An Introduction to Quasisymmetric Schur Functions: Hopf by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

An advent to Quasisymmetric Schur Functions is aimed toward researchers and graduate scholars in algebraic combinatorics. The target of this monograph is twofold. the 1st target is to supply a reference textual content for the fundamental idea of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric services and connections among them. the second one target is to provide a survey of effects with appreciate to an exhilarating new foundation of the Hopf algebra of quasisymmetric services, whose combinatorics is similar to that of the well known Schur functions.

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Then B is a bialgebra if 1. Δ and ε are algebra morphisms or equivalently, 2. m and u are coalgebra morphisms. We are now ready to define a Hopf algebra. 4. Let (H , m, u, Δ , ε ) be a bialgebra. Then H is a Hopf algebra if there is a linear map S : H → H such that m ◦ (S ⊗ id) ◦ Δ = u ◦ ε = m ◦ (id ⊗ S) ◦ Δ . Thus, in Sweedler notation, S satisfies ∑ S(h1)h2 = ε (h)1 = ∑ h1S(h2) for all h ∈ H , where 1 is the identity element of H . The map S is called the antipode of H . A subset I ⊆ H is a Hopf ideal if it is both an ideal and coideal, and S(I ) ⊆ I .

In [34]. As we will see, they are closely connected to both quasisymmetric and symmetric functions. Throughout we use the notation used in [11], which evokes the relationship with symmetric functions. 1. The Hopf algebra of noncommutative symmetric functions, denoted by NSym, is Q e1 , e2 , . . generated by noncommuting indeterminates en of degree n with the operations of the next subsection. 4 The Hopf algebra of noncommutative symmetric functions 45 This definition is analogous to the earlier fundamental theorem of symmetric functions, which showed that we can regard Sym as the algebra Q[e1 , e2 , .

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 Δ : C → C ⊗ C and counit or augmentation ε : C → R, such that the following diagrams commute. 1 Hopf algebra basic theory 21 i. Coassociativity property Δ C ✲ C⊗C Δ Δ ⊗ id ❄ C⊗C ❄ ✲ C⊗C⊗C id ⊗ Δ ii. Counitary property C ✠ 1⊗ R⊗ C ❅ ❅ ❅ ⊗1 ❅ ❅ ❘ Δ C ⊗R ❅ ■ ✒ ❅ ε ⊗ id id ⊗ ε ❅ ❅ ❅ ❄ C⊗C In both diagrams, id is the identity map on C .

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