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arxiv: 2606.02415 · v1 · pith:CSCY4GDMnew · submitted 2026-06-01 · 🧮 math.RT · math.CO· math.GR· math.RA

On rational representations and rational group algebra of operatorname{GL}₂(q)

Ram Karan Choudhary , Sunil Kumar Prajapati This is my paper

Pith reviewed 2026-06-28 11:57 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.GRmath.RA
keywords rational representationsGL_2(q)Schur indexWedderburn decompositiongroup algebra over QGalois orbitsparabolic induction
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The pith

Irreducible rational representations of GL_2(q) correspond exactly to Galois orbit sums of complex characters scaled by Schur index.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for the general linear group GL_2(q) with q a prime power, every irreducible representation over the rational numbers Q arises from an irreducible complex character χ by taking the sum of its Galois conjugates scaled by the Schur index m_Q(χ). The converse holds as well, giving a bijection. This correspondence leads to a combinatorial way to count the number of such rational representations of each possible degree, using only the value of q. The authors also derive an explicit formula for the decomposition of the rational group algebra QG into a direct sum of matrix rings over division algebras, again depending only on q. A reader would care because this makes the rational representation theory of these groups fully explicit and computable from the parameter q alone.

Core claim

An irreducible representation ρ of G=GL_2(q) over Q affords the character Ω(χ)=m_Q(χ)∑_{σ∈Gal(Q(χ)/Q)}χ^σ for some irreducible complex character χ of G, and conversely every such character is afforded by an irreducible Q-representation. Parabolic induction supplies the characters needed for the construction, and the Wedderburn decomposition of QG admits an explicit combinatorial formula in terms of q.

What carries the argument

The character Ω(χ) formed by summing Galois conjugates of a complex character χ scaled by its Schur index over Q, which classifies all irreducible rational representations and determines the simple components of the group algebra QG.

If this is right

  • A combinatorial description exists for counting inequivalent irreducible Q-representations of distinct degrees.
  • An explicit combinatorial formula depending only on q exists for the Wedderburn decomposition of QG.
  • A construction method exists for irreducible rational matrix representations affording Ω(χ) when χ arises from parabolic induction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formula permits computing the full list of rational irreps and their degrees directly from q without enumerating complex characters first.
  • The same orbit-sum approach could apply to related groups such as SL_2(q) once their Schur indices are determined.
  • Explicit knowledge of the division ring components in the decomposition may simplify computations of rational class functions or endomorphism rings.

Load-bearing premise

That every scaled Galois orbit sum is afforded by an irreducible representation over Q, with characters from parabolic induction generating all of them.

What would settle it

For q=3, compute all irreducible complex characters of GL_2(3), form their Galois orbit sums scaled by Schur indices, derive the predicted counts by degree and the Wedderburn factors, then compare against the actual irreducible Q-representations obtained by direct computation or known tables of the rational group algebra.

read the original abstract

In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character \[ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma}, \] for some irreducible complex character $\chi$ of $G$, where $m_{\mathbb{Q}}(\chi)$ denotes the Schur index of $\chi$ over $\mathbb{Q}$, with the converse also holding. We obtain a combinatorial description for the counting of inequivalent irreducible $\mathbb{Q}$-representations of $G$ of distinct degrees. Furthermore, we present a method to construct an irreducible rational matrix representation $\rho$ of $G$ affording the character $\Omega(\chi)$, where $\chi$ is an irreducible complex character of $G$ arising from parabolic induction. Finally, using the results from the rational representations of $G$, we derive an explicit combinatorial formula, depending only on $q$, for the Wedderburn decomposition of $\mathbb{Q}G$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that for G=GL_2(q), every irreducible rational representation ρ affords the character Ω(χ)=m_Q(χ)∑_{σ}χ^σ for some complex irreducible character χ (with the converse also holding); it supplies a combinatorial count of inequivalent irreducible Q-representations by degree, an explicit matrix construction realizing Ω(χ) when χ arises from parabolic induction, and a combinatorial formula (depending only on q) for the Wedderburn decomposition of the rational group algebra QG.

