Canonicalmonomial

Canonicalmonomial is a term used in algebraic combinatorics and invariant theory to denote a distinguished monomial selected from a symmetry class or from a polynomial to provide a canonical representative. The basic idea is to pick a single monomial from among those considered equivalent under a given relation, such as the action of a symmetry group on the variables or the leading term under a fixed monomial order.

Construction and definition: Fix a monomial order, for example lexicographic order with a specified variable ranking. Suppose a symmetry group G, such as the permutation group on the variables, acts on the set of monomials. The orbit of a monomial x^a under G consists of all monomials obtained by permuting the exponents according to G. The canonicalmonomial of that orbit is defined as the unique monomial in the orbit that is minimal (or maximal) under the chosen order. More generally, for a given polynomial f, the leading monomial lm(f) with respect to the same order often serves as the canonicalmonomial associated with f.

Example: In two variables x and y with the swap symmetry, the orbit of x^2 y includes x^2 y and x y^2; using lex order with x > y, the canonicalmonomial is x^2 y.

Applications and notes: Canonicalmonomials help compare polynomials and monomial ideals under symmetry, assist in defining invariants, and provide stable representatives for algorithms in symbolic computation. The specific canonicalmonomial depends on the chosen symmetry and the term order, so different contexts may yield different representatives. See also monomial, term order, Gröbner basis, orbit, and invariant theory.