By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An advent to Quasisymmetric Schur Functions is geared toward researchers and graduate scholars in algebraic combinatorics. The objective of this monograph is twofold. the 1st objective is to supply a reference textual content for the elemental idea of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric services and connections among them. the second one aim is to provide a survey of effects with admire to a thrilling new foundation of the Hopf algebra of quasisymmetric capabilities, whose combinatorics is comparable to that of the popular Schur functions.
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Extra info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
Replace k2 by k1 , that is, k1 bumps k2 . 4. Repeat the previous steps with k2 and row 2, k3 and row 3, etc. As before, the set of cells whose values are modified by the insertion, including the final cell added, is called the insertion path, and the final cell is called the new cell. 2. If we insert 5, then we have 1 3 6 7 1 2 2 4 3 5 4 ←5 = 1 2 3 2 6 7 1 3 2 4 4 5 5 2 where the bold cells indicate the insertion path. Given a type of insertion and list of positive integers σ = σ1 · · · σn , we define the P-tableau, or insertion tableau, or rectification of σ , denoted by P(σ ), to be (· · · ((0/ ← σ1 ) ← σ2 ) · · · ) ← σn .
And the set of invariants under this action is QSym. 3 The Hopf algebra of quasisymmetric functions 33 The set of all quasisymmetric functions with the operations of the next subsection forms a graded Hopf algebra QSym = QSymn n 0 spanned by the following functions, suggested by the definition of QSym. 4. Let α = (α1 , . . , αk ) be a composition. Then the monomial quasisymmetric function Mα is defined by α Mα = ∑ xαi11 · · · xikk , where the sum is over all k-tuples (i1 , . . , ik ) of indices i1 < · · · < ik .
We include its proof as the equivalent statements are useful to know. 6. Given an SYT T , we have comp(Γˇ (T ))) = comp(T )r . Proof. Suppose T is an SYT with n cells. The following statements are equivalent. 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.
An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg