[Paper Review] Construction of supercharacter theories of finite groups

Anders O. F. Hendrickson|ArXiv.org|May 21, 2009

Coding theory and cryptography13 references3 citations

TL;DR

This paper introduces five new constructions for supercharacter theories of finite groups—direct product, lattice-theoretic join, two types of products over normal subgroups, and duality for abelian groups—demonstrating that these operations generate all supercharacter theories for certain infinite families of finite groups, including cyclic $p$-groups of odd order and cyclic groups of order $pq$ and $pqr$. The work extends the framework of Diaconis and Isaacs by providing systematic methods to build new theories from existing ones.

ABSTRACT

Much can be learned about a finite group from its character table, but sometimes that table can be difficult to compute. Supercharacter theories are generalizations of character theory defined by P. Diaconis and I.M. Isaacs, in which certain (possibly reducible) characters called supercharacters take the place of the irreducible characters, and a certain coarser partition of the group takes the place of the conjugacy classes. We present five new ways to construct new supercharacter theories out of supercharacter theories already known to exist, including a direct product, a lattice-theoretic join, two products over normal subgroups, and a duality for supercharacter theories of abelian groups.

Motivation & Objective

• To develop systematic methods for constructing new supercharacter theories from known ones, addressing the difficulty of computing full character tables in complex finite groups.
• To generalize existing supercharacter theory constructions by introducing new operations such as the $*$-product, $ riangle$-product, and duality.
• To demonstrate that these new constructions, combined with prior methods, suffice to generate all supercharacter theories for specific infinite families of finite groups.
• To clarify the structural relationships between supercharacter theories via lattice operations and duality, particularly in abelian and nilpotent groups.

Proposed method

• The direct product construction combines supercharacter theories of two groups $G$ and $H$ to form a theory on $G \times H$, preserving compatibility of superclasses and supercharacters.
• The lattice-theoretic join operation combines two supercharacter theories by taking the coarsest common coarsening of their superclass partitions and corresponding character partitions.
• The $*$-product is defined over a normal subgroup $M \trianglelefteq G$, using the isomorphism between $\operatorname{Irr}(M)$ and $\operatorname{Irr}(G)/\operatorname{Irr}(G/M)$ to lift theories from $M$ and $G/M$ to $G$, with a duality condition.
• The $\triangle$-product generalizes the $*$-product to two normal subgroups $N$ and $M$, using a dual isomorphism to relate $\operatorname{Irr}(N)$ and $\operatorname{Irr}(M)$, ensuring compatibility of the resulting theory.
• Duality is applied to abelian groups, where the supercharacter theory of $G$ is mapped to a theory on $\operatorname{Irr}(G)$, preserving structure under the isomorphism $\maltese: \operatorname{Irr}(M) \to \operatorname{Irr}(G)/\operatorname{Irr}(G/M)$.
• The paper proves that if a supercharacter theory $\sf{E}$ on $G$ is a $*$-product over $M$, then its dual $\widehat{\sf{E}}$ is a $*$-product over $\operatorname{Irr}(G/M)$, establishing a duality between constructions.

Experimental results

Research questions

• RQ1 Can new supercharacter theories be systematically constructed from existing ones using algebraic and lattice-theoretic operations?
• RQ2 How do the $*$-product and $\triangle$-product constructions generalize the known product structures in supercharacter theory?
• RQ3 What is the role of duality in relating supercharacter theories of a group and its character group, particularly in abelian groups?
• RQ4 Under what conditions does the $*$-product construction yield a valid supercharacter theory on a finite group with a normal subgroup?
• RQ5 Can the combination of these new constructions and prior methods from Diaconis and Isaacs generate all supercharacter theories for specific infinite families of finite groups?

Key findings

• The direct product of two supercharacter theories on groups $G$ and $H$ yields a valid supercharacter theory on $G \times H$, with $|\sf{C} \times \sf{D}| = |\sf{C}| \cdot |\sf{D}|$.
• The lattice-theoretic join of two supercharacter theories produces a new theory whose superclass partition is the meet of the original partitions, and whose character partition is the join.
• The $*$-product construction over a normal subgroup $M$ of $G$ is valid if the dual of the theory on $G$ is a $*$-product over $\operatorname{Irr}(G/M)$, as shown in Corollary 11.6.
• The $\triangle$-product generalizes the $*$-product to two normal subgroups $N$ and $M$, and a supercharacter theory $\sf{E}$ on $G$ is a $\triangle$-product over $N$ and $M$ if and only if its dual $\widehat{\sf{E}}$ is a $\triangle$-product over $\widetilde{N}$ and $\widetilde{M}$.
• The duality operation $\widehat{\sf{E}}$ maps a supercharacter theory on $G$ to a theory on $\operatorname{Irr}(G)$, and when $G$ is abelian, this duality preserves the $*$-product structure.
• The paper proves that the five new constructions—direct product, join, $*$-product, $\triangle$-product, and duality—when combined with prior methods, suffice to generate all supercharacter theories for cyclic $p$-groups of odd order and cyclic groups of order $pq$ and $pqr$.