Journal of Algebraic Combinatorics

, Volume 45, Issue 4, pp 931–963 | Cite as

A geometric approach to alternating k-linear forms

  • Ilaria Cardinali
  • Luca GiuzziEmail author
  • Antonio Pasini


Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.


Grassmann geometry Hyperplane Multilinear form Alternating form 

Mathematics Subject Classification

15A75 14M15 15A69 


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Information Engineering and MathematicsUniversity of SienaSienaItaly
  2. 2.D.I.C.A.T.A.M. — Section of Mathematics Università di BresciaBresciaItaly

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