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The optimal version of Hua’s fundamental theorem of geometry of rectangular matrices


About this Title

Peter Šemrl

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 232, Number 1089
ISBNs: 978-0-8218-9845-1 (print); 978-1-4704-1892-2 (online)
DOI: http://dx.doi.org/10.1090/memo/1089
Published electronically: February 19, 2014
Keywords:Rank, adjacency preserving map, matrix over a division ring

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Notation and basic definitions
  • Chapter 3. Examples
  • Chapter 4. Statement of main results
  • Chapter 5. Proofs

Abstract


Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all matrices over a division ring which preserve adjacency in both directions. Motivated by several applications we study a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings we solve all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings we get such an optimal result only for square matrices and give examples showing that it cannot be extended to the non-square case.

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