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Real irreducible representations of SL(2,q) and their fixed point dimensions for cyclic subgroups

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Pith's one-line read Real irreducible characters of SL(2,q) and all cyclic fixed-point dimensions are determined.

desk verdict Useful reference tables for fixed point dimensions of cyclic subgroups of SL(2,q), with a few typo-level errors that should be fixed before publication. read the letter →

arxiv 1908.08848 v1 pith:7P3MSQGR submitted 2019-08-23 math.RT

Piotr Mizerka This is my paper · ORCID
classification math.RT MSC 20C1520C33
keywords realirreduciblerepresentationsFrobenius-SchurindicatorSL(2q)speciallineargroupfixedpointdimensioncyclicsubgroupscharactertableroot-of-unitysums
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper settles, for $q$ an odd prime, the real representation theory of the finite group $\operatorname{SL}(2,q)$ in its entirety. It identifies exactly which complex irreducible characters are characters of real irreducible representations: $1$, $\psi$, the even-indexed $\chi$ and $\theta$ families, doubled odd-indexed members, and, depending on $q$ modulo 4, the exceptional characters or their real parts. It then computes, for every cyclic subgroup, the dimension of the fixed-point subspace of each such representation, using the classical averaging formula over the subgroup. The result is a complete, closed-form description that reduces the answer to parity and greatest-common-divisor data of the generator exponents. A sympathetic reader would care because these dimensions are the invariants needed to understand the real representation ring and restrictions to cyclic subgroups, with no case-by-case numerical computation left undone.

What carries the argument

The engine of the proof is the Frobenius–Schur indicator of a complex irreducible character, evaluated by a formula that sums the character over the squares of elements of the group, rewritten as a weighted sum over conjugacy classes. The sign of this indicator decides whether the corresponding real representation is the character itself (indicator 1), the doubled character $2\chi$ (indicator $-1$), or the real-part character $2\operatorname{Re}\chi$ (indicator 0). For $\operatorname{SL}(2,q)$, the indicator sums are worked out with the root-of-unity notation $\nu_r^s = \zeta_r^s + \zeta_r^{-s}$, and the whole computation reduces to parity bookkeeping about which conjugacy classes contain squares of the generators $c$, $d$, $a$, $b$, and the central element $z$.

What would settle it

For $q=7$, the claimed value of $\dim (2\operatorname{Re}\eta_1)^{\langle b\rangle}$ with $b$ of order 8 is 2. Recompute this dimension directly from the complex character table: average $\eta_1+\eta_2$ over the eight powers of $b$, i.e. evaluate $\frac18\sum_{k=0}^{7}(2\operatorname{Re}\eta_1)(b^k)$. If the result differs from 2, the one-line parity assertion in Lemma 7 that a certain sub-sum equals 4 is wrong.

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Extended reading notes

Core claim

The central claim is a complete classification plus a dimension formula. Using the Frobenius–Schur indicator computed by class sums, the paper shows that the real irreducible characters are exactly those listed in Lemma 3: the trivial and Steinberg characters, the even-index characters $\chi_{2i}$ and $\theta_{2j}$, the doubled odd-index characters $2\chi_{2i'+1}$ and $2\theta_{2j'+1}$, and, for $q \equiv 1 \pmod 4$, $\xi_1$, $\xi_2$, $2\eta_1$, $2\eta_2$, or for $q \equiv 3 \pmod 4$, the real parts $2\operatorname{Re}\xi_1$ and $2\operatorname{Re}\eta_1$. For each such character and each cyclic subgroup generated by a representative of a conjugacy class, the fixed-point dimension is computed by averaging the character over the subgroup; Lemmas 4–7 and the three summary tables list all resulting values in terms of $(q-1,l)$ and $(q+1,m)$, with the parity of the ratios deciding which case applies.

Load-bearing premise

The computation is only as reliable as the imported complex character table of $\operatorname{SL}(2,q)$, and the most delicate internal step is Lemma 7's claim, for $q \equiv 3 \pmod 4$ and odd $m$, that a certain alternating sum of character values on powers of $b^m$ equals exactly 4, with the parity justification given in a single line.

