Canonical Representation of Polynomial Expressions with Indices

Abstract

Computer algebra methods are widely employed in various branches of mathematics, physics, and other sciences. Simplification of algebraic expressions with indices is one of the important problems. Tensor expressions are the most typical example of these expressions. This paper briefly describes some basic methods for reducing expressions with indices to canonical form. The focus is placed on taking into account the properties of symmetries with respect to various permutations of indices in elementary symbols, symmetries associated with renaming summation indices, and general linear relationships among them. This paper also gives a definition of canonical representation for polynomial (multiplicative) expressions of variables with abstract indices that results from averaging the initial expression over the action of some finite group (signature stabilizer). In practice, e.g., for expressions of Riemann curvature tensors, the proposed algorithms show high efficiency.