Primitive central idempotents of a finite group over a field

Jagmohan Tanti; Satyam Shukla

REVIEW 4 major objections 4 minor 15 references

Primitive central idempotents of a finite group over a field

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For irreducible characters whose nonzero values have eigenvalues of a single order, the primitive central idempotent of a finite group algebra equals a normalized Möbius-weighted sum over elements, with weight $\mu(d(g))/\varphi(d(g))$ on…

desk verdict The finite-field extension of Bakshi–Passi's idempotent formula is false: for G=C3 and F=F7, Theorem 3.3 returns e(χ2)+e(χ3) instead of e(χ2). read the letter →

arxiv 2608.03371 v1 pith:J4F2OCGU submitted 2026-08-04 math.RT math.RA

Satyam Shukla , Jagmohan Tanti This is my paper · ORCID

classification math.RTmath.RA MSC 20C0520C1516S34

keywords primitivecentralidempotentgroupalgebrafinitefieldirreduciblecharacterMöbiusfunctionEulerphicyclotomictracestar-cleanring

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

The paper aims to compute the primitive central idempotents of a semi-simple finite group algebra $F[G]$ for an irreducible complex character $\chi$ with the property that, for every $g \in G$, either $\chi(g)=0$ or all eigenvalues of $\rho(g)$ have the same order. The proposed answer is a normalized weighted sum over the non-vanishing elements $g$: the weight of $g$ is $\mu(d(g))/\varphi(d(g))$, where $d(g)$ is the order of $g$ modulo $\ker \chi$. If correct, this turns the idempotent attached to $\chi$ into a purely combinatorial object, computable from the zero set of $\chi$ and element orders, without knowing the explicit character values. This matters because primitive central idempotents are the atoms of the Wedderburn decomposition of group algebras, and explicit formulas for them underpin questions about ideals, clean rings, and the structure of finite group algebras.

What carries the argument

The load-bearing identity is the cyclotomic trace evaluation: for an element $g$ all of whose eigenvalues have order $d$, $\sum_{\sigma\in\mathrm{Gal}(F(\zeta_n)/F)}\sigma(\chi(g)) = \mu(d)\,\chi(1)\,\varphi(n)/\varphi(d)$, where $n=|G|$ and $d$ is the order of $g$ modulo $\ker\chi$. This reduces the Galois average that defines $e_F(\chi)$ to the coefficient $\mu(d(g))/\varphi(d(g))$ on each group element, so the whole idempotent becomes a weighted sum over the non-vanishing support of $\chi$. The second mechanism is Property $\varrho$ itself, which guarantees that for each $g$ either $\chi(g)=0$ or all eigenvalues of $\rho(g)$ share one order, so the trace identity applies uniformly across the whole group.

What would settle it

Compute the primitive central idempotents of $F_7[C_3]$, where $G=\langle a\rangle$ has order $3$. Since $7\equiv 1\pmod 3$, the field $F_7$ already contains the cube roots of unity, so $[F_7(\zeta_3):F_7]=1$ and the key trace identity of Lemma 3.2 fails. The theorem's formula assigns the same element to both non-trivial characters, and direct calculation shows this element equals the sum of the two non-trivial primitive central idempotents, not a primitive central idempotent; this disproves the claim as stated for arbitrary finite fields.

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Extended reading notes

Core claim

The paper's central claim, Theorem 3.3, is an explicit formula for the primitive central idempotent $e_F$ of $F[G]$ associated with an irreducible complex character $\chi$ that satisfies Property $\varrho$: for every $g \in G$, either $\chi(g)=0$ or all eigenvalues of $\rho(g)$ have the same order. Writing $d(g)$ for the order of the image of $g$ in $G/\ker \chi$, the theorem states that $e_F$ equals a normalized weighted sum of the elements $g$ with $\chi(g)\neq 0$, where the weight of $g$ is $\mu(d(g))/\varphi(d(g))$ and the normalization is the reciprocal of the sum of the squared weights. The proof starts from the standard identity $e_F(\chi)=\sum_{\sigma} \sigma(e(\chi))$ over the Galois group of the character field, applies a cyclotomic trace lemma to replace each $\sigma(\chi(g))$ by $\mu(d(g))\,\chi(1)\,\varphi(n)/\varphi(d(g))$, and uses idempotency to fix the scalar; the resulting formula depends on $\chi$ only through its kernel and its zero set, not on the actual character values.

Load-bearing premise

The entire calculation assumes the base field $F$ is a subfield of the cyclotomic field $\mathbb{Q}(\zeta_n)$ with $[F(\zeta_n):F]=\varphi(n)$, so that the Galois trace of a primitive $d$-th root of unity is $\mu(d)\varphi(n)/\varphi(d)$; for a finite field that already contains those roots this identity is false and the formula returns a sum over Galois-conjugate characters rather than the primitive idempotent of one character.

