Complex Representations of Groups and Involutions of its Automorphisms

Ajay Kumar Shukla; Rahul Dixit; Venkata Subbaiah Yerrapati

arxiv: 2605.23195 · v1 · pith:2LLBIFZHnew · submitted 2026-05-22 · 🧮 math.RT · math.CO· math.GR

Complex Representations of Groups and Involutions of its Automorphisms

Venkata Subbaiah Yerrapati , Rahul Dixit , Ajay Kumar Shukla This is my paper

Pith reviewed 2026-05-25 03:07 UTC · model grok-4.3

classification 🧮 math.RT math.COmath.GR

keywords finite groupsirreducible characterstwisted involutionsautomorphismssymmetric groupcomplex representationsrepresentation theory

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The pith

The sum of irreducible character degrees of a finite group relates to the number of twisted involutions associated with its automorphisms.

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

The paper establishes a relationship between the sum of the degrees of irreducible characters of a finite group and the number of twisted involutions linked to the group's automorphisms. Algorithmic frameworks are developed to compute these quantities for inner automorphisms and for the symmetric group. As an application, a criterion is given for identifying which groups have complex irreducible representations, along with exploration of resulting structural properties.

Core claim

We establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. Algorithmic frameworks are developed for evaluating these quantities in the context of inner automorphisms and the symmetric group S_n. This provides a criterion for identifying groups that possess complex irreducible representations.

What carries the argument

The relationship between the sum of irreducible character degrees and the number of twisted involutions associated with automorphisms, which connects representation degrees to a counting of special involutory elements under group automorphisms.

If this is right

Algorithmic methods exist to evaluate the sum and the number for inner automorphisms.
Computations apply to the symmetric group S_n.
A criterion identifies groups with complex non-real irreducible representations.
Structural consequences follow from the established relationship.

Where Pith is reading between the lines

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

This could lead to new classification tools for groups based on the reality of their representations.
Similar relationships might exist for other classes of automorphisms beyond inner ones.
The criterion may be tested on small groups to verify its utility in detecting non-real representations.

Load-bearing premise

The definitions and properties of twisted involutions associated with automorphisms are well-defined and the relationship holds for finite groups without additional restrictions.

What would settle it

For a concrete finite group such as S_3, compute both the sum of its irreducible character degrees and the number of twisted involutions from its automorphisms, then verify whether the stated relationship holds exactly.

read the original abstract

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these quantities in the context of inner automorphisms and the symmetric group $\mathfrak{S}_n$. As an application, we provide a criterion for identifying groups that possess complex (non-real) irreducible representations and explore the structural consequences arising from these results.

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.

Desk Editor's Note private letter to a colleague

Abstract claims a link between sum of character degrees and twisted involutions from automorphisms plus algorithms for S_n, but without the full paper the novelty and correctness stay uncheckable.

read the letter

The main claim is a relationship between the sum of irreducible character degrees and the number of twisted involutions tied to automorphisms of a finite group, plus algorithmic frameworks for inner automorphisms and for S_n, with an application to a criterion for groups that have non-real irreps. The algorithmic part for S_n is the piece that could actually be useful if it gives a practical way to compute or bound these quantities. The criterion for complex representations follows directly from the main relationship and looks like a straightforward consequence rather than a deep new insight. What the paper does well is to focus on concrete computation in a well-studied family like the symmetric groups, where character tables and involution counts are already computable by other means. If the algorithms are explicit and efficient, that could save time for people running calculations in GAP or similar systems. The soft spots are bigger. The abstract states the relationship without any derivation, small example, or comparison to existing results on involutions in Aut(G) or on the Frobenius-Schur indicator. It is therefore impossible to tell whether this is a new observation or a repackaging of known facts about character sums and fixed-point-free involutions. The term 'twisted involutions' also needs a clear definition; if it is non-standard, the paper must show it behaves well under the group operations. Because the full manuscript was not supplied, these points cannot be checked. This work is aimed at specialists in computational representation theory of finite groups who already work with S_n or inner automorphisms. A serious referee could evaluate the algorithms and the criterion once the proofs and examples are in hand. I would send it to peer review rather than desk-reject, since the topic is legitimate and the claims are in principle verifiable.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. It develops algorithmic frameworks for evaluating these quantities in the context of inner automorphisms and the symmetric group S_n, and applies the results to provide a criterion for identifying groups that possess complex (non-real) irreducible representations.

Significance. If the relationship and algorithms are rigorously established, the work could offer a novel link between character sums and automorphism structures, with potential utility for computational group theory and representation classification. The algorithmic development for S_n and inner automorphisms is a concrete strength if accompanied by verifiable implementations or proofs.

major comments (1)

The full manuscript text is not supplied in the available review context (only the abstract is provided), so the definitions of twisted involutions, the precise statement of the claimed relationship, and any supporting derivations or algorithms cannot be examined for correctness or gaps.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The sole major comment concerns the availability of the full manuscript in the review context. We address this below.

read point-by-point responses

Referee: The full manuscript text is not supplied in the available review context (only the abstract is provided), so the definitions of twisted involutions, the precise statement of the claimed relationship, and any supporting derivations or algorithms cannot be examined for correctness or gaps.

