Embeddings of orthogonal Grassmannians (Q2016991)

In the paper under review, the author gives a concise survey of recent developments on embeddings of orthogonal Grassmannians. After a brief review of basic definitions and notions of projective and Veronesean embeddings of point-line geometries, the author restricts himself to Grassmannians \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) of buildings of types \(B_n\) and \(D_n\) and seven associated embeddings. Some of the embeddings are known to be isomorphic in case the characteristic of the underlying field is not 2. Section 4 of the paper is mostly devoted to discussing the various embeddings in case of characteristic 2. Sketches of proofs are provided in order to convey the flavour of the arguments. The results of this section as well as of the following section on universality of embeddings are a summary of three papers by \textit{I. Cardinali} and \textit{A. Pasini} [J. Algebr. Comb. 38, No. 4, 863--888 (2013; Zbl 1297.14053); J. Comb. Theory, Ser. A 120, No. 6, 1328--1350 (2013; Zbl 1278.05052)] and [J. Group Theory 17, No. 4, 559--588 (2014; Zbl 1320.20041)]. The last section deals with universality of Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\). These are obtained from the fundamental dominant weights for the root system of types \(B_n\) and \(D_n\). \textit{A. Kasikova} and \textit{E. Shult} [J. Algebra 238, No. 1, 265--291 (2001; Zbl 0988.51001)] showed that most of these point-geometries admit universal projective embeddings. The author conjectures that the Weyl embeddings of \(\mathcal B_{n,k}\) and \(\mathcal D_{n,k}\) are universal for \(k = 2, \dots, n-1\); the ones when \(k = 1\) are known to be universal. He then considers the cases \(k = 2\) and 3 under additional assumptions and outlines a proof of universality in these situations.