Editorial extensions

If this is right

For characters of degree $\sqrt{[G:Z(\chi)]}$, the idempotent simplifies to a Möbius-weighted sum over the centre-to-kernel section, and when that section is a $p$-group, to the difference $\widehat{\ker\chi} - \widehat{H}$ where $H/\ker\chi$ is the subgroup of order $p$.
If $G/Z(\chi)$ is abelian, every primitive central idempotent is of the form $E_{\ker\chi,Z(\chi)}$; for nilpotent groups of class at most $2$ this gives a bijection between primitive central idempotents and normal subgroups $N$ with $Z(G/N)$ cyclic.
For CM $p-1$-groups, the whole set of primitive central idempotents is exhausted by $\widehat{G}$ and differences $\widehat{N}-\widehat{H}$, so the Wedderburn decomposition has a closed form.
Because the weights depend only on $d(g)$ and the zero set of $\chi$, computing an idempotent requires no explicit character table values, only the orders of elements modulo $\ker\chi$.
If every primitive central idempotent is self-adjoint and $*$-clean, the entire group algebra $F[G]$ is $*$-clean; the formula gives a concrete way to check this condition in the covered families.

Reading between the lines

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

When $F$ already contains the relevant roots of unity, the trace identity degenerates and the same Möbius expression appears to recover the sum of the primitive idempotents over the Galois orbit of $\chi$; dividing by the orbit size is a natural correction that would restore primitivity.
A testable extension is to replace the full cyclotomic trace with the relative trace $F(\zeta_n)/F(\chi)$, which should adapt the formula to fields of smaller cyclotomic degree and to non-maximal intermediate fields.
The same Möbius-weight structure may carry over to semisimple group algebras over local or $p$-adic fields with unramified cyclotomic extensions, where traces are computable from residue fields alone.
For families such as extraspecial $p$-groups, combining the kernel-based formula with the classification of irreducible characters could yield a full explicit Wedderburn decomposition that is currently only known piecemeal.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the paper's only genuinely new claim is Theorem 3.3, and that claim is false over finite fields. The rational-group-algebra version is due to Bakshi and Passi, which the paper cites; the intended contribution is the extension from Q to finite fields. That extension fails.

What the paper does well: it organizes the material cleanly, states property ϱ carefully, and the corollaries for CM p−1 groups and nilpotent groups of class ≤2 would be natural if the main theorem held. The references to the rational case are appropriate. This is not a lazy paper; the flaw is decisive but technical.

The problem is Lemma 3.2. It computes the trace from F(ζ_n) to F as [F(ζ_n):F(ζ_n^{n/d})][F(ζ_n^{n/d}):F], then reads these degrees as φ(n)/φ(d) and φ(n/d). That is only valid when [F(ζ_d):F]=φ(d), which holds over Q but not over a finite field. For F=F_q, [F_q(ζ_d):F_q]=ord_d(q). The proof of Theorem 3.3 depends on exactly this step, so the theorem inherits the false assumption.

A concrete check, in the paper's own setting: G=C3, F=F7. Since 7≡1 mod 3, F7 contains primitive cube roots. For a nontrivial character χ2, the true primitive central idempotent in F7[C3] is e(χ2)=1/3(1+4a+2a^2). The paper's formula, which only uses d(g), gives 1/3(2−a−a^2), and this equals e(χ2)+e(χ3), not e(χ2). The formula cannot distinguish the two conjugate characters because it never uses their values, only the orders of elements modulo the kernel. Corollaries 3.4–3.6 and Section 4 all rest on Theorem 3.3, so they fall with it. Proposition 3.1 also has independent gaps in its proof, but those are secondary.

Who this is for: someone wanting a quick reminder of the Bakshi–Passi argument might find the exposition useful, but they should cite the original. As a finite-field result it should not appear in the literature as stated.

Recommendation: desk reject. The central theorem has a small, explicit counterexample, so referee time would not change the outcome. If the authors restrict to rational coefficients, the result is known; if they want finite fields, they need a different trace argument, and the formula will need to depend on the field through q-cyclotomic classes.

Referee Report

4 major / 4 minor

Summary. The paper claims to compute the primitive central idempotents of the semi-simple group algebra F[G] over a finite field F=F_q, for irreducible complex characters χ satisfying a property ϱ: for every g, either χ(g)=0 or all eigenvalues of ρ(g) have the same order. The main result, Theorem 3.3, expresses e_F(χ) as a weighted sum over group elements using only the Möbius function, Euler phi function, and d(g), the order of g modulo ker(χ). The proof is based on Galois descent from the complex group algebra, with a key trace computation in Lemma 3.2. Several corollaries are then derived for characters of degree sqrt([G:Z(χ)]), for characters with G/Z(χ) abelian, and for CM p−1-groups and nilpotent class ≤2 groups; the paper also contains results on ∗-clean group algebras.

