Secant Dimensions of Minimal Orbits: Computations and Conjectures

Abstract

We present an algorithm for computing the dimensions of higher secant varieties of minimal orbits. Experiments with this algorithm lead to many conjectures on secant dimensions, especially of Grassmannians and Segre products. For these two classes of minimal orbits we give a short proof of the relation, known from the work of Ehrenborg, Catalisano-Geramita-Gimigliano, and Sturmfels-Sullivant, between the existence of certain codes and non-defectiveness of certain higher secant varieties.

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

“…When it comes to Segre-Veronese varieties with only two factors, there are many results by many authors, which allow us to get a quite complete picture, described by the following conjectures, as stated in [121]. The above conjectures are based on results that can be found in [123,131,93,128,99,122,126,130,112,129,134,125,124,132,133,127,121].…”

“…Let k 2. Then, the secant variety σ s (G(k, n)) has the expected dimension except for the following cases: actual codimension expected codimension σ 3 (G(2, 6)) 1 0 σ 3 (G(3, 7)) 20 19 σ 4 (G(3, 7)) 6 2 σ 4 (G(2, 8)) 10 8 In [112], they proved the conjecture for n 15 (the case n 14 can be found in [113]). The conjecture has been proven to hold for s 6 (see [111]) and later for s 12 in [114].…”

“…Back to the results on dimensions of secant varieties of Grassmannians: in [112], a tropical geometry approach is involved. In [111], as was done by Alexander and Hirshowitz for the symmetric case, the authors needed to introduce a specialization technique, by placing a certain number of points on sub-Grassmannians and by using induction.…”

Skew-symmetric tensor decomposition

“…, while the generators in degree 2 are described by (7). This leads to the following algorithm for the skew-symmetric tensor decomposition of a tensor v ∈…”

Secants of Lagrangian Grassmannians

“…We quote from [5,Conjecture 4.1], yet such a conjecture was long before believed to be true. Conjecture 1.3 Let k ≥ 3.…”