Secant dimensions of minimal orbits: computations and conjectures Let G be a connected reductive complex algebraic group and let V be a nontrivial irreducible module for G. It is well-known that the projective space ℙV contains a unique (Zariski)-closed orbit X consisting of the highest-weight lines and called the minimal orbit. Denote by C⊂V the affine cone over X. Moreover, denote by kX the k-th secant variety of k-secant (k-1)-planes to X, and by kC its affine cone. In the paper under review, the authors present an algorithm, called sedimo and available for download from jdraisma/, which allow to compute dimkC. After dealing with some implementation issues, they list some conjectures about secant dimensions especially regarding Grassmannians and Segre products. Moreover, for these classes of minimal orbits, the authors give a proof of the relation between the existence of certain codes and non-defectiveness of certain higher secant varieties.

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  1. Ballico, Edoardo: Set evincing the ranks with respect to an embedded variety (symmetric tensor rank and tensor rank) (2018)
  2. Bernardi, Alessandra; Carlini, Enrico; Catalisano, Maria Virginia; Gimigliano, Alessandro; Oneto, Alessandro: The Hitchhiker guide to: secant varieties and tensor decomposition (2018)
  3. Bernardi, Alessandra; Vanzo, Davide: A new class of non-identifiable skew-symmetric tensors (2018)
  4. Bernardi, Alessandra; Daleo, Noah S.; Hauenstein, Jonathan D.; Mourrain, Bernard: Tensor decomposition and homotopy continuation (2017)
  5. Abo, Hirotachi; Brambilla, Maria Chiara: On the dimensions of secant varieties of Segre-Veronese varieties (2013)
  6. Abo, Hirotachi; Wan, Jia: On Waring’s problem for systems of skew-symmetric forms (2013)
  7. Abo, Hirotachi; Brambilla, Maria Chiara: New examples of defective secant varieties of Segre-Veronese varieties (2012)
  8. Abo, Hirotachi; Ottaviani, Giorgio; Peterson, Chris: Non-defectivity of Grassmannians of planes (2012)
  9. Boralevi, Ada; Buczyński, Jarosław: Secants of Lagrangian Grassmannians (2011)
  10. Abo, Hirotachi: On non-defectivity of certain Segre-Veronese varieties (2010)
  11. Abo, Hirotachi; Brambilla, Maria Chiara: Secant varieties of Segre-Veronese varieties (\mathbbP^m \times\mathbbP^n) embedded by (\mathcalO(1,2)) (2009)
  12. Draisma, Jan: A tropical approach to secant dimensions (2008)
  13. Baur, Karin; Draisma, Jan; de Graaf, Willem A.: Secant dimensions of minimal orbits: computations and conjectures (2007)

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