By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An creation to Quasisymmetric Schur Functions is geared toward researchers and graduate scholars in algebraic combinatorics. The aim of this monograph is twofold. the 1st objective is to supply a reference textual content for the elemental idea of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric capabilities and connections among them. the second one target is to offer a survey of effects with recognize 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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Additional info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
Unit u : R → H ∗ given by u(r) = rεH , where εH is the counit of H (thus εH is the identity element of H ∗ ), 24 3 Hopf algebras 3. coproduct Δ : H ∗ → H ∗ ⊗ H ∗ given by Δ( f ) = ρ −1 ( f ◦ mH ), where ρ is the invertible linear map defined on H ∗ ⊗ H ∗ by ρ ( f ⊗ g) (h1 ⊗ h2) = f (h1 )g(h2 ) and mH is the product of H , 4. counit ε : H ∗ → R given by ε ( f ) = f (1H ), where 1H is the identity element of H , 5. antipode S : H ∗ → H ∗ given by S( f ) = f ◦ SH , where SH is the antipode of H . The Hopf algebra H ∗ is called the graded Hopf dual of H .
1. 2. 3. 4. 5. i ∈ Des(T ). i + 1 is weakly to the left of i in T . n − i is weakly to the left of n − i + 1 in Γˇ (T ). n − i + 1 is weakly to the right of n − i in Γˇ (T ). n − i ∈ Des(Γˇ (T )). This establishes the claim. 6 Schensted insertion Schensted insertion is an algorithm with many interesting combinatorial properties and applications to representation theory. For further details see [31,72,81]. We will also use this algorithm and the variation below in Chapter 5. In particular, Schensted insertion inserts a positive integer k1 into a semistandard or standard Young tableau T and is denoted by T ← k1 .
The converse is also true: Given a fundamental quasisymmetric function Fα , we can always find a labelled chain (w, γ ) such that Fα = F(w, γ ). Indeed, let w be the chain with order w1 < · · · < wn , where n = |α |. If i1 < i2 < · · · are the elements of set(α ) and j1 < j2 < · · · are the elements of [n] − set(α ), let γ respectively map wi1 , wi2 , . . → n, n − 1, . . and w j1 , w j2 , . . → 1, 2, . . Then D(w, γ ) = set(α ), since 1. i, i + 1 ∈ set(α ) implies γ (wi ) > γ (wi+1 ), 2. j, j + 1 ∈ [n] − set(α ) implies γ (w j ) < γ (w j+1 ), 3.