Significance. If the main formula were correct, it would give a simple and apparently field-independent combinatorial expression for primitive central idempotents of finite group algebras over finite fields, which would be a useful tool for Wedderburn decomposition and related questions. The paper is not accompanied by machine-checked proofs or reproducible code, and the central claim is in fact false as stated: the key trace lemma assumes cyclotomic degree formulas that hold over Q but fail for finite fields already containing the relevant roots of unity. A concrete counterexample is supplied below. Because the main theorem and all of its corollaries and applications rely on this trace step, the manuscript cannot be accepted in its present form.

major comments (4)

[Section 3, Lemma 3.2] Lemma 3.2 is false over a finite field. The proof uses [F(ζ):F(ζ^{n/d})][F(ζ^{n/d}):F] = φ(n)/φ(d), which assumes [F(ζ_d):F] = φ(d). For F=F_q, however, [F_q(ζ_d):F_q] = ord_d(q), the multiplicative order of q modulo d. When q ≡ 1 mod d, the trace is the identity map, so the claimed value μ(d)χ(1)φ(n)/φ(d) is wrong; the error is not a technical gap but a wrong domain assumption in the paper's own finite-field setting.
[Theorem 3.3 and Section 3, Eq. (3.1)] Theorem 3.3 is refuted by G=C_3 and F=F_7. Since 7 ≡ 1 mod 3, F_7 contains the primitive cube roots. For a nontrivial character χ_2, the true primitive central idempotent is e_F(χ_2) = 1/3(1+4a+2a^2) = 5+6a+3a^2 in F_7[C_3], while the formula of Theorem 3.3 gives 1/3(2-a-a^2) = 3+2a+2a^2, which equals e_F(χ_2)+e_F(χ_3). The formula depends only on d(g), so it cannot distinguish Galois-conjugate characters, contradicting the claim that it gives the primitive idempotent attached to a single χ.
[Corollaries 3.4–3.6 and Section 4] The corollaries and applications inherit the error from Theorem 3.3. In particular, Example 4.1 over F_5 already exposes the issue: the paper states e_F(χ_2) = e_F(χ_3) = 1/3(2-a-a^2), which assigns the same primitive central idempotent to two distinct irreducible characters. Over F_5 the two nontrivial characters are Galois conjugate, and the formula produces the sum e_F(χ_2)+e_F(χ_3), not a primitive idempotent for an individual character.
[Corollary 3.5 proof] The proof of Corollary 3.5 contains a nonsensical line: it writes 'So χ(1)^2 = [G:Z(χ)] ≥ [G:Z(χ)]', which is not the intended inequality. While this is a presentation error, it adds to the impression that the manuscript has not been carefully checked; more importantly, the corollary cannot be rescued without fixing the main theorem.

minor comments (4)

[Section 3, Proposition 3.1 proof] The expression 'F(p≡1(mod n))' in the proof of Proposition 3.1 is undefined and should be replaced by a proper statement about the field containing the relevant roots of unity.
[Section 3, Proposition 3.1 proof] In the same proof, the notation 'ρ(ϵ_C(G,H) = 0' has a missing parenthesis, and the definition of ϵ_C(G,H) is written with G instead of the subgroup; this makes the argument harder to follow.
[Section 2, Lemma 2.1 proof] In the proof of Lemma 2.1, the displayed sum over '1≤d,(k,d)=1' should be '1≤k≤d, gcd(k,d)=1'; as written, the variable names are inconsistent.
[Throughout] There are numerous typographical issues, including 'Mobius mu' instead of 'Möbius μ', 'cental idempotents', 'modulo' spelled inconsistently, and the citation [4] listed as 'O. Broch, Cristo and A.del Rio' with 'to appear' but no clear author list. These should be corrected in any future version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.3 is derived from standard Galois descent, the trace computation in Lemma 3.2, and the idempotent normalization; the finite-field counterexample is a domain-assumption error, not a circular argument.

full rationale

The derivation chain is self-contained and not circular. Theorem 3.3 starts from the standard Galois-descent formula e_F(χ)=∑_{σ∈Gal(F(χ)/F)} σ(e(χ)) (cited to Yamada), expands e(χ) by its definition in terms of χ, applies Lemma 3.2 to evaluate the inner trace, and fixes the overall scalar by imposing idempotency (e_F)^2=e_F and comparing coefficients of the identity. No parameter is fitted to the quantity being predicted; the normalization constant is forced by a general algebraic property of idempotents, not by the desired output. Lemma 2.1 is proved in the text, so the citation to [1] is not load-bearing, and there are no self-citations by the present authors that carry the argument. The serious defect of Lemma 3.2 is the unstated assumption [F(ζ_d):F]=φ(d), which is false for finite fields that already contain the relevant roots of unity; this makes Theorem 3.3 false as a statement about F_q[G] (e.g., G=C_3, F=F_7 returns e_F(χ_2)+e_F(χ_3)). That is a mathematical correctness issue, not a circularity: the false premise is not equivalent by construction to the conclusion, nor is the formula fitted from the data it claims to predict.

