By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg
An creation 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 concept of Hopf algebras, specifically the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric capabilities and connections among them. the second one objective is to provide 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 popular Schur functions.
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Extra info for An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux
K ) and a k-tuple I = (i1 , . . , ik ) of positive integers i1 < · · · < ik , let xαI α denote the monomial xαi11 · · · xikk . A quasisymmetric function is a formal power series f ∈ Q[[x1 , x2 , . ]] such that 1. f has finite degree, 2. f is invariant under the action of S∞ on Q[[x1 , x2 , . I is defined to be the k-tuple obtained by arranging the numbers σ (i1 ), . . , σ (ik ) in increasing order. That is, σ . f = f where the action of σ is extended by linearity. 3. If f = x21 x2 + x21 x3 + x22 x3 and σ ∈ S2 , then σ .
K ) n. Furthermore we say a formal power series has finite degree if each monomial has degree at most m for some nonnegative integer m, and is homogeneous of degree n if each monomial has degree n. 1. A symmetric function is a formal power series f ∈ Q[[x1 , x2 , . ]] such that 1. The degree of f is finite. α 2. For every composition (α1 , . . , αk ), all monomials xαi11 · · · xikk in f with distinct indices i1 , . . , ik have the same coefficient. Let n be a nonnegative integer, then recall that a permutation of [n] is a bijection σ : [n] → [n], which we may write as an n-tuple σ (1) · · · σ (n).
Equivalently we can think of a symmetric function as follows. 2. A symmetric function is a formal power series f ∈ Q[[x1 , x2 , . ]] such that 1. f has finite degree, 2. f is invariant under the action of S∞ on Q[[x1 , x2 , . (xαi11 · · · xαikk ) = xασ1(i ) · · · xασk(i ) . 1 k That is, σ . f = f when the action of σ is extended by linearity. 3. If f = x21 + x22 + x1 x2 and σ ∈ S2 , then σ . f = x21 + x22 + x1x2 = f . The set of all symmetric functions with the operations of the next subsection forms a graded Hopf algebra Sym = Symn n 0 spanned by the following functions, strongly suggested by the definition of Sym.