Editorial extensions

If this is right

  • The real character table is now fully explicit for every odd prime $q$, so any real representation of $\operatorname{SL}(2,q)$ can be decomposed by reading off its character against the listed rows.
  • For any cyclic subgroup, the fixed-point dimension of any real irreducible representation is given by a closed formula in terms of $\gcd(q-1,l)$ or $\gcd(q+1,m)$, with no remaining case analysis beyond the parity of $l$ and $m$.
  • The mod-4 dichotomy is sharp: when $q \equiv 1 \pmod 4$ every conjugacy class is real and the exceptional characters $\xi_1$, $\xi_2$, $2\eta_1$, $2\eta_2$ are real irreducibles; when $q \equiv 3 \pmod 4$ the real parts $2\operatorname{Re}\xi_1$ and $2\operatorname{Re}\eta_1$ take their place.
  • The tables for $\langle c\rangle$, $\langle d\rangle$, $\langle zc\rangle$, $\langle zd\rangle$ give immediate invariant dimensions such as 1, 2, 4, or 0 for the standard representations, which can be used to count fixed vectors in permutation or tensor constructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: the same averaging recipe applies verbatim to any subgroup of $\operatorname{SL}(2,q)$, not only cyclic ones, once its conjugacy classes are known; the tables here are the cyclic case of that general formula.
  • Beyond the paper: the mod-4 dichotomy suggests that for $\operatorname{PSL}(2,q)$ the real character pattern will be governed by a different central-fusion rule, so a similar table could be derived by taking $z$-invariants of these characters.
  • Beyond the paper: a direct numerical check for $q=5,7,11$ against the complex character table would validate every entry of the summary tables; the only genuinely delicate row is the $q \equiv 3 \pmod 4$, odd-$m$ entry for $2\operatorname{Re}\eta_1$.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This paper computes the real irreducible characters of SL(2,q) for an odd prime q, starting from the known complex character table of Dornhoff and applying Frobenius–Schur theory together with character averaging. It then computes, for every real irreducible representation and every cyclic subgroup generated by a conjugacy-class representative, the dimension of the fixed-point subspace. The main results are collected in explicit character tables and in Lemmas 4–7, with summary tables at the end of Section 2.

Significance. If the results are correct, the paper provides a complete and explicit reference for the real representation theory of SL(2,q) and for fixed-point dimensions of cyclic subgroups. The derivations are standard, transparent, and appear to be reproducible: there are no fitted parameters, no circular arguments, and the formulas can be checked by hand on small primes such as q=3 and q=5. The value is primarily as a convenient reference rather than as a major theoretical advance; the paper fills a small gap in the literature with elementary tools.

major comments (1)
  1. [Section 2, Lemma 6] The statement "dimψH = (q−1,l)" is incorrect for the case where (q−1)/(q−1,l) is odd. The correct value, as computed by the proof's own intermediate expression and as listed in the summary table, is (q−1,l)+1. Concretely, for q=7 and l=2, H=⟨a^2⟩ has order 3, and ψ takes values 7,1,1 on the elements of H, so the fixed-point dimension is 3=(7−1,2)+1, not 2. The proof's final equality also contains the same arithmetic slip: the displayed product d/(q−1)·(q+(q−1)/d−1) simplifies to d+1, not d. Please correct both the lemma statement and that final equality.
minor comments (5)
  1. [Section 2, real character table for q≡1 mod 4] In the table following Lemma 3, the last row is labeled "η2" but its entries are those of 2η2 (the values on 1 and z coincide with those of 2η1, and the c,d entries are twice the corresponding η2 entries). The factor 2 should appear in the row label.
  2. [Section 2, real character table for q≡3 mod 4] In the second table for q≡3 (mod 4), the second row is labeled "2 Reξ2". The entries shown are those of 2Reη1=η1+η2 (degree q−1, values q−1 on z, −1 on c and d, 0 on a^l, and 2(−1)^{m+1} on b^m). There is no real irreducible character "2 Reξ2" distinct from "2 Reξ1"; the label should be "2 Reη1".
  3. [Section 2, Lemma 6] In the statement of Lemma 6, the clause "dimξH1 = dimξK2 = (q+1,m)/2" contains a typo: the first subscript should be K, so it should read "dimξK1 = dimξK2 = (q+1,m)/2". The proof itself uses the correct labels.
  4. [Section 2, Lemma 7] The parity argument for q≡3 (mod 4) and m odd is very compressed. The claim that the sum over non-central powers of b^m contributes 4 is correct—the powers split into pairs with equal signs—but an explanatory sentence describing this pairing would improve readability and reduce the risk of reader error.
  5. [Section 1, proof of Lemma 3] In the inequalities showing ι(χ2i)−ι(χ2i'+1)>0, the bounds use |ν|≤2 and then replace each term by 4 in the worst case. This is valid, but the displayed inequalities with strict ">" are technically only "≥" at the intermediate step; a brief comment on the strictness or a change to "≥" would avoid a minor logical gap.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper derives real character tables and fixed-point dimensions from an external standard character table via Frobenius–Schur theory and character averaging.

full rationale

The paper's derivation chain is self-contained against an external input: the complex character table and conjugacy class data are imported from Dornhoff [1, Theorem 38.1], a standard reference, and all later results are obtained by applying textbook Frobenius–Schur indicator formulas (Serre [4]) and elementary character averaging. No parameter is fitted, no target result is assumed, and no load-bearing claim is justified by a self-citation. The indicators in Lemma 3 are computed directly from the imported table, and the fixed-point dimensions in Lemmas 4–7 are computed by averaging the resulting real character values over cyclic subgroups. The only notable defect is an internal typo-level inconsistency in Lemma 6: the statement lists dim ψ^H = (q−1,l) while the proof and summary table give (q−1,l)+1; this affects a displayed formula, not the method, and is not a circularity. Overall, the derivation is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted, and no new entities are invented. The central claim rests on the imported character table of SL(2,q), on standard Frobenius-Schur theory, on the fixed point averaging formula, and on routine root-of-unity evaluations. The computation is therefore honest but depends on the correctness of the cited character table.

