \(V(\lambda_k)\) of \(\mathrm{SO}(2n+1,\mathbb F)\), and the Weyl embedding is the associated embedding \(\widetilde\varepsilon_k : \Delta_k \rightarrow \mathrm{PG}(V(\lambda_k))\). Set \(G = \mathrm{SO}(2n+1,\mathbb F)\) and, for \(k \leq n\), let \(\langle \varepsilon_k(\Delta_k) \rangle \subset W_k\) be the submodule generated by the image of \(\varepsilon_k\), which is a quotient of \(V(\lambda_k)\). As a consequence of the irreducibility of \(V(\lambda_k)\), the \(G\)-modules \(\langle \varepsilon_k(\Delta_k) \rangle\) and \(V(\lambda_k)\) are isomorphic if \(\mathrm{char}(\mathbb F) \neq 2\). An easy proof of this fact is given in the first theorem of the paper, without making use the irreducibility of \(V(\lambda_k)\) and relying only on elementary properties of quadratic forms in odd characteristic. If instead \(\mathrm{char}(\mathbb F) = 2\), then the authors show that \(\mathrm{dim}(V(\lambda_k)) - \mathrm{dim} \langle \varepsilon_k (\Delta_k) \rangle = {2n+1 \choose k-2}\). In the second part of the paper the authors study the universality of Grassmann and Weyl embeddings, as defined by \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)]. Given \(k < n\), the Weyl embedding \(\widetilde\varepsilon_k\) is called universal if all embeddings of \(\Delta_k\) defined over the same division ring as \(\widetilde\varepsilon_k\) are quotients of \(\widetilde\varepsilon_k\), it is a well known fact that \(\widetilde \varepsilon_1\) is universal for any field \(\mathbb{F}\). In the last theorem of the paper, assuming that \(\mathbb F\) is a perfect field of positive characteristic or a number field, the authors prove that \(\widetilde\varepsilon_2\) is universal provided that \(n > 2\), and that \(\widetilde\varepsilon_3\) is universal provided that \(n > 3\) and \(\mathbb F \neq \mathbb F_2\). Then they conjecture that the Weyl embedding \(\widetilde\varepsilon_k\) is always universal, for any \(k < n\) and for any field \(\mathbb F\).