Transitivity in wreath products with symmetric groups

Kai-Uwe Schmidt; Lukas Klawuhn

arxiv: 2409.20495 · v3 · submitted 2024-09-30 · 🧮 math.CO · math.GR· math.RT

Transitivity in wreath products with symmetric groups

Lukas Klawuhn , Kai-Uwe Schmidt This is my paper

Pith reviewed 2026-05-23 20:16 UTC · model grok-4.3

classification 🧮 math.CO math.GRmath.RT

keywords transitive subsetswreath productssymmetric groupsassociation schemesdesignsLivingstone-Wagner theoremCharlier polynomialspermutation groups

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The pith

Transitive subsets of the wreath product G wr S_n admit a character-theoretic characterization and generalize the Livingstone-Wagner theorem.

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

The paper extends the natural notion of transitivity from subgroups to subsets of a permutation group to the case where the group is a wreath product G wr S_n with G finite abelian. It provides structural characterizations of such transitive subsets via the character theory of the wreath product and interprets them as designs in the conjugacy class association scheme. Explicit constructions are given, a generalization of the Livingstone-Wagner theorem is proved, and connections to Charlier polynomials are established to study codes and designs in the special case C_r wr S_n. These results extend earlier work that was known only for the symmetric group S_n itself.

Core claim

Transitive subsets of G wr S_n for finite abelian G can be characterized using the character theory of the wreath product and interpreted as designs in its conjugacy class association scheme; this includes a generalization of the Livingstone-Wagner theorem together with explicit constructions of such subsets, and the Charlier polynomials yield further information on codes and designs when G is cyclic.

What carries the argument

Character-theoretic characterization of transitive subsets in G wr S_n, which enables their interpretation as designs in the conjugacy class association scheme.

If this is right

Explicit constructions of transitive sets exist inside G wr S_n for any finite abelian G.
The Livingstone-Wagner theorem extends from S_n to the wreath product setting.
Charlier polynomials give information on codes and designs inside C_r wr S_n.
Results previously known only for the symmetric group S_n now hold for the larger class of wreath products G wr S_n.

Where Pith is reading between the lines

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

The design interpretation may yield new parameter bounds or existence criteria for combinatorial objects inside these wreath products.
The character approach could be tested on the hyperoctahedral case G = C_2 to recover or improve known results on signed permutations.
The link to orthogonal polynomials suggests possible extensions to other families of polynomials for different choices of G.

Load-bearing premise

The natural extension of transitivity from subgroups to subsets continues to hold and admits a character-theoretic characterization when the ambient group is replaced by the wreath product G wr S_n.

What would settle it

An explicit subset of G wr S_n that meets the definition of transitivity yet fails to satisfy the character-theoretic conditions derived in the paper, or a counterexample to the stated generalization of the Livingstone-Wagner theorem.

read the original abstract

It is known that the notion of a transitive subgroup of a permutation group $P$ extends naturally to the subsets of $P$. We study transitive subsets of the wreath product $G \wr S_n$, where $G$ is a finite abelian group. This includes the hyperoctahedral group for $G=C_2$. We give structural characterisations of transitive subsets using the character theory of $G \wr S_n$ and interpret such subsets as designs in the conjugacy class association scheme of $G \wr S_n$. In particular, we prove a generalisation of the Livingstone-Wagner theorem and give explicit constructions of transitive sets. Moreover, we establish connections to orthogonal polynomials, namely the Charlier polynomials, and use them to study codes and designs in $C_r \wr S_n$. Many of our results extend results about the symmetric group $S_n$.

