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arxiv: 2604.16078 · v1 · submitted 2026-04-17 · 🧮 math.RT · math.CO· math.GR· math.RA

Wedderburn decomposition of the rational group algebras of operatorname{SL}₂(q) and operatorname{PSL}₂(q)

Ram Karan Choudhary , Saikat Panja This is my paper

Pith reviewed 2026-05-10 07:23 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.GRmath.RA
keywords Wedderburn decompositionrational group algebraSL(2,q)PSL(2,q)simple modulescombinatorial formulasSchur index
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The pith

Explicit formulas depending only on q give the Wedderburn decomposition of the rational group algebras of SL_2(q) and PSL_2(q).

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

The paper derives combinatorial expressions in the single parameter q that completely describe how the rational group algebra Q[SL_2(q)] decomposes as a direct sum of matrix rings over division algebras, and likewise for Q[PSL_2(q)]. It further supplies formulas that count the distinct simple modules over these algebras in each possible dimension. A reader interested in representation theory would care because the formulas turn the abstract Wedderburn structure into something that can be written down immediately once q is known, without separate case-by-case analysis of characters or Schur indices for each finite group.

Core claim

The Wedderburn decomposition of Q[SL_2(q)] and of Q[PSL_2(q)] consists of an explicit direct sum of full matrix algebras over rational division algebras whose sizes and multiplicities are given by combinatorial functions of q alone; the same functions determine the exact number of pairwise non-isomorphic simple QG-modules in each dimension.

What carries the argument

Combinatorial formulas, derived from the known classification of irreducible rational representations, that encode the dimensions, Schur indices, and multiplicities appearing in the decomposition.

If this is right

  • The Artinian structure of these group algebras becomes fully explicit and computable for every prime power q.
  • The possible dimensions of simple rational modules for SL_2(q) and PSL_2(q) are listed by closed formulas rather than by ad-hoc character tables.
  • The number of distinct simple modules of each dimension follows directly from the same combinatorial data used for the decomposition.
  • Any further invariant that depends only on the Wedderburn components, such as the number of indecomposable projective modules, can now be read off from q.

Where Pith is reading between the lines

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

  • The explicit dependence on q alone suggests that similar closed-form decompositions may exist for other families of finite groups of Lie type once their rational character tables are known.
  • These formulas could be used to test conjectures about the distribution of Schur indices or the rationality of representations for large q without enumerating conjugacy classes.
  • The same counting technique may simplify calculations of the rational representation ring or the Grothendieck ring for these groups.

Load-bearing premise

The formulas assume that the complete list of irreducible representations over the rationals and their associated division algebras can be captured by closed combinatorial expressions in q with no additional hidden case distinctions.

What would settle it

For a concrete small prime power such as q=5, compute the actual Wedderburn decomposition of Q[SL_2(5)] by enumerating the rational irreducibles and their endomorphism algebras, then check whether the dimensions and multiplicities match the paper's explicit formulas.

read the original abstract

In this article, we derive explicit combinatorial formulas, depending only on $q$, for the Wedderburn decomposition of the rational group algebras of the finite linear groups $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$. Furthermore, we also determine the number of pairwise non-isomorphic simple $\mathbb Q G$-modules of each possible dimension for $G$ being either $\operatorname{SL}_2(q)$ or $\operatorname{PSL}_2(q)$.

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

0 major / 2 minor

Summary. The manuscript derives explicit combinatorial formulas depending only on q for the Wedderburn decomposition of the rational group algebras of SL_2(q) and PSL_2(q). It further determines the number of pairwise non-isomorphic simple QG-modules of each possible dimension.

Significance. If the derivations hold, the explicit formulas provide a complete, parameter-free description of the rational Wedderburn decomposition for these groups, building directly on the classical complex character tables, Galois orbits, and Schur indices. This is a useful contribution to the representation theory of finite groups of Lie type, as the combinatorial expressions allow direct computation of the isomorphism type of QG as a product of matrix rings over division algebras and the associated module dimensions without case-by-case computation beyond the formulas themselves.

minor comments (2)
  1. [§3] §3, after the statement of Theorem 3.2: the formula for the number of 1-dimensional simple modules could include an explicit cross-reference to the contribution of the trivial representation and the sign representation when q is odd.
  2. [§5] §5: the final count of simple modules of dimension (q+1)/2 for PSL_2(q) is stated combinatorially, but a brief remark confirming that the Schur index is 1 in all cases would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The referee's description accurately reflects the explicit combinatorial formulas we derive for the Wedderburn decompositions of Q[SL_2(q)] and Q[PSL_2(q)], as well as the counts of simple modules by dimension.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives explicit formulas for the rational Wedderburn decomposition and simple module counts by assembling the classical complex character tables of SL_2(q) and PSL_2(q), applying Galois orbits on irreducible characters, and computing Schur indices via standard methods from representation theory. These inputs are external, tabulated in the literature, and independent of the present work; the resulting combinatorial expressions in q are genuine outputs rather than renamings or fits. No self-definitional steps, no fitted parameters presented as predictions, and no load-bearing self-citations appear in the derivation chain. The manuscript is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no free parameters, axioms, or invented entities are identifiable; the work appears to rest on standard facts from the representation theory of finite groups of Lie type.

