Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices

Australas. J. Comb. Volume 88(3) (2024), Pages 282-293 It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with eigenva...

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Bibliographic Details
Main Author Narita, Kazumasa
Format Journal Article
LanguageEnglish
Published 14.04.2022
Subjects
Mathematics - Combinatorics
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Summary:Australas. J. Comb. Volume 88(3) (2024), Pages 282-293 It is known that the complex Grassmannian of $k$-dimensional subspaces can be identified with the set of projection matrices of rank $k$. It is also classically known that the convex hull of this set is the set of Hermitian matrices with eigenvalues between $0$ and $1$ and summing to $k$. We give a new proof of this fact. We also give an existence theorem for a certain combinatorial class of hypermatrices by a similar argument. This existence theorem can be rewritten into an existence theorem for a uniform weighted hypergraph with given weighted degree sequence.
DOI:10.48550/arxiv.2204.06953