Non-injective Inductions and Restrictions of Modules over Finite Groups: Extending Classical Results

Główne pojęcia

This note generalizes the concepts of induction and restriction of modules over finite groups to non-injective group homomorphisms, proving key properties like transitivity, Frobenius reciprocity, and Mackey's formula in this extended context.

Streszczenie

Li, C., & Tian, Y. (2024). Non-injective inductions and restrictions of modules over finite groups. arXiv:2410.22742v1 [math.RT].

This note aims to extend the classical concepts of induction and restriction of modules over finite groups from injective to non-injective group homomorphisms. The authors investigate whether key properties such as transitivity, Frobenius reciprocity, and Mackey's formula still hold in this generalized setting.

Kluczowe wnioski z

Non-injective inductions and restrictions of modules over finite groups

by Conghui Li, ... o arxiv.org 10-31-2024

https://arxiv.org/pdf/2410.22742.pdf

Głębsze pytania

How can the concept of non-injective inductions and restrictions of modules be applied to study the representation theory of infinite groups or Lie algebras?

Extending the concepts of non-injective inductions and restrictions to infinite groups and Lie algebras presents significant challenges but also exciting possibilities. Here's a breakdown of potential applications and hurdles:
Infinite Groups:

Challenges:

Topological Considerations: For infinite groups, we often work with topological groups and continuous representations. Defining induction and restriction for non-injective continuous homomorphisms requires careful consideration of topologies on the induced and restricted modules.
Finiteness Conditions: Many classical results in finite group representation theory rely on finiteness conditions (e.g., finite index subgroups). Adapting these results to infinite groups necessitates finding suitable analogs of these conditions.
Structure of Subgroups: The structure of subgroups in infinite groups can be vastly more complex than in the finite case. This complexity makes it challenging to establish general results like Mackey's formula.

Potential Applications:

Representations of Locally Compact Groups: Non-injective homomorphisms naturally arise in the study of locally compact groups. Adapting these concepts could lead to new insights into their representation theory.
Unitary Representations: The theory of unitary representations of infinite groups is of fundamental importance. Exploring how non-injective induction and restriction behave in this context could be fruitful.
Lie Algebras:

Challenges:

Infinite Dimensionality: Lie algebras and their representations are often infinite-dimensional, requiring tools from functional analysis.
Structure of Subalgebras: Similar to infinite groups, the structure of Lie subalgebras can be intricate.

Potential Applications:

Representation Decomposition: Non-injective homomorphisms between Lie algebras could provide new techniques for decomposing representations into smaller, more manageable pieces.
Connections to Physics: Lie algebras and their representations play a crucial role in theoretical physics. New tools in this area could have implications for understanding physical systems.
General Strategies:

Focus on Specific Cases: Instead of aiming for completely general results, focusing on specific classes of infinite groups or Lie algebras with well-behaved non-injective homomorphisms might be more tractable.
Category-Theoretic Approach: Employing category theory could provide a more abstract and flexible framework for studying non-injective induction and restriction in these broader contexts.

Could there be alternative formulations of Mackey's formula for non-injective homomorphisms that provide different insights or computational advantages?

While the Mackey's formula presented in the paper provides a valuable generalization for non-injective homomorphisms, exploring alternative formulations is an interesting avenue for research. Here are some possibilities:

Exploiting Kernel Structure: The current formula relies on the double coset decomposition. Alternative decompositions of the groups involved, perhaps leveraging the structure of the kernels of the homomorphisms, might lead to different expressions. For instance, if the kernels have a specific normal subgroup relationship, a formula based on a different coset decomposition might emerge.

Category-Theoretic Interpretation: Mackey's formula has a natural interpretation in terms of adjoint functors and fiber products in the category of group representations. Seeking alternative categorical constructions, such as different limits or colimits, could yield new formulas with distinct interpretations.

Special Cases and Simplifications: Investigating special cases, such as when the homomorphisms have certain properties (e.g., one being a normal inclusion), might lead to simplified versions of the formula that are easier to compute or provide more specific insights.

Duality and Adjunction: Exploring the interplay between induction, restriction, and their adjoint functors in a more general categorical setting could reveal new relationships and potentially lead to dual versions of Mackey's formula.

Computational Efficiency: The current formula involves a sum over double cosets, which can be computationally expensive. Alternative formulations might offer computational advantages in certain situations, especially when dealing with large groups or specific types of homomorphisms.

What are the implications of this work for understanding the deeper connections between category theory and representation theory, particularly in the context of module categories?

This work on non-injective inductions and restrictions strengthens the deep connections between category theory and representation theory, particularly within the realm of module categories:

Module Categories as a Natural Setting: The paper implicitly highlights the importance of module categories. Module categories over a ring form a natural setting for studying representations, and the functors of induction and restriction are key players in understanding the relationships between these categories.

Adjunction and Universal Properties: The Frobenius reciprocity theorem, even in this generalized setting, emphasizes the role of adjoint functors. Induction and restriction, when viewed as functors between appropriate module categories, form an adjoint pair. This adjoint relationship captures a fundamental duality between these operations and underscores the power of universal properties in representation theory.

Mackey's Formula and Categorical Constructions: Mackey's formula, with its connection to pullbacks (a type of limit in category theory), suggests that other categorical constructions might have meaningful interpretations in representation theory. This opens avenues for exploring how more advanced category theory can be applied to understand representations.

Generalization and Abstraction: The extension to non-injective homomorphisms encourages a more abstract and general viewpoint. Instead of focusing solely on subgroups, we can consider arbitrary homomorphisms and the corresponding functors they induce between module categories. This broader perspective aligns well with the philosophy of category theory, which emphasizes relationships and structures over specific objects.

Potential for Further Development: This work lays the groundwork for further exploration of the interplay between category theory and representation theory. For example, studying non-injective induction and restriction in the context of derived categories or other categorical settings could lead to new insights and connections.