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

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 objective of this monograph is twofold. the 1st target is to supply a reference textual content for the elemental concept of Hopf algebras, particularly the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric features and connections among them. the second one objective is to provide a survey of effects with appreciate to an exhilarating new foundation of the Hopf algebra of quasisymmetric services, whose combinatorics is comparable to that of the well known Schur functions.

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Ik ) of distinct indices that yield distinct monomials. We define m0/ = 1. 5. We have m(2,1) = x21 x12 + x22 x11 + x21 x13 + x23 x11 + x21 x14 + x24 x11 + x22 x13 + x23 x12 + · · · . Moreover, since the mλ are independent we have Symn = span{mλ | λ n}. The basis of monomial symmetric functions is not the only interesting and useful basis. 6. Let n be a nonnegative integer. Then the n-th elementary symmetric function, denoted by en , is defined by en = m(1n ) = ∑ i1 <···

Associative property m ⊗ id A ⊗A ⊗A ✲ A ⊗A id ⊗ m m ❄ A ⊗A m ❄ ✲ A ii. Unitary property A ⊗A ✒ u ⊗ id R⊗A ❅ ■ ❅ ❅ id ⊗ u ❅ ❅ m A ⊗R ❅ ❅ ❅ ❅ ✠ ❘ ❄ ❅ A In both diagrams, id is the identity map on A . The two lower maps in diagram ii. are given by scalar multiplication. A map f : A → A , where (A , m , u ) is another algebra over R, is an algebra morphism if f ◦ m = m ◦ ( f ⊗ f ) and f ◦ u = u . We shall frequently write ab instead of m(a ⊗ b). The algebra A has identity element 1A = u(1R ), where 1R is the identity element of R.

We define m0/ = 1. 5. We have m(2,1) = x21 x12 + x22 x11 + x21 x13 + x23 x11 + x21 x14 + x24 x11 + x22 x13 + x23 x12 + · · · . Moreover, since the mλ are independent we have Symn = span{mλ | λ n}. The basis of monomial symmetric functions is not the only interesting and useful basis. 6. Let n be a nonnegative integer. Then the n-th elementary symmetric function, denoted by en , is defined by en = m(1n ) = ∑ i1 <···

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