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Additional info for An Introduction to Quasisymmetric Schur Functions (September 26, 2012)
Are given by scalar multiplication. A map f : A → A , where (A , m , u ) is another algebra over R, is an algebra morphism if f ◦ m = m ◦ ( f ⊗ f ) and f ◦ u = u . We shall frequently write ab instead of m(a ⊗ b). The algebra A has identity element 1A = u(1R ), where 1R is the identity element of R. The unit u is always given by u(r) = r1A for all r ∈ R. A coalgebra is defined by reversing the arrows in the diagrams that define an algebra. 2. A coalgebra over R is an R-module C together with R-linear maps coproduct or comultiplication ∆ : C → C ⊗ C and counit or augmentation ε : C → R, such that the following diagrams commute.
19. The weight enumerator F(P, γ) of a labelled poset (P, γ) is a quasisymmetric function. Proof. Let (α1 , . . , αm ) be a composition of |P| and (k1 , . . , km ) a sequence of positive integers k1 < · · · < km . The coefficient of the monomial xkα11 · · · xkαmm in F(P, γ) is the number of (P, γ)-partitions that map αi elements of P to ki . 38 3 Hopf algebras Let f be such a (P, γ)-partition and suppose that (l1 , . . , lm ) is another sequence of positive integers l1 < · · · < lm . It is easy to see that the map φ ( f ), defined by setting (φ ( f ))(p) = li if f (p) = ki , is a (P, γ)-partition that maps αi elements of P to li .
3). The Hopf algebras that we shall study are infinite-dimensional, graded and connected, with each component of the direct sum having finite dimension. Associated with each such Hopf algebra is another Hopf algebra of interest to us. 3]. 8. Let H = n 0 H n be a connected, graded Hopf algebra over R, such that each homogeneous component H n is finite-dimensional. Define the module H ∗ by H∗= (H n )∗ , n 0 where (H denotes the set of all linear maps f : H n → R. ∗ Then H is a Hopf algebra with n )∗ 1.