pith-pipeline@v0.9.0 · 5387 in / 984 out tokens · 49392 ms · 2026-05-10T07:23:32.153006+00:00 · methodology

discussion (0)

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Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Rational group algebras of generalized strongly monomial groups: primitive idempotents and units

    Gurmeet K. Bakshi, Jyoti Garg, and Gabriela Olteanu. “Rational group algebras of generalized strongly monomial groups: primitive idempotents and units”. In:Math. Comp.93.350 (2024), pp. 3027–3058.DOI: 10.1090/mcom/3937[ 2]

    work page doi:10.1090/mcom/3937 2024

  2. [2]

    The rational group algebra of a normally monomial group

    Gurmeet K. Bakshi and Sugandha Maheshwary. “The rational group algebra of a normally monomial group”. In:J. Pure Appl. Algebra218.9 (2014), pp. 1583–1593.DOI:10.1016/j.jpaa.2013.12.010[ 2]

    work page doi:10.1016/j.jpaa.2013.12.010 2014

  3. [3]

    Rational group algebras of Caminap-groups

    Ram Karan Choudhary and Sunil Kumar Prajapati. “Rational group algebras of Caminap-groups”. In: Arch. Math. (Basel)125.3 (2025), pp. 247–258.DOI:10.1007/s00013-025-02142-w[ 2]

    work page doi:10.1007/s00013-025-02142-w 2025

  4. [4]

    Rational representations and rational group algebra of VZp-groups

    Ram Karan Choudhary and Sunil Kumar Prajapati. “Rational representations and rational group algebra of VZp-groups”. In:J. Aust. Math. Soc.118.1 (2025), pp. 1–30.DOI:10.1017/S1446788724000132[ 2]

    work page doi:10.1017/s1446788724000132 2025

  5. [5]

    A combinatorial formula for the Wedderburn decom- position of rational group algebras and the rational representations of ordinary metacyclicp-groups

    Ram Karan Choudhary and Sunil Kumar Prajapati. “A combinatorial formula for the Wedderburn decom- position of rational group algebras and the rational representations of ordinary metacyclicp-groups”. In: Algebr. Represent. Theory29.1 (2026), pp. 85–109.DOI:10.1007/s10468-025-10371-4[ 2]

    work page doi:10.1007/s10468-025-10371-4 2026

  6. [6]

    A combinatorial formula for the Wedderburn decom- position of rational group algebras of split metacyclicp-groups

    Ram Karan Choudhary and Sunil Kumar Prajapati. “A combinatorial formula for the Wedderburn decom- position of rational group algebras of split metacyclicp-groups”. In:J. Algebra Appl.(2026), (to appear). DOI:doi.org/10.1142/S0219498826500684[ 2]

    work page doi:10.1142/s0219498826500684 2026

  7. [7]

    A combinatorial technique for the Wedderburn decom- position of rational group algebras of nested GVZp-groups

    Ram Karan Choudhary and Sunil Kumar Prajapati. “A combinatorial technique for the Wedderburn decom- position of rational group algebras of nested GVZp-groups”. In:J. Algebraic Combin.63.39 (2026), (to appear).DOI:doi.org/10.1007/s10801-026-01527-6[ 2]

    work page doi:10.1007/s10801-026-01527-6 2026

  8. [8]

    On matrix representations of groups of orderp 5 over Q

    Ram Karan Choudhary and Sunil Kumar Prajapati. “On matrix representations of groups of orderp 5 over Q”. In:J. Group Theory(2026), (to appear) [ 2]. [9]The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.13.0. https://www.gap-system.org. 2024 [ 2]

    work page 2026

  9. [9]

    The isomorphism problem for rational group algebras of finite metacyclic groups

    `Angel Garc´ıa-Bl´azquez and ´Angel del R´ıo. “The isomorphism problem for rational group algebras of finite metacyclic groups”. In:J. Pure Appl. Algebra229.6 (2025), Paper No. 107951, 36.DOI:10 . 1016 / j . jpaa.2025.107951[ 2]. 18 REFERENCES

    work page arXiv 2025

  10. [10]

    Meinolf Geck and Gunter Malle.The character theory of finite groups of Lie type. V ol. 187. Cambridge Studies in Advanced Mathematics. A guided tour. Cambridge University Press, Cambridge, 2020, pp. ix+394 [ 5]

    work page 2020

  11. [11]

