The Hopf monoid on nonnesting supercharacters of pattern groups

Abstract

We construct supercharacter theories for a collection of unipotent matrix groups and produce a Hopf monoid from the supercharacters. These supercharacter theories are coarser than those defined by Diaconis--Isaacs for algebra groups and have supercharacters and superclasses indexed by nonnesting labeled set partitions. We compute the supercharacter tables and describe the product and coproduct of the Hopf monoid combinatorially. We also show that this Hopf monoid is free.

arXiv commentFinal version; DOI added

References in corpus (6)

• Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras
• Hopf monoids from class functions on unitriangular matrices
• Representations of Finite Unipotent Linear Groups by the Method of Clusters
• Supercharacters of unipotent groups defined by involutions
• A supercharacter table decomposition via power-sum symmetric functions
• Actions and identities on set partitions

Cited by in corpus (5)

• Hopf Algebras and Markov Chains
• Supercharacter theories constructed by the method of little groups
• The combinatorics of ${\mathrm{GL}}_n$ generalized Gelfand--Graev characters
• Supercharacter theories of dihedral groups
• Positive Self-Dual Hopf Algebras of Galois Characters