Computing Unitary Discriminants of Characters for Finite Groups

Core Concepts

This paper presents new methods for computing the unitary discriminants of characters for finite groups, particularly focusing on those with even degree and indicator o.

Abstract

• Bibliographic Information: Nebe, G. (2024). Unitary discriminants of characters. arXiv preprint arXiv:2411.06235v1.
• Research Objective: This paper aims to develop and implement computational methods for determining the unitary discriminants of characters for finite groups, extending previous work on orthogonal discriminants. This is particularly relevant for characters with even degree and indicator o, which have well-defined unitary discriminants.
• Methodology: The paper leverages various character-theoretic techniques, including modular reduction, restriction, induction, tensor products, and symmetrizations. A key innovation is the introduction of "unitary condensation," which allows for computations on characters of large degree by relating them to the discriminants of invariant Hermitian forms. This method utilizes a suitable automorphism to define an "α-discriminant" within the real subfield of the character field, which can be computed using skew-adjoint units in the α-fixed algebra.
• Key Findings: The paper establishes a theoretical framework for computing unitary discriminants and demonstrates its effectiveness through computational examples. It shows that all primes dividing the unitary discriminant also divide the group order. The paper also highlights the importance of unitary stability, a property ensuring the well-definedness of the unitary discriminant.
• Main Conclusions: The methods presented provide a systematic approach to computing unitary discriminants, offering valuable insights into the representation theory of finite groups. The author suggests that these techniques can be further developed and applied to a wider range of groups, including those of Lie type.
• Significance: This research significantly contributes to computational group theory by providing tools for analyzing the structure and properties of finite groups through their characters. The computed unitary discriminants offer valuable information about the representations of these groups, which has implications for various areas of mathematics and physics.
• Limitations and Future Research: The paper primarily focuses on ordinary characters and suggests further exploration of unitary discriminants for Brauer characters. Additionally, the computational complexity of the methods for larger groups and more complicated character tables may require further optimization and development of more efficient algorithms.

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Unitary discriminants of characters

by Gabriele Neb... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06235.pdf

Deeper Inquiries

How can the methods presented in the paper be extended or adapted to compute unitary discriminants for other families of finite groups, such as sporadic groups beyond the Harada-Norton group?

Extending the computation of unitary discriminants to larger sporadic groups, or other families of finite groups, presents several challenges and requires a multi-pronged approach:
1. Computational Limits: The primary hurdle is computational complexity. As group size increases, directly constructing representations and invariant forms becomes infeasible. The paper already utilizes techniques like modular reduction and condensation to mitigate this, but further refinements are necessary:
* **Improved Condensation:** Exploring different automorphisms α for condensation (Section 8.2) could yield smaller α-fixed algebras, making computations more manageable.
* **Block-Theoretic Approaches:** Leveraging the block structure of the group algebra modulo different primes can provide information about the distribution of irreducible constituents in modular reductions. This can be particularly useful for blocks with specific properties, like cyclic defect groups.
* **Exploiting Subgroup Structure:** Systematically analyzing restrictions to subgroups with known unitary discriminants (Remark 5.10) can provide valuable constraints. This requires identifying suitable subgroups where computations are easier and the restriction remains unitary stable.

2. Character Table Information: The methods heavily rely on character table data. For larger groups, this information might be incomplete or unavailable:
* **Character Table Completion:** Efforts to complete character tables of larger sporadic groups are crucial. This often involves a combination of theoretical techniques and sophisticated computations.
* **Fusion Control:** Understanding how characters of subgroups fuse into characters of the larger group is essential for effectively using restriction and induction arguments.

3. Generalizing Existing Results: The paper showcases specific examples (e.g., U3(7), 3.ON). Generalizing these insights to broader families requires:
* **Lie Type Groups:** For groups like U3(q), understanding the representation theory over finite fields and its connection to ordinary characters is key. Results on generic character tables can guide the analysis.
* **Sporadic Group Families:** Identifying patterns and relationships within families of sporadic groups (e.g., Mathieu groups, Conway groups) might lead to more general techniques for computing unitary discriminants.

4. Algorithmic Improvements: Developing more efficient algorithms for:
* **Computing with Lattices:** Efficiently computing with lattices over number fields, particularly finding invariant lattices and analyzing their reductions modulo primes, is crucial.
* **Determining Discriminant Algebras:** Finding faster ways to determine the discriminant algebra of a Hermitian form, especially for large dimensions, would significantly speed up computations.