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

Claims a generalization of Livingstone-Wagner to transitive subsets in G wr S_n plus Charlier polynomial links, but only the abstract exists so nothing is verified.

read the letter

The paper announces a generalization of the Livingstone-Wagner theorem on transitive subsets from S_n to the wreath product G wr S_n for finite abelian G, including the hyperoctahedral case when G is C_2. It gives character-theoretic characterizations of these subsets, interprets them as designs in the conjugacy class association scheme, supplies explicit constructions, and connects the C_r case to Charlier polynomials for work on codes and designs. Several results are presented as extensions of the symmetric group literature. That is the actual content on offer. The move to wreath products and the polynomial angle are the pieces that go past the cited prior work. The abstract frames the approach as standard character theory plus a natural extension of transitivity, with no free parameters or invented objects visible. The main limitation is that only the abstract is supplied, so there are no derivations, no explicit character calculations, and no way to check whether the claimed characterizations or the polynomial applications actually hold without extra conditions. The central assumption that transitivity extends in the same way to the wreath product case remains untested in what we have. This is narrow combinatorial group theory and design theory. A reader already working on permutation groups, association schemes, or orthogonal polynomials in this setting might extract something usable if the full proofs are sound. It is the sort of specialized extension that deserves a referee to examine the details rather than a desk rejection.

Referee Report

2 major / 0 minor

Summary. The paper studies transitive subsets of the wreath product G wr S_n for finite abelian G (including the hyperoctahedral group when G=C_2). It claims to give structural characterisations of such subsets via the character theory of G wr S_n, interpret them as designs in the conjugacy class association scheme, prove a generalisation of the Livingstone-Wagner theorem, provide explicit constructions of transitive sets, establish connections to Charlier polynomials for codes and designs in C_r wr S_n, and extend several results known for the symmetric group S_n.

Significance. If substantiated, the work would extend classical results on transitivity and designs from S_n to wreath products, offering character-theoretic tools and explicit constructions that could inform the study of association schemes, permutation group actions, and combinatorial codes in these groups. The link to Charlier polynomials provides a potential bridge to orthogonal polynomial techniques in design theory.

major comments (2)

Abstract: The central claim of proving a generalisation of the Livingstone-Wagner theorem and giving explicit constructions of transitive sets cannot be assessed, as the manuscript supplies only the abstract with no derivations, character tables, or construction details provided.
Abstract: The asserted character-theoretic characterisation of transitive subsets and their interpretation as designs in the conjugacy class association scheme of G wr S_n is stated without any indication of the specific characters, idempotents, or parameters used, preventing verification of the claimed structural results.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their report. The provided manuscript consists solely of the abstract, which summarizes results without including proofs or tables. We address the major comments point by point below.

read point-by-point responses

Referee: Abstract: The central claim of proving a generalisation of the Livingstone-Wagner theorem and giving explicit constructions of transitive sets cannot be assessed, as the manuscript supplies only the abstract with no derivations, character tables, or construction details provided.

Authors: The observation is correct: the text supplied here is only the abstract and therefore contains none of the requested derivations, tables or constructions. The full arXiv version contains the character-theoretic proofs of the generalised Livingstone-Wagner theorem together with the explicit constructions, but these cannot be reproduced from the material available for this response. revision: no
Referee: Abstract: The asserted character-theoretic characterisation of transitive subsets and their interpretation as designs in the conjugacy class association scheme of G wr S_n is stated without any indication of the specific characters, idempotents, or parameters used, preventing verification of the claimed structural results.

Authors: Again, the abstract alone supplies no characters, idempotents or parameters. The full manuscript develops these via the character theory of the wreath product and interprets the subsets as designs in the conjugacy-class scheme, but the details are not present in the text provided here. revision: no

standing simulated objections not resolved

Full derivations, character tables, idempotents, parameters and explicit constructions are absent from the only text available (the abstract), so the referee's request for verification cannot be met with concrete evidence.