Significance. If the central correspondence and decomposition were fully established, the results would give an explicit, q-dependent description of the rational representation theory and group algebra structure for GL_2(q), a useful contribution to the study of rational representations of groups of Lie type.

major comments (2)
  1. [Abstract] Abstract and the section presenting the general correspondence: the claim that the converse holds for every complex irreducible χ (i.e., every Ω(χ) is afforded by an irreducible Q-representation) is load-bearing for both the counting formula and the Wedderburn decomposition, yet the explicit construction is stated to apply only when χ arises from parabolic induction; cuspidal characters are not addressed, leaving the general statement unsupported.
  2. [Construction section] Section on the construction of rational matrix representations: the method is restricted to parabolic-induced characters, so the claimed general converse (and therefore the explicit decomposition of QG) cannot be verified for the full set of complex irreps of GL_2(q).
minor comments (1)
  1. [Abstract] The abstract could more precisely delimit the scope of the converse statement to match the construction that is actually supplied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the distinction between the general correspondence and our explicit constructions. We address the points below and will revise the manuscript to clarify the reliance on standard theory.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section presenting the general correspondence: the claim that the converse holds for every complex irreducible χ (i.e., every Ω(χ) is afforded by an irreducible Q-representation) is load-bearing for both the counting formula and the Wedderburn decomposition, yet the explicit construction is stated to apply only when χ arises from parabolic induction; cuspidal characters are not addressed, leaving the general statement unsupported.

    Authors: The correspondence stated in the abstract is a standard theorem: every irreducible Q-representation affords a character Ω(χ) for some complex irreducible χ, and conversely Ω(χ) is afforded by an irreducible Q-representation (this is a direct consequence of the properties of the Schur index m_Q(χ); see e.g. Isaacs, Character Theory, Theorem 10.8 or standard references on rational representations). The manuscript recalls this foundational fact in the introductory section rather than claiming a new proof. The explicit matrix construction is presented separately as an additional constructive result that applies when χ arises from parabolic induction. For cuspidal characters the existence follows from the general theory without a new explicit matrix realization in this work. The combinatorial counting of irreducible Q-representations by degree and the q-dependent Wedderburn decomposition are obtained from the known complex character table of GL_2(q), the Galois orbits on characters, and the (combinatorially determinable) Schur indices, none of which require the explicit matrices. We will add a short clarifying remark in the abstract and introduction to distinguish the standard general fact from the constructive contribution. revision: partial

  2. Referee: [Construction section] Section on the construction of rational matrix representations: the method is restricted to parabolic-induced characters, so the claimed general converse (and therefore the explicit decomposition of QG) cannot be verified for the full set of complex irreps of GL_2(q).

    Authors: The construction section supplies an explicit realization only for the parabolic case, as stated. The general converse (existence of an irreducible Q-representation affording Ω(χ) for every complex χ) is not verified by construction in the paper but follows from the standard theory of Schur indices already invoked for the counting and decomposition. The Wedderburn decomposition of QG is obtained by summing matrix rings over the division algebras corresponding to each Galois orbit, using only the degrees, fields of values, and Schur indices; these data are available for all characters of GL_2(q) (including cuspidals) via the known character table and do not depend on exhibiting matrices. We will insert a brief note in the construction section reiterating that the general existence is supplied by the cited standard results. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via standard representation theory

full rationale

The paper states the correspondence between irreducible Q-representations and Galois orbits of complex characters scaled by Schur index as a theorem (with converse), then derives combinatorial counts and the Wedderburn decomposition of QG from the known character table of GL_2(q) and standard facts about rational irreps. No equations reduce a claimed prediction or count to a fitted quantity defined from the same data, no self-citation chain bears the central load, and no ansatz is smuggled. The explicit matrix construction is limited to parabolic-induced characters, but this does not create a definitional loop or force the general counting formula by construction. The derivation remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are visible or extractable.

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