2015/12/23 by Mark Pankov, Pankov, Mark
Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1512.07517
openalex publication_date 2015/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a complex Hilbert space of finite dimension n≥ 3. Denote by \mathcal Gk(H) the Grassmannian consisting of k-dimensional subspaces of H. Every orthogonal apartment of \mathcal Gk(H) is defined by a certain orthogonal base of H and consists of all k-dimensional subspaces spanned by subsets of this base. For n≠ 2k (except the case when n=6 and k is equal to 2 or 4) we show that every bijective transformation of \mathcal Gk(H) sending orthogonal apartments to orthogonal apartments is induced by an unitary or conjugate-unitary operator on H. The second result is the following: if n=2k≥ 8 and f is a bijective transformation of \mathcal Gk(H) such that f and f-1 send orthogonal apartments to orthogonal apartments then there is an unitary or conjugate-unitary operator U such that for every X∈ \mathcal Gk(H) we have f(X)=U(X) or f(X) coincides with the orthogonal complement of U(X).