In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable number of variables.
This ring can be realized as a specific limit of the rings of quasisymmetric polynomials in n variables, as n goes to infinity.
This ring serves as universal structure in which relations between quasisymmetric polynomials can be expressed in a way independent of the number n of variables (but its elements are neither polynomials nor functions).
The ring of quasisymmetric functions, denoted QSym, can be defined over any commutative ring R such as the integers.
Quasisymmetric functions are power series of bounded degree in variables
is equal to the coefficient of the monomial
for any strictly increasing sequence of positive integers
indexing the variables and any positive integer sequence
Both symmetric and quasisymmetric polynomials may be characterized in terms of actions of the symmetric group
permutes variables, changing a polynomial
conditionally permutes variables, changing a polynomial
[2][3] Those polynomials unchanged by all such conditional swaps form the subring of quasisymmetric polynomials.
is the polynomial The simplest symmetric polynomial containing these monomials is QSym is a graded R-algebra, decomposing as where
-span of all quasisymmetric functions that are homogeneous of degree
and all formal power series The fundamental basis consists
is the ring of rational numbers, one has Then one can define the algebra of symmetric functions
as the subalgebra of QSym spanned by the monomial symmetric functions
which rearrange to the integer partition
Other important bases for quasisymmetric functions include the basis of quasisymmetric Schur functions,[4] the "type I" and "type II" quasisymmetric power sums,[5] and bases related to enumeration in matroids.
[6][7] Quasisymmetric functions have been applied in enumerative combinatorics, symmetric function theory, representation theory, and number theory.
Applications of quasisymmetric functions include enumeration of P-partitions,[8][9] permutations,[10][11][12][13] tableaux,[14] chains of posets,[14][15] reduced decompositions in finite Coxeter groups (via Stanley symmetric functions),[14] and parking functions.
[16] In symmetric function theory and representation theory, applications include the study of Schubert polynomials,[17][18] Macdonald polynomials,[19] Hecke algebras,[20] and Kazhdan–Lusztig polynomials.
[21] Often quasisymmetric functions provide a powerful bridge between combinatorial structures and symmetric functions.
As a graded Hopf algebra, the dual of the ring of quasisymmetric functions is the ring of noncommutative symmetric functions.
The ring of quasisymmetric functions is the terminal object in category of graded Hopf algebras with a single character.
[22] Hence any such Hopf algebra has a morphism to the ring of quasisymmetric functions.
[23] The Malvenuto–Reutenauer algebra[24] is a Hopf algebra based on permutations that relates the rings of symmetric functions, quasisymmetric functions, and noncommutative symmetric functions, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram.
The duality between QSym and NSym mentioned above is reflected in the main diagonal of this diagram.
Many related Hopf algebras were constructed from Hopf monoids in the category of species by Aguiar and Majahan.
[25] One can also construct the ring of quasisymmetric functions in noncommuting variables.