This bachelor thesis aims to give an introduction to various Hopf algebras that arise in combinatorics, with a view towards symmetric functions. We begin by covering the algebraic background needed to define Hopf algebras, including a discussion of the algebra-coalgebra duality. Takeuchi's formula for the antipode is stated and proved. It is then generalised to incidence Hopf algebras. This is followed by a discussion of the Hopf algebra of symmetric functions. It is shown that the Hopf algebra of symmetric functions is self-dual. We also show that the graded dual of the Hopf algebra of quasisymmetric functions is the Hopf algebra of non-commutative symmetric functions. Relations to the Hopf algebra of symmetric functions in non-commuting variables are emphasised. Finally, we state and prove the Aguiar-Bergeron-Sottile universality theorem.
Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:kth-348646 |
Date | January 2024 |
Creators | Dahlgren, Isabel |
Publisher | KTH, Skolan för teknikvetenskap (SCI) |
Source Sets | DiVA Archive at Upsalla University |
Language | English |
Detected Language | English |
Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
Format | application/pdf |
Rights | info:eu-repo/semantics/openAccess |
Relation | TRITA-SCI-GRU ; 2024:250 |
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