By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

ISBN-10: 1461472997

ISBN-13: 9781461472995

ISBN-10: 1461473004

ISBN-13: 9781461473008

*An advent to Quasisymmetric Schur Functions* is aimed toward researchers and graduate scholars in algebraic combinatorics. The objective of this monograph is twofold. the 1st aim is to supply a reference textual content for the elemental concept of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric capabilities and connections among them. the second one objective is to offer a survey of effects with admire to a thrilling new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is similar to that of the well known Schur functions.

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**Extra resources for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux**

**Sample text**

Are given by (1⊗)(c) = 1 ⊗ c and (⊗1)(c) = c ⊗ 1 for c ∈ C. We may omit the indexing required to express a coproduct Δ (c) as an element of C ⊗ C and use Sweedler notation to write Δ (c) = ∑ c1 ⊗ c2 . Thus, in Sweedler notation, diagrams i. and ii. 1) ∑ ε (c1 )c2 = c = ∑ ε (c2 )c1 . 2) We say the coproduct Δ is cocommutative if c ⊗ c is a term of Δ (c) whenever c ⊗ c is. A submodule I ⊆ C is a coideal if Δ (I ) ⊆ I ⊗ C + C ⊗ I and ε (I ) = {0}. 22 3 Hopf algebras A map f : C → C , where (C , Δ , ε ) is another coalgebra over R, is a coalgebra morphism if Δ ◦ f = ( f ⊗ f ) ◦ Δ and ε = ε ◦ f .

12) P where the sum is over all paths P in the (x, y) plane from (0, 0) to ( (α ), (β )) with steps (1, 0), (0, 1) and (1, 1). Given two permutations σ = σ (1) · · · σ (n) ∈ Sn and τ = τ (1) · · · τ (m) ∈ Sm , we say a shuffle of σ and τ is a permutation in Sn+m such that σ (i) appears to the right of σ (i − 1) and to the left of σ (i + 1) for all 2 i n − 1 and similarly, τ (i) + n appears to the right of τ (i − 1) + n and to the left of τ (i + 1) + n for all 2 i m − 1. We denote by σ τ the set of all shuffles of σ and τ .

Our presentation is based on that in [69]. Other references are [68] and [84]. Throughout this section, let R be a commutative ring with identity element. We remind the reader that an R-module is defined in the same way as a vector space, except that the field of scalars is replaced by the ring R. 1. An algebra over R is an R-module A together with R-linear maps product or multiplication m : A ⊗ A → A and unit u : R → A , such that the following diagrams commute. K. 1007/978-1-4614-7300-8 3, © Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg 2013 19 20 3 Hopf algebras i.

### An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

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