Regular Article
On Identities Satisfied by Minors of a Matrix

https://doi.org/10.1006/aima.1993.1030Get rights and content
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Abstract

The algebraic relations between minors of a generic matrix are of great interest in many domains (invariant theory, representations of the symmetric group, algebraic geometry, Schur functions, etc.). We show that most of these relations result easily from a particular identity, which was stated by H. W. Turnbull in 1909 but has remained little known since. The introduction of Cayley (or Peano) algebras by G.-C. Rota throws light on this identity. Indeed, Turnbull′s formula turns out to be nothing but the expression of associativity and anticommutativity for the second product (the "meet operation") of these algebras. A constant problem when dealing with minors′ identities is the choice of appropriate notations. The difficulty of grasping the cumbersome symbols used by many authors of the last century is no doubt responsible for the present neglect of this field. We propose to code these identities by permutations of letters in tableaux. As an illustration of these principles we derive explicitly a great number of classical theorems due to Sylvester, Bazin, Reiss, Picquet, Muir, etc., and we give also several new formulae which generalize some of these theorems. However, we do not try to give an exhaustive list of identities. It seems to us more interesting to show that these tools provide an effective calculus on minors, which enables anyone to find the formulae, old or new, suitable for a particular application.

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