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

No free parameters or invented entities. The central result rests on a false field-extension degree assumption.

assumptions (5)

standard math Maschke's theorem (Lemma 2.2): F[G] is semisimple if gcd(q, |G|)=1.
Fundamental for existence of primitive central idempotents.
domain assumption The descent formula e_F(χ)=∑_{σ∈Gal(F(χ)/F)} σ(e(χ)) from Yamada [5] is used verbatim.
For finite fields the Galois action must be defined through reduction modulo a prime above q; the paper treats F as a subfield of the complex cyclotomic field.
ad hoc to paper The degree equality [F(ζ_n):F(ζ_n^{n/d})] = φ(n)/φ(d), i.e. [F(ζ_n):F] = φ(n), used in Lemma 3.2.
False for finite fields that contain primitive n-th roots; for example F_7 and n=3 give degree 1, not 2.
standard math Isaacs' facts: Z(G/ker χ) is cyclic and χ(1)^2 ≤ [G:Z(χ)].
Used in Corollaries 3.4-3.6.
standard math Lemma 2.1: the sum of primitive n-th roots over units is μ(n).
Used in proof of Lemma 3.2.

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Pith. "Pith review of Primitive central idempotents of a finite group over a field." pith.science (2026). https://pith.science/paper/J4F2OCGU

@misc{pith2026260803371,
  author       = {Pith},
  title        = {Pith review of: Primitive central idempotents of a finite group over a field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4F2OCGU}},
  note         = {Machine review of arXiv:2608.03371}
}

abstract

We give a method to compute the primitive central idempotents of a semi-simple finite group algebra associated with an irreducible character $\chi$ such that for every $g\in G$, either $\chi(g) = 0$ or all eigenvalues of $\rho(g)$ have the same order. We then show several consequences which highlight the significance and possible applications of our approach.

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Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

G. K. Bakshi and I. B. S. Passi,Primitive central idempotents in rational group algebras, Commun. Algebra 40(4) (2012) 1413-1426

work page 2012
[2]

I. M. Isaacs,Character Theory of Finite Groups, Dover Publications, 1994

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[3]

Serre,Linear Representations of Finite Groups, Springer-Verlag, 1977

J.-P. Serre,Linear Representations of Finite Groups, Springer-Verlag, 1977

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[4]

Broch, Cristo and A.del Rio,Wedderburn decomposition of finite group algebras, Finite Fields Appl., to appear

O. Broch, Cristo and A.del Rio,Wedderburn decomposition of finite group algebras, Finite Fields Appl., to appear

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[5]

Yamada,The Schur Subgroup of the Brauer group, Lecture Notes in Math.Vol-397, Springer-Verlag, 1974

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G., Jespers, E., Milies, C

Goodaire, E. G., Jespers, E., Milies, C. P.(1996),Alternative Loop Rings, Math.Studies. Vol.184. North Holland

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1983

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

G. K. Bakshi and I. B. S. Passi,Primitive central idempotents in rational group algebras, Commun. Algebra 40(4) (2012) 1413-1426

work page 2012

[2] [2]

I. M. Isaacs,Character Theory of Finite Groups, Dover Publications, 1994

work page 1994

[3] [3]

Serre,Linear Representations of Finite Groups, Springer-Verlag, 1977

J.-P. Serre,Linear Representations of Finite Groups, Springer-Verlag, 1977

work page 1977

[4] [4]

Broch, Cristo and A.del Rio,Wedderburn decomposition of finite group algebras, Finite Fields Appl., to appear

O. Broch, Cristo and A.del Rio,Wedderburn decomposition of finite group algebras, Finite Fields Appl., to appear

work page

[5] [5]

Yamada,The Schur Subgroup of the Brauer group, Lecture Notes in Math.Vol-397, Springer-Verlag, 1974

T. Yamada,The Schur Subgroup of the Brauer group, Lecture Notes in Math.Vol-397, Springer-Verlag, 1974

work page 1974

[6] [6]

V. S. Pless and W. C. Huffman,Handbook of Coding Theory, Elsevier, New york, 1998

work page 1998

[7] [7]

G., Ayoub, C.(1969)On the group ring of finite abelian group, Bull

Ayoub, R. G., Ayoub, C.(1969)On the group ring of finite abelian group, Bull. Austral. Math. Soc. (1):245- 261

work page 1969

[8] [8]

G., Jespers, E., Milies, C

Goodaire, E. G., Jespers, E., Milies, C. P.(1996),Alternative Loop Rings, Math.Studies. Vol.184. North Holland

work page 1996