Gröbner bases and the immersion of real flag manifolds in Euclidean space (Q2777531)

In [Trans. Am. Math. Soc. 213, 305-314 (1975; Zbl 0312.55020)] \textit{K. Y. Lam} gave upper bounds for the immersion dimension or flag manifolds in the real, complex, and quaternionic cases. \textit{R.~E. Stong} [Proc. Am. Math. Soc. 88, 708-710 (1983; Zbl 0532.57020)] showed that these dimensions will be best possible for the (real) flag manifolds \(F=F(n_1,\dots ,n_s)\) in certain cases. This theorem is substantially generalized in Remark 5.3 of the present paper: If \(w=\{n_1,\dots ,n_s\}\) can be partitioned as \(w=w_1 \sqcup w_2\sqcup \dots \sqcup w_r,\) where \(w_1,w_2 \not = \emptyset \), andNEWLINENEWLINENEWLINE1) \(|\Sigma _{m\in w_1} m - \Sigma _{m\in w_2} m|\leq 1,\)NEWLINENEWLINENEWLINE2) for \(i\geq 3\), \(k\in w_i \Rightarrow k\leq \Sigma _{m\in w_1} m+ \dots + \Sigma _{m\in w_{i-1}} m+ 1\),NEWLINENEWLINENEWLINEthen Lam's result is best possible for \(F\) (the case \(r=3\) is Stong's theorem). NEWLINENEWLINENEWLINEThe main tool used is the Stiefel-Whitney classes. The introduction of Gröbner bases as well as their implementation with MAPLE software to compute these in many cases is another interesting feature. Finally, the adjoint representation \(Ad\) (together with its restriction to suitable subgroups, in particular the diagonal subgroup \(D(n) \subset O(n)\)), as well as the induced map \(B Ad\) on the classifying spaces and its action \((B Ad)^*\) in cohomology, is the principal technique used to handle the Stiefel-Whitney classes of the normal bundle. In particular, Theorem 4.1 is obtained in this way: The \(k\)th Stiefel-Whitney class of the normal bundle of \(F\) is zero if the \(k\)th elementary symmetric polynomial \(\tau _k(t_{12},\dots ,t_{n-1 n})\), evaluated on the elements \(t_{ij}=x_i+x_j\), belongs to the ideal \({\mathcal I}\) generated by the elementary symmetric polynomials in \({\mathbb Z}_2[x_1,\dots , x_n]\) (as usual \(n=n_1+\dots +n_s\)).