The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

代数群的超字符理论为我们提供了非交换变量中对称函数的Hopf代数的表示理论实现。底层表示理论框架配备了两个规范基础，其中之一在对称函数方面是全新的。本文通过考虑更大类别的组同时对这一Hopf结构进行了概括，同时也将表示理论限制为更易于组合的组。使用模式组的正常晶格超字符理论，我们不仅获得了第三规范基础，而且还能够计算相应的Hopf monoid中的众多结构常数，包括新碱基的副产物和对映体。