2024/05/08 by Lim, Lek-Heng, Ye, Ke · 1 citation
#05E14 #14E25 #14F45 #Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2405.05128
The degree of the Grassmannian with respect to the Plücker embedding is well-known. However, the Plücker embedding, while ubiquitous in pure mathematics, is almost never used in applied mathematics. In applied mathematics, the Grassmannian is usually embedded as projection matrices Gr(k,ℝn) ≅ \P ∈ ℝn × n : P\scriptscriptstyleT = P = P2, tr(P) = k\ or as involution matrices Gr(k,ℝn) ≅ \X ∈ ℝn × n : X\scriptscriptstyleT = X, X2 = I, tr(X)=2k - n\. We will determine an explicit expression for the degree of the Grassmannian with respect to these embeddings. In so doing, we resolved a conjecture of Devriendt, Friedman, Reinke, and Sturmfels about the degree of Gr(2, ℝn) and in fact generalized it to Gr(k, ℝn). We also proved a set theoretic variant of another conjecture of theirs about the limit of Gr(k,ℝn) in the sense of Gröbner degneration.