    On the automorphism groups of rational group algebras of metacyclic groups

    Allen Herman. “On the automorphism groups of rational group algebras of metacyclic groups”. In:Comm. Algebra25.7 (1997), pp. 2085–2097.DOI:10.1080/00927879708825973[ 1]

    work page doi:10.1080/00927879708825973 1997

  12. [12]

    Martin Isaacs.Character theory of finite groups

    I. Martin Isaacs.Character theory of finite groups. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. Dover Publications, Inc., New York, 1994, pp. xii+303 [ 4]

    work page 1976

  13. [13]

    Simple components ofQ[SL(2,q)]

    G. J. Janusz. “Simple components ofQ[SL(2,q)]”. In:Comm. Algebra1 (1974), pp. 1–22.DOI:10.1080/ 00927877408548606[ 7, 13, 14]

    work page 1974

  14. [14]

    Rational group algebras of finite groups: from idem- potents to units of integral group rings

    Eric Jespers, Gabriela Olteanu, and ´Angel del R´ıo. “Rational group algebras of finite groups: from idem- potents to units of integral group rings”. In:Algebr. Represent. Theory15.2 (2012), pp. 359–377.DOI: 10.1007/s10468-010-9244-4[ 2]

    work page doi:10.1007/s10468-010-9244-4 2012

  15. [15]

    A structure theorem for the unit group of the integral group ring of some finite groups

    Eric Jespers and Angel del R ´ıo. “A structure theorem for the unit group of the integral group ring of some finite groups”. In:J. Reine Angew. Math.521 (2000), pp. 99–117.DOI:10.1515/crll.2000.032[ 1]

    work page doi:10.1515/crll.2000.032 2000

  16. [16]

    Eric Jespers and ´Angel del R´ıo.Group ring groups. Vol. 1. Orders and generic constructions of units. De Gruyter Graduate. De Gruyter, Berlin, 2016, pp. xii+447 [ 8, 10, 15]

    work page 2016

  17. [17]

    Isomorphism of group algebras of metacyclic groups over the field of rational numbers

    Nako Nachev and Yordan Epitropov. “Isomorphism of group algebras of metacyclic groups over the field of rational numbers”. In:J. Algebra680 (2025), pp. 58–69.DOI:10.1016/j.jalgebra.2025.04.040[ 2]

    work page doi:10.1016/j.jalgebra.2025.04.040 2025

  18. [18]

    Computing the Wedderburn decomposition of group algebras by the Brauer-Witt the- orem

    Gabriela Olteanu. “Computing the Wedderburn decomposition of group algebras by the Brauer-Witt the- orem”. In:Math. Comp.76.258 (2007), pp. 1073–1087.DOI:10 . 1090 / S0025 - 5718 - 07 - 01957 - 6[ 2]

    work page 2007

  19. [19]

    Abelian group algebras of finite order

    Sam Perlis and Gordon L. Walker. “Abelian group algebras of finite order”. In:Trans. Amer. Math. Soc.68 (1950), pp. 420–426.DOI:10.2307/1990406[ 2]

    work page doi:10.2307/1990406 1950

  20. [20]

    The Schur index in the theory of group representations

    Irving Reiner. “The Schur index in the theory of group representations”. In:Michigan Math. J.8 (1961), pp. 39–47 [ 4]

    work page 1961

  21. [21]

    Construction of units in integral group rings of finite nilpotent groups

    J ¨urgen Ritter and Sudarshan K. Sehgal. “Construction of units in integral group rings of finite nilpotent groups”. In:Trans. Amer. Math. Soc.324.2 (1991), pp. 603–621.DOI:10.2307/2001734[ 1]

    work page doi:10.2307/2001734 1991

  22. [22]

    Serre.A course in arithmetic

    J.-P. Serre.A course in arithmetic. V ol. No. 7. Graduate Texts in Mathematics. Translated from the French. Springer-Verlag, New York-Heidelberg, 1973, pp. viii+115 [ 14]

    work page 1973

  23. [23]

    Schur indices of irreducible characters of SL(2,q)

    M. A. Shahabi Shojaei. “Schur indices of irreducible characters of SL(2,q)”. In:Arch. Math. (Basel)40.3 (1983), pp. 221–231.DOI:10.1007/BF01192774[ 11]

    work page doi:10.1007/bf01192774 1983

  24. [24]

    Toshihiko Yamada.The Schur subgroup of the Brauer group. V ol. V ol. 397. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1974, pp. v+159 [ 1]. Email address, (Choudhary):ramkchoudhary1997@gmail.com, ram.choudhary@iiserpune.ac.in INDIANINSTITUTE OFSCIENCEEDUCATION ANDRESEARCHPUNE, DR. HOMIBHABHAROAD, PASHAN, PUNE411008, INDIA Email address...

    work page 1974