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Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices

Kazumasa Narita

Abstract

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.

Convex Hulls of Grassmannians and Combinatorics of Symmetric Hypermatrices

Abstract

It is known that the complex Grassmannian of -dimensional subspaces can be identified with the set of projection matrices of rank . It is also classically known that the convex hull of this set is the set of Hermitian matrices with eigenvalues between and and summing to . 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.
Paper Structure (4 sections, 13 theorems, 40 equations)

This paper contains 4 sections, 13 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Majorization
  3. Convex Hulls of Grassmannians
  4. Symmetric Hypermatrices

Key Result

Theorem 1.1

(FW) The convex hull of the embedded Grassmannian $G_{k}(\mathbf{C}^{n})$ in $HM(n)$ is and the set of extreme points of $\Omega$ is $G_{k}(\mathbf{C}^{n})$.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • ...and 8 more