The degree of the Grassmannian with respect to the Pl & uuml;cker embedding is well-known. However, the Pl & uuml;cker 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' ) =similar to {P is an element of R'x' :PT = P = P2, tr(P) = k} or as involution matrices Gr(k,R') =similar to {X is an element of R'x' : XT = 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, R') and in fact generalized it to Gr(k, R'). We also proved a set-theoretic variant of another conjecture of theirs about the limit of Gr(k,R') in the sense of Gr & ouml;bner degeneration. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Publication:

ADVANCES IN MATHEMATICS

http://dx.doi.org/10.1016/j.aim.2025.110459

Author:

Lek-Heng Lim

Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL 60637-1514, United States of America

E-mail addresses: lekheng@uchicago.edu

Ke Ye

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Corresponding author

E-mail addresses: keyk@amss.ac.cn