Combinatorial aspects of simplification of algebraic expressions

Abstract

A possible way of developing computer algebra systems is an "education" of the system. By "education" we mean here supplying the system with the capacity for handling new classes of mathematical objects. Tensors or, more generally, functions of several variables are an examples of such objects of practical interest. The main features of these objects which makes nontrivial the problem of incorporating them into a computer algebra system, are the symmetry properties exhibited by tensors (functions) we meet in practice. The simplification of expressions contaning tensor monomials involves two operations: monomial identification and monomial pattern matching. The tensor identification problem when dummy summation indices are included and the tensor pattern matching problem when the pattern has free variables may both be reformulated as combinatorial problems. The first one is reducible to the double coset problem for the permutation group and the second, to a variant of the general combinatorial object isomorphism problem. The backtrack algorithms for solving the above two problems are based on some well-known ideas in computational group theory. A recursive variant of the tensor identification algorithm for a restricted class of symmetry groups and the algorithm implementation are discussed briefly.