Could there be alternative approaches, perhaps drawing from different areas of mathematics like algebraic geometry or topology, that offer new perspectives or computational advantages for determining unitary discriminants?

Yes, exploring alternative approaches from other areas of mathematics could provide fresh insights and computational advantages for determining unitary discriminants:
1. Algebraic Geometry:
* **Representation Varieties:** Representations of a finite group G can be viewed as points in an algebraic variety. Studying the geometry of these varieties, particularly their rational points over number fields, might shed light on the existence and properties of invariant Hermitian forms.
* **Character Sheaves:** This sophisticated theory, which blends representation theory with algebraic geometry, provides a geometric framework for studying characters. It might offer tools to analyze the behavior of characters under various operations, potentially leading to new methods for computing discriminants.

2. Topology:
* **Equivariant K-Theory:** This theory studies vector bundles with group actions. The unitary discriminant can be interpreted as an element in a suitable equivariant K-theory group. Topological tools might provide alternative ways to compute or relate these invariants.
* **Cohomology of Classifying Spaces:** The representation theory of a finite group G is intimately connected to the topology of its classifying space BG. Cohomological invariants of BG might encode information about unitary discriminants.

3. Number Theory:
* **Class Field Theory:** The discriminant algebra of a Hermitian form is a central simple algebra, which is classified by class field theory. Deeper connections with class field theory might provide new perspectives on the arithmetic properties of unitary discriminants.
* **Modular Forms:** In some cases, traces of representations of finite groups can be related to coefficients of modular forms. Exploring these connections might offer alternative ways to compute or study unitary discriminants.

4. Computational Homological Algebra:
* **Representation Theory over Finite Fields:** Developing more sophisticated algorithms for computing with representations over finite fields, including determining decompositions, invariant forms, and Schur indices, would be beneficial.

These alternative approaches are currently areas of active research, and their full potential for computing unitary discriminants is yet to be fully explored. However, they offer promising avenues for gaining a deeper understanding of these invariants and developing more efficient computational methods.

What are the potential applications of these computed unitary discriminants in other areas of mathematics or physics, such as coding theory, cryptography, or quantum information theory?

The computation of unitary discriminants, while a challenging problem in representation theory, has the potential to impact several other areas of mathematics and physics:
1. Coding Theory:
* **Lattice Codes:** Unitary lattices, derived from Hermitian forms, are used to construct lattice codes for wireless communication. The discriminant of the lattice directly influences the coding gain and error-correction capabilities of the code. Knowing the unitary discriminants of representations can aid in designing codes with better performance.
* **Space-Time Codes:** In multiple-antenna communication systems, space-time codes based on division algebras are employed. The discriminant algebra of a Hermitian form is closely related to the structure of division algebras. Understanding unitary discriminants can lead to the discovery of new space-time codes with improved reliability.

2. Cryptography:
* **Lattice-Based Cryptography:** The security of many lattice-based cryptographic schemes relies on the difficulty of computational problems related to lattices, such as the shortest vector problem and the closest vector problem. The discriminant of a lattice plays a role in the complexity of these problems. Unitary discriminants could potentially be used to analyze the security of such cryptosystems or to design new ones.
* **Isogeny-Based Cryptography:** This emerging area of cryptography relies on the difficulty of computing isogenies between elliptic curves. The endomorphism rings of elliptic curves are often orders in quaternion algebras, which are closely related to discriminant algebras of Hermitian forms. Unitary discriminants might provide insights into the structure of these endomorphism rings and have implications for the security of isogeny-based cryptography.

3. Quantum Information Theory:
* **Quantum Codes:** Unitary representations and their associated invariant Hermitian forms are fundamental in the construction of quantum error-correcting codes. The discriminant of the form can influence the properties of the code, such as its distance and error-correction capability. Understanding unitary discriminants could aid in designing more efficient and robust quantum codes.
* **Topological Quantum Computing:** Some proposals for topological quantum computers rely on the properties of anyons, which are exotic quasiparticles with non-abelian statistics. The mathematical framework for describing anyons involves representation theory and often involves unitary representations. Unitary discriminants might play a role in understanding the properties of anyons and their potential for quantum computation.

4. Number Theory:
* **Inverse Galois Problem:** The inverse Galois problem asks whether every finite group can be realized as the Galois group of a Galois extension of the rational numbers. The computation of unitary discriminants can provide information about the splitting behavior of primes in number fields, which is relevant to the inverse Galois problem.

These are just a few potential applications, and further research may reveal even more connections and uses for unitary discriminants in various fields. The interplay between representation theory and these areas is an active and fruitful area of research.