Isometric embeddings of Johnson graphs in Grassmann graphs
Mark Pankov
DOI: 10.1007/s10801-010-0258-0
Abstract
Let V be an n-dimensional vector space (4\leq n<\infty ) and let G k( V) {\mathcal{G}}_{k}(V) be the Grassmannian formed by all k-dimensional subspaces of V. The corresponding Grassmann graph will be denoted by Γ k ( V). We describe all isometric embeddings of Johnson graphs J( l, m), 1< m< l - 1 in Γ k ( V), 1< k< n - 1 (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of J( n, k) in Γ k ( V) is an apartment of G k( V) {\mathcal{G}}_{k}(V) if and only if n=2 k. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in Γ k ( V), 1< k< n - 1.
Pages: 555–570
Keywords: keywords Johnson graph; Grassmann graph; building; apartment
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References
1. Baer, R.: Linear Algebra and Projective Geometry. Academic Press, New York (1952)
2. Brown, K.: Buildings. Springer, Berlin (1989)
3. Chow, W.L.: On the geometry of algebraic homogeneous spaces. Ann. Math. 50, 32-67 (1949)
4. Cooperstein, B.N., Shult, E.E.: Frames and bases of Lie incidence geometries. J. Geom. 60, 17-46 (1997)
5. Cooperstein, B.N., Kasikova, A., Shult, E.E.: Witt-type theorems for Grassmannians and Lie incidence geometries. Adv. Geom. 5, 15-36 (2005)
6. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries. Eur. J. Comb. 28, 1493-1529 (2007)
7. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries II. Adv. Geom. 9, 45-84 (2009)
8. Pankov, M.: Grassmannians of Classical Buildings. Algebra and Discrete Math. Series, vol.
2. World Scientific, Singapore (2010)
9. Pasini, A.: Diagram Geometries. Clarendon Press, Oxford (1994)
10. Tits, J.: Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Mathematics, vol. 386.
2. Brown, K.: Buildings. Springer, Berlin (1989)
3. Chow, W.L.: On the geometry of algebraic homogeneous spaces. Ann. Math. 50, 32-67 (1949)
4. Cooperstein, B.N., Shult, E.E.: Frames and bases of Lie incidence geometries. J. Geom. 60, 17-46 (1997)
5. Cooperstein, B.N., Kasikova, A., Shult, E.E.: Witt-type theorems for Grassmannians and Lie incidence geometries. Adv. Geom. 5, 15-36 (2005)
6. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries. Eur. J. Comb. 28, 1493-1529 (2007)
7. Kasikova, A.: Characterization of some subgraphs of point-collinearity graphs of building geometries II. Adv. Geom. 9, 45-84 (2009)
8. Pankov, M.: Grassmannians of Classical Buildings. Algebra and Discrete Math. Series, vol.
2. World Scientific, Singapore (2010)
9. Pasini, A.: Diagram Geometries. Clarendon Press, Oxford (1994)
10. Tits, J.: Buildings of Spherical Type and Finite BN-pairs. Lecture Notes in Mathematics, vol. 386.
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