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By Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg

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The Kλ µ are known as Kostka numbers. 11. We have s(2,1) = m(2,1) + 2m(1,1,1) from the SSYTs 3 2 1 1 1 2 2 1 3 2 2 3 1 or equivalently from the following SSRTs. 8 the straight shape λ was replaced by a skew shape λ /µ, then a function would still be defined. These functions also play a role in the theory of symmetric functions. 12. Let λ /µ be a skew shape. Then the skew Schur function sλ /µ is defined to be s0/ = 1 and sλ /µ = ∑ xT T where the sum is over all SSYTs (or equivalently SSRTs) T of shape λ /µ.

Let (B, m, u) be an algebra and (B, ∆ , ε) a coalgebra. Then B is a bialgebra if 1. ∆ and ε are algebra morphisms or equivalently, 2. m and u are coalgebra morphisms. We are now ready to define a Hopf algebra. 4. Let (H , m, u, ∆ , ε) be a bialgebra. Then H is a Hopf algebra if there is a linear map S : H → H such that m ◦ (S ⊗ id) ◦ ∆ = u ◦ ε = m ◦ (id ⊗ S) ◦ ∆ . Thus, in Sweedler notation, S satisfies ∑ S(h1 )h2 = ε(h)1 = ∑ h1 S(h2 ) for all h ∈ H , where 1 is the identity element of H . The map S is called the antipode of H .

1 2 1 1 2 1 3 2 2 3 2 1 Schur functions can also be expressed in terms of SSYTs or SSRTs when expanded in the basis of monomial symmetric functions. 10. Let λ n. Then sλ = ∑ Kλ µ mλ µ n where s0/ = 1 and Kλ µ is the number of SSYTs (or equivalently SSRTs) T satisfying sh(T ) = λ and cont(T ) = µ. The Kλ µ are known as Kostka numbers. 11. We have s(2,1) = m(2,1) + 2m(1,1,1) from the SSYTs 3 2 1 1 1 2 2 1 3 2 2 3 1 or equivalently from the following SSRTs. 8 the straight shape λ was replaced by a skew shape λ /µ, then a function would still be defined.

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An Introduction to Quasisymmetric Schur Functions (September 26, 2012) by Kurt Luoto, Stefan Mykytiuk, Stephanie van Willigenburg


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