Noncommutative symmetric functions are a generalisation of symmetric functions. Many concepts and ideas extend from symmetric functions to noncommutative symmetric functions and the way in which they extend sheds light on their behaviour for ordinary symmetric functions.
Noncommutative symmetric functions also arise in their own right as interesting objects of study.
The graded Hopf algebra of noncommutative symmetric functions, , is defined in the following way.
As an algebra, it is , the free algebra in countably many indeterminants over .
The comultiplication is given by , where .
The counit is for .
The antipode is , where is a word in with and .
The degree of is .
The object represents a functor from not-necessarily-commutative rings to groups given by sending to formal power series in with leading term , .