Quasideterminants, I

Introduction A notion of quasideterminants for matrices over a noncommutative skew-field was introduced in [GR], [GR1], [GR2]. It proved its effectiveness for many areas including noncommutative symmetric functions [GKLLRT], noncommutative The main property of quasideterminants is a " heredity principle " : let A be a square matrix over a skew-field and (A ij) be its block decomposition into subma-trices of A. Consider A ij 's as elements of a matrix X. Then a quasideterminant of matrix X will be a matrix B again, and (under a natural assumption) a quaside-terminant of B will be equal to a suitable quasideterminant of A. This principle is not valid for commutative determinants, because they are not defined for block-matrices. Quasideterminants are not analogues of commutative determinants but rather a ratio of determinants of n × n-matrices to determinants of their (n − 1) × (n − 1)-submatrices. In fact, for a matrix over a commutative algebra a quasideterminant is equal to a correspondence ratio. Many noncommutative areas of mathematics (Ore rings, rings of differential operators, theory of factors, " quantum mathematics " , Clifford algebras, etc) were developed separately from each other. Our approach shows an advantage of working with totally noncommutative variables (over free rings and skew-fields). It leads us to a big variety of results, and their specialization to different noncommutative areas implies known theorems with additional information. The price you pay for this is a huge number of inversions in rational noncom-mutative expressions measured by their height which cannot be reduced. But this invariant (inversion height) shows us a " degree of noncommutativity " and it is of a great interest by itself. Our experience shows that dealing with noncommutative objects one should not imitate the classical commutative mathematics, but follow " the way it is " starting with basics. In this paper we consider mainly two such problems: noncommu-tative Plücker coordinates (as a background of a noncommutative geometry) and noncommutative Bezout and Vieta theorems (as a background of noncommuta-tive algebra). We apply these results to the theory of noncommutative symmetric functions started in [GKLLRT]. The first chapter of this paper contains basic definitions and properties of quaside-terminants. We consider Plücker coordinates in Chapter II. Chapter III is devoted to Bezout and Vieta theorems and also noncommutative symmetric functions. In