[Paper Review] Reciprocity laws for representations of finite groups

Sunil K. Chebolu, C. M. Reis+1 more|arXiv (Cornell University)|Nov 19, 2009

Finite Group Theory Research10 references3 citations

TL;DR

This paper establishes explicit reciprocity laws for representations of finite metacyclic groups—specifically semidirect products of two cyclic groups—by constructing representations over finite fields and determining their realizability using group invariants and field-theoretic conditions. The key contribution is a constructive reciprocity law that reduces the question of whether a representation is realizable over a finite field to a finite number of computations, recovering both the quadratic reciprocity law and Sylvester's theorem as special cases.

ABSTRACT

Much has been written on reciprocity laws in number theory and their connections with group representations. In this paper we explore more on these connections. We prove a "reciprocity Law" for certain specific representations of semidirect products of two cyclic groups which is in complete analogy with classical reciprocity laws in number theory. In fact, we show that the celebrated quadratic reciprocity law is a direct consequence of our main theorem applied to a specific group. As another consequence of our main theorem we also recover a classical theorem of Sylvester. Our main focus is on explicit constructions of representations over sufficiently small fields. These investigations give further evidence that there is still much unexplored territory in connections between number theory and group representations, even at an elementary level.

Motivation & Objective

• To develop a constructive reciprocity law for representations of finite metacyclic groups over finite fields.
• To establish necessary and sufficient conditions for the realizability of induced representations over finite fields, reducing infinite field questions to finite computations.
• To demonstrate that classical number-theoretic results, such as quadratic reciprocity and Sylvester's theorem, emerge as special cases of the main theorem.
• To provide explicit matrix constructions of representations over small finite fields, avoiding reliance on abstract invariants like the Schur index.
• To explore deeper connections between group representation theory and number theory at an elementary level, suggesting a broader theory of reciprocity in finite group representations.

Proposed method

• The paper studies split semidirect products $ G = igracevert a,b igracevert a^m = 1 = b^n, b^{-1}ab = a^k \rangle $, where $ k $ has order $ t $ modulo $ m $.
• It induces a representation $ \rho^G $ from a 1-dimensional character $ \rho(a) = \zeta $, where $ \zeta $ is a primitive $ m $th root of unity in $ \overline{\mathbb{F}}_s $.
• Using the order of $ k $ modulo $ m $, the paper determines irreducibility of $ \rho^G $: it is irreducible iff $ |k|_m = n $, the order of $ b $.
• When $ |k|_m = t < n $, the representation decomposes into $ r = n/t $ pairwise inequivalent irreducible components, each realizable via companion matrices over $ \overline{\mathbb{F}}_s $.
• For realizability over $ \mathbb{F}_q $, the paper derives conditions based on whether $ q \equiv k^j \pmod{m} $ for some $ j $, and uses field norm computations to construct explicit matrices over $ \mathbb{F}_q $.
• The method involves solving norm equations in extension fields and using the Chinese Remainder Theorem to construct transition matrices, enabling explicit matrix realizations over $ \mathbb{F}_q $.

Experimental results

Research questions

• RQ1 Under what conditions is the induced representation $ \rho^G $ of a metacyclic group realizable over a finite field $ \mathbb{F}_q $?
• RQ2 How can the realizability of such representations be determined via a finite number of computations, analogous to classical reciprocity laws?
• RQ3 In what way do classical number-theoretic results like quadratic reciprocity and Sylvester's theorem arise as special cases of the main reciprocity theorem?
• RQ4 Can the realizability condition $ q \equiv k^j \pmod{m} $ be both necessary and sufficient for $ \rho^G $ to be realizable over $ \mathbb{F}_q $, even when $ |b| \neq |k|_m $?
• RQ5 What are the implications of this constructive approach for extending reciprocity laws beyond finite fields and to other classes of finite or algebraic groups?

Key findings

• The representation $ \rho^G $ is irreducible over $ \overline{\mathbb{F}}_s $ if and only if the order of $ k $ modulo $ m $ equals the order of $ b $ in $ G $.
• When $ |k|_m = t < n $, $ \rho^G $ decomposes into $ r = n/t $ pairwise inequivalent irreducible representations over $ \overline{\mathbb{F}}_s $, each of dimension $ t $.
• For $ \mathbb{F}_q $, the representation $ \rho^G $ is completely realizable if and only if $ q \equiv k^j \pmod{m} $ for some $ j $, provided $ (q, |G|) = 1 $.
• Explicit matrix realizations of $ \rho^G $ over $ \mathbb{F}_q $ are constructed using solutions to norm equations and companion matrix forms, with examples computed over $ \mathbb{F}_{19} $.
• The quadratic reciprocity law is recovered as a direct consequence of the main theorem by choosing a specific group $ G $ and representation $ \rho^G $.
• Sylvester's classical theorem on the sum of two squares is also recovered as a consequence of the main result, demonstrating the broad applicability of the reciprocity framework.