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Preprint
Hopf Algebras in Combinatorics
arXiv.org
29 Sep 2014
Abstract
These notes -- originating from a one-semester class by their second author
at the University of Minnesota -- survey some of the most important Hopf
algebras appearing in combinatorics. After introducing coalgebras, bialgebras
and Hopf algebras in general, we study the Hopf algebra of symmetric functions,
including Zelevinsky's axiomatic characterization of it as a "positive
self-adjoint Hopf algebra" and its application to the representation theory of
symmetric and (briefly) finite general linear groups. The notes then continue
with the quasisymmetric and the noncommutative symmetric functions, some Hopf
algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer
Hopf algebra of permutations. Among the results surveyed are the
Littlewood-Richardson rule and other symmetric function identities,
Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition
enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory
of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial
freeness of QSym. The notes are written with a graduate student reader in mind,
being mostly self-contained but requiring a good familiarity with multilinear
algebra and -- for the representation-theory applications -- basic group
representation theory.
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Details
- Title
- Hopf Algebras in Combinatorics
- Creators
- Darij GrinbergVictor Reiner
- Publication Details
- arXiv.org
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862238104721