Shifted combinatorial Hopf algebras from $K$-theory
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| Materias: | |
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| _version_ | 1866910677897052160 |
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| author | Marberg, Eric |
| author_facet | Marberg, Eric |
| contents | In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_01092 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Shifted combinatorial Hopf algebras from $K$-theory Marberg, Eric Combinatorics K-Theory and Homology Rings and Algebras In prior joint work with Lewis, we developed a theory of enriched set-valued $P$-partitions to construct a $K$-theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of $K$-theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new $K$-theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures. |
| title | Shifted combinatorial Hopf algebras from $K$-theory |
| topic | Combinatorics K-Theory and Homology Rings and Algebras |
| url | https://arxiv.org/abs/2211.01092 |