Authors: The complete manuscript, including precise definitions of twisted involutions, the stated relationship between the sum of irreducible character degrees and the number of twisted involutions, all derivations, algorithmic frameworks for inner automorphisms and S_n, and the criterion for complex representations, is available in full on arXiv:2605.23195. The abstract in the review materials is only a summary; the submitted paper contains the full details required for verification. revision: no

Circularity Check

0 steps flagged

No circularity detectable; abstract contains no derivations or equations

full rationale

The supplied document consists solely of the abstract, which asserts a relationship between the sum of irreducible character degrees and the number of twisted involutions but provides no equations, proofs, definitions of twisted involutions, algorithmic frameworks, or citations. No load-bearing steps exist that could reduce by construction to inputs, self-citations, or fitted parameters. The derivation chain is therefore inaccessible and cannot be flagged as circular under the required rules that demand explicit quotes and reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no access to definitions, assumptions, or derivations in the paper.

pith-pipeline@v0.9.0 · 5602 in / 953 out tokens · 13448 ms · 2026-05-25T03:07:37.809208+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Number of elements mapped to their inverses under this homomorphism:

C. Ryan Vinroot,Twisted Frobenius-Schur indicators of finite symplectic groups, J. Algebra293(2005), no. 1, 279–311. MR2173976 AppendixA.Computational Verification via Magma In this appendix, we provide the Magma [1] code used to verify Theorem 3.5 and Theorem 3.9 for outer automorphisms ofS 6. The following script calculatesT(G) and compares it against|S...

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[1] [1]

Wieb Bosma, John Cannon, and Catherine Playoust,The Magma algebra system. I. The user language, J. Symbolic Comput.24(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR1484478

work page 1997

[2] [2]

Algebra278(2004), no

Daniel Bump and David Ginzburg,Generalized Frobenius-Schur numbers, J. Algebra278(2004), no. 1, 294–

work page 2004

[3] [3]

Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli,Mackey’s theory ofτ-conjugate representations for finite groups, Jpn. J. Math.10(2015), no. 1, 43–96. MR3320995

work page 2015

[4] [4]

Chowla, I

S. Chowla, I. N. Herstein, and W. K. Moore,On recursions connected with symmetric groups. I, Canad. J. Math.3(1951), 328–334. MR41849

work page 1951

[5] [5]

Frobenius and I

G. Frobenius and I. Schur, ¨Uber die reellen Darstellungen der endlichen Gruppen., Berl. Ber.1906(1906), 186–208

work page 1906

[6] [6]

35, Cambridge University Press, Cambridge, 1997

William Fulton,Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR1464693

work page 1997

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129, Springer-Verlag, New York, 1991

William Fulton and Joe Harris,Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course, Readings in Mathematics. MR1153249

work page 1991

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Gow,Properties of the characters of the finite general linear group related to the transpose-inverse involution, Proc

R. Gow,Properties of the characters of the finite general linear group related to the transpose-inverse involution, Proc. London Math. Soc. (3)47(1983), no. 3, 493–506. MR716800

work page 1983

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Algebra96 (1985), no

,Real representations of the finite orthogonal and symplectic groups of odd characteristic, J. Algebra96 (1985), no. 1, 249–274. MR808851

work page 1985

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J.19(1990), no

Noriaki Kawanaka and Hiroshi Matsuyama,A twisted version of the Frobenius-Schur indicator and multiplicity- free permutation representations, Hokkaido Math. J.19(1990), no. 3, 495–508. MR1078503

work page 1990

[11] [11]

J.19(1990), no

,A twisted version of the Frobenius-Schur indicator and multiplicity-free permutation representations, Hokkaido Math. J.19(1990), no. 3, 495–508. MR1078564 COMPLEX REPRESENTATIONS OF GROUPS AND INVOLUTIONS OF ITS AUTOMORPHISMS 23

work page 1990

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E. I. Khukhro and V. D. Mazurov (eds.),The Kourovka notebook, Twentieth, Sobolev Institute of Mathe- matics. Russian Academy of Sciences. Siberian Branch, Novosibirsk, 2022. Unsolved problems in group theory. MR4842130

work page 2022

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Math.7(1955), 159–168

Leo Moser and Max Wyman,On solutions ofx d = 1in symmetric groups, Canadian J. Math.7(1955), 159–168. MR68564

work page 1955

[14] [14]

Rotman,An introduction to the theory of groups, Fourth, Graduate Texts in Mathematics, vol

Joseph J. Rotman,An introduction to the theory of groups, Fourth, Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR1307623

work page 1995

[15] [15]

Sagan,The symmetric group, Second, Graduate Texts in Mathematics, vol

Bruce E. Sagan,The symmetric group, Second, Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR1824028

work page 2001

[16] [16]

Number of elements mapped to their inverses under this homomorphism:

C. Ryan Vinroot,Twisted Frobenius-Schur indicators of finite symplectic groups, J. Algebra293(2005), no. 1, 279–311. MR2173976 AppendixA.Computational Verification via Magma In this appendix, we provide the Magma [1] code used to verify Theorem 3.5 and Theorem 3.9 for outer automorphisms ofS 6. The following script calculatesT(G) and compares it against|S...

work page 2005