Circularity Check

0 steps flagged

No circularity detectable from abstract alone

full rationale

Only the abstract is supplied, which states that the work studies transitive subsets of G wr S_n using character theory of the wreath product, proves a generalization of the Livingstone-Wagner theorem, and extends results about S_n. It explicitly builds on a 'known' natural extension of transitivity and standard character theory without presenting any equations, fitted parameters, self-citations, or derivation steps. No load-bearing claim reduces to its own inputs by construction, and the abstract gives no indication of self-referential definitions or predictions forced by prior fits. The derivation chain cannot be walked, so no circularity is exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from representation theory of finite groups and the known extension of transitivity to subsets; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math Character theory of finite groups applies to wreath products G wr S_n
Invoked for structural characterizations.
domain assumption Transitivity extends naturally to subsets of permutation groups
Stated as known at the outset of the abstract.

pith-pipeline@v0.9.0 · 5648 in / 1260 out tokens · 27860 ms · 2026-05-23T20:16:48.444951+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Bannai and T

E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes , The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984

work page 1984
[2]

Bannai, Y

E. Bannai, Y. Nakata, T. Okuda, and D. Zhao, Explicit construction of exact unitary designs , Adv. Math. 405 (2022), Paper No. 108457, 1–25

work page 2022
[3]

Blake, G

I. Blake, G. Cohen, and M. Deza, Coding with Permutations , Information and Control 43 (1979), 1–19

work page 1979
[4]

R. C. Bose, On the Application of the Properties of Galois Fields to the P roblem of Construction of Hyper-Graeco-Latin Squares, Sankhy¯ a: The Indian Journal of Statistics 3 (1938), 323–338

work page 1938
[5]

P. J. Cameron, Permutation Groups , Cambridge University Press, 1999

work page 1999
[6]

Chow and T

C.-O. Chow and T. Mansour, Asymptotic probability distributions of some permutation statistics for the wreath product Cr ≀Sn, Online J. Anal. Comb. 7 (2012), 1–14

work page 2012
[7]

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover Publications, 1973

work page 1973
[8]

Delsarte, An algebraic approach to the association schemes of coding t heory, Philips Research Reports Supplements 10 (1973), 1–97

P. Delsarte, An algebraic approach to the association schemes of coding t heory, Philips Research Reports Supplements 10 (1973), 1–97

work page 1973
[9]

J. D. Dixon and B. Mortimer, Permutation Groups , Springer, 1996

work page 1996
[10]

Ernst and K.-U

A. Ernst and K.-U. Schmidt, Transitivity in ﬁnite general linear groups , Math. Z. 307 (2024), 1–25

work page 2024
[11]

Hall, Projective planes, Trans

M. Hall, Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277

work page 1943
[12]

D. R. Hughes and F. C. Piper, Projective Planes, Springer, 1973

work page 1973
[13]

J. E. Humphreys, Reﬂection Groups and Coxeter Groups , Cambridge University Press, 1990

work page 1990
[14]

Ito, Designs in a coset geometry: Delsarte theory revisited , European J

T. Ito, Designs in a coset geometry: Delsarte theory revisited , European J. Combin. 25 (2004), no. 2, 229–238

work page 2004
[15]

Koekoek and R

R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomial s and its q-analogue, Tech. Report 98, Delft University of Technology, 1998

work page 1998
[16]

Kuperberg, S

G. Kuperberg, S. Lovett, and R. Peled, Probabilistic existence of regular combinatorial structu res, Geom. Funct. Anal. 27 (2017), no. 4, 919–972

work page 2017
[17]

Livingstone and A

D. Livingstone and A. Wagner, Transitivity of ﬁnite permutation groups on unordered sets , Math- ematische Zeitschrift 90 (1965), 393–403

work page 1965
[18]

I. G. Macdonald, Symmetric functions and Hall polynomials , 2nd ed., Oxford University Press, 1995

work page 1995
[19]

W. J. Martin and B. E. Sagan, A new notion of transitivity for groups and sets of permutati ons, J. London Math. Soc. (2) 73 (2006), no. 1, 1–13

work page 2006
[20]

S. J. Mayer, On the characters of the Weyl group of type C, Journal of Algebra 33 (1975), 59–67

work page 1975
[21]

B. E. Sagan, The Symmetric Group , 2nd ed., Springer, 2001

work page 2001
[22]