assumptions (5)
  • domain assumption Complex character table and conjugacy class set of SL(2,q) as stated in Dornhoff [1, Theorem 38.1].
    Every later statement, Lemmas 2 through 7, uses the class representatives, class sizes, and character values from this table; it is quoted without proof.
  • standard math Frobenius-Schur indicator classification: indicators 1, 0, and -1 give real, conjugate-pair, and doubled real irreducible representations.
    Used in Lemma 3 to convert the complex character table into the real irreducible characters.
  • standard math Fixed point dimension formula dim V^H = (1/|H|) times the sum of the character over H.
    This is equation (1) in Lemma 4 and is the engine for all fixed point dimension computations.
  • domain assumption Every cyclic subgroup of SL(2,q) is conjugate to a subgroup generated by one of 1, z, c, d, zc, zd, a^l, or b^m.
    Used implicitly in Section 2 to restrict the fixed point dimension analysis to the representatives handled in Lemmas 4 through 7.
  • standard math Root-of-unity summation identity: the sum of zeta_n^{s j} over j=1 to n-1 equals -1 when n does not divide s.
    Used repeatedly in Lemmas 6 and 7 to evaluate character sums over the cyclic subgroups H and K.

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Cite this review

Pith. "Pith review of Real irreducible representations of SL(2,q) and their fixed point dimensions for cyclic subgroups." pith.science (2026). https://pith.science/paper/7P3MSQGR

@misc{pith2026190808848,
  author       = {Pith},
  title        = {Pith review of: Real irreducible representations of SL(2,q) and their fixed point dimensions for cyclic subgroups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7P3MSQGR}},
  note         = {Machine review of arXiv:1908.08848}
}
abstract

We compute the characters of real irreducible representations of SL(2,q), the special linear group on q letters, for an odd prime $q$. Moreover, we give the dimensions of these irreducible representations under the actions of cyclic subgroups of SL(2,q).

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

  1. [1]

    Real irreducible representations of SL 2(q)5 Lemma 1. The conjugacy classes of SL 2(q) are given by representatives6 1,z,c,d,zc,zd,a l,b m, where, for some ν ∈ Fx q with ⟨ν⟩ = Fx q ,7 1 = ( 1 0 0 1 ) z = ( −1 0 0 −1 ) c = ( 1 0 1 1 ) d = (1 0 ν 1 ) a = (ν 0 0 ν−1 ) b 8 whereb is any element of order q+1 and 1 ≤l ≤ (q−3)/2, 1 ≤m ≤ (q−1)/2.9 The conjugacy c...

    work page 2019

  2. [2]

    ι(χ) = 1 if χ is a character of some real irreducible representation94

    work page

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    ι(χ) = 0 if 2 Re χ = χ + χ is a character of some real irreducible95 representation (in this case χ takes some non-real values)96

    work page

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    ι(χ) = −1 if 2 χ is a character of some real irreducible representation97 (int this case χ takes all real values).98 Moreover, every character of some real irreducible representa tion is obtained99 in that way. Hence, we only have to show that ι(1) = ι(ψ) = ι(χ2i) = ι(θ2j) =100 1, ι(χ2i+1) = ι(θ2j+1) = −1 and ι(ξ1) = ι(ξ2) = 1, ι(η1) = ι(η2) = −1 in case1...

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    Obvio usly, dimV ⟨1⟩ =109 dimV for any irreducible RG-module V

    Fixed point dimensions for cyclic subgroups106 Since subgroups’ fixed point subspaces’ dimensions depend only on the107 conjugacy classes of subgroups, it suffices to find the dimensions o f cyclic sub-108 groups generated by conjugacy classes representatives. Obvio usly, dimV ⟨1⟩ =109 dimV for any irreducible RG-module V . The folllowing tables are the real1...

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    = 2, dim(2 χ2i+1)⟨c⟩ = 1 q ((2q + 2) + (q − 1) · 2) = 4, dim θ⟨c⟩ 2j = 1 q ((q −134

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    We know that ck = ( 1 0 k 1 )

    + (q − 1) · (−1)) = 0, dim(2 θ2j+1)⟨c⟩ = 1 q ((2q − 2) + (q − 1) · (−2)) = 0135 and, if q ≡ 3 (mod 4), then dim(2 Re ξ1)⟨c⟩ = 1 q ((q + 1) + (q − 1) · 1) = 2,136 dim(2 Reη1)⟨c⟩ = 1 q ((q − 1) + (q − 1) · (−1)) = 0.137 For the case q ≡ 1 (mod 4), we have to show a little bit more since it is not the case that ξi andηi are equal on c andd. We know that ck =...

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    L. Dornhoff, Group representation theory. Part A: Ordinary r epresen-197 tation theory, Marcel Dekker, Inc., New York, 1971. Pure and Ap plied198 Mathematics, 7.199

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