quasisymmetric

Quasisymmetric is a term used in several areas of mathematics to describe a weakening of symmetry that preserves relative structure rather than exact values. In one-dimensional geometric terms, a homeomorphism f: R → R is quasisymmetric if there exists a constant M ≥ 1 such that for all x ∈ R and all t > 0, the inequality |f(x+t) − f(x)| ≤ M |f(x) − f(x−t)| holds. More generally, quasisymmetric maps between metric spaces are controlled by a distortion function η: [0, ∞) → [0, ∞) so that the relative sizes of adjacent intervals are uniformly preserved. Quasisymmetric maps generalize bi-Lipschitz maps and are central to quasiconformal geometry, Teichmüller theory, and geometric group theory, with stability under composition and limits.

Quasisymmetric functions form a family of formal power series in infinitely many variables that generalize symmetric functions. They were introduced by Ira Gessel in 1984. A formal power series F in variables x1, x2, … is quasisymmetric if the coefficients of certain monomials depend only on the relative order of the indices, not on their exact values. The space QSym of quasisymmetric functions is a graded Hopf algebra containing the symmetric functions as a subalgebra. It has standard bases indexed by compositions, notably the monomial basis Mα and the fundamental basis Fα. Quasisymmetric functions arise in the study of P-partitions, permutation statistics, and representation theory, and they are in duality with the noncommutative symmetric functions NSym.