A geometric approach to alternating k-linear forms

Abstract

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.

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Correspondence to Luca Giuzzi.

Appendix: The polynomial \(\Delta (u_1,\ldots ,u_8)\)

Appendix: The polynomial \(\Delta (u_1,\ldots ,u_8)\)

We present in detail the outcome of the computation of the polynomial \(\Delta (c_1,\ldots ,c_8)\) of Sect. 4. This result has been obtained by a straightforward implementation of the procedure outlined in Subsection 4.2 using the computer algebra system [16]. The actual computation, in particular factoring the discriminant of the quadrics, took, for each polynomial being considered, slightly more than 3 h on a multiprocessor XEON E7540 machine and has necessitated a maximum of approximately 22 Gb of RAM.

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Cardinali, I., Giuzzi, L. & Pasini, A. A geometric approach to alternating k-linear forms. J Algebr Comb 45, 931–963 (2017). https://doi.org/10.1007/s10801-016-0730-6

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Keywords

  • Grassmann geometry
  • Hyperplane
  • Multilinear form
  • Alternating form

Mathematics Subject Classification

  • 15A75
  • 14M15
  • 15A69