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September 5, 2025

On the Hopf superalgebra of symmetric functions in superspace

Key Points

The Hopf superalgebra of quasi‐symmetric functions in superspace serves as a terminal object in its category, simplifying combinatorial analysis.
It demonstrates relationships between symmetric functions and combinatorial hopf superalgebras, implicating new areas of study in algebra and combinatorics.
A new analogue for chromatic symmetric functions is developed, expanding the understanding of graphs in the context of hopf superalgebras.
This research enriches mathematical frameworks and may enhance applications in combinatorial theory and algebraic structures.

Abstract

Abstract We introduce a superspace analogue of combinatorial Hopf algebras (Aguiar–Bergeron–Sottile, Compos. Math. 142 (2006), 1–30), and show that the Hopf superalgebra of quasi‐symmetric (resp., symmetric) functions in superspace (Fishel–Lapointe–Pinto, J. Combin. Theory Ser. A 166 (2019), 144–170) is a terminal object in the category of all (resp., cocommutative) combinatorial Hopf superalgebras. We also introduce a superspace analogue of chromatic symmetric functions of graphs (Stanley, Adv. Math. 111 (1995), 166–194) using the chromatic Hopf superalgebra of two‐colored graphs.

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