The Grassmannian of affine subspaces

Abstract

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, R n ) ∼ = {P ∈ R n×n : P T = P = P 2 , tr(P ) = k} or as involution matrices Gr(k, R n ) ∼ = {X ∈ R n×n : X T = X, X 2 = 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-Sturmfels about the degree Gr(2, R n ) and in fact generalized it to Gr(k, R n ). We also proved a set theoretic variant of another conjecture of Devriendt-Friedman-Sturmfels about the limit of Gr(k, R n ) in the sense of Gröbner degneration.

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