Serre, Linear Representations of Finite Groups , Springer, 1977

J.-P. Serre, Linear Representations of Finite Groups , Springer, 1977. Department of Mathematics, P aderborn University, W arburger Str. 100, 33098 P ader- born, Germany. Email address : klawuhn@math.upb.de

work page 1977

[1] [1]

Bannai and T

E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes , The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984

work page 1984

[2] [2]

Bannai, Y

E. Bannai, Y. Nakata, T. Okuda, and D. Zhao, Explicit construction of exact unitary designs , Adv. Math. 405 (2022), Paper No. 108457, 1–25

work page 2022

[3] [3]

Blake, G

I. Blake, G. Cohen, and M. Deza, Coding with Permutations , Information and Control 43 (1979), 1–19

work page 1979

[4] [4]

R. C. Bose, On the Application of the Properties of Galois Fields to the P roblem of Construction of Hyper-Graeco-Latin Squares, Sankhy¯ a: The Indian Journal of Statistics 3 (1938), 323–338

work page 1938

[5] [5]

P. J. Cameron, Permutation Groups , Cambridge University Press, 1999

work page 1999

[6] [6]

Chow and T

C.-O. Chow and T. Mansour, Asymptotic probability distributions of some permutation statistics for the wreath product Cr ≀Sn, Online J. Anal. Comb. 7 (2012), 1–14

work page 2012

[7] [7]

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover Publications, 1973

work page 1973

[8] [8]

Delsarte, An algebraic approach to the association schemes of coding t heory, Philips Research Reports Supplements 10 (1973), 1–97

P. Delsarte, An algebraic approach to the association schemes of coding t heory, Philips Research Reports Supplements 10 (1973), 1–97

work page 1973

[9] [9]

J. D. Dixon and B. Mortimer, Permutation Groups , Springer, 1996

work page 1996

[10] [10]

Ernst and K.-U

A. Ernst and K.-U. Schmidt, Transitivity in ﬁnite general linear groups , Math. Z. 307 (2024), 1–25

work page 2024

[11] [11]

Hall, Projective planes, Trans

M. Hall, Projective planes, Trans. Amer. Math. Soc. 54 (1943), 229–277

work page 1943

[12] [12]

D. R. Hughes and F. C. Piper, Projective Planes, Springer, 1973

work page 1973

[13] [13]

J. E. Humphreys, Reﬂection Groups and Coxeter Groups , Cambridge University Press, 1990

work page 1990

[14] [14]

Ito, Designs in a coset geometry: Delsarte theory revisited , European J

T. Ito, Designs in a coset geometry: Delsarte theory revisited , European J. Combin. 25 (2004), no. 2, 229–238

work page 2004

[15] [15]

Koekoek and R

R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomial s and its q-analogue, Tech. Report 98, Delft University of Technology, 1998

work page 1998

[16] [16]

Kuperberg, S

G. Kuperberg, S. Lovett, and R. Peled, Probabilistic existence of regular combinatorial structu res, Geom. Funct. Anal. 27 (2017), no. 4, 919–972

work page 2017

[17] [17]

Livingstone and A

D. Livingstone and A. Wagner, Transitivity of ﬁnite permutation groups on unordered sets , Math- ematische Zeitschrift 90 (1965), 393–403

work page 1965

[18] [18]

I. G. Macdonald, Symmetric functions and Hall polynomials , 2nd ed., Oxford University Press, 1995

work page 1995

[19] [19]

W. J. Martin and B. E. Sagan, A new notion of transitivity for groups and sets of permutati ons, J. London Math. Soc. (2) 73 (2006), no. 1, 1–13

work page 2006

[20] [20]

S. J. Mayer, On the characters of the Weyl group of type C, Journal of Algebra 33 (1975), 59–67

work page 1975

[21] [21]

B. E. Sagan, The Symmetric Group , 2nd ed., Springer, 2001

work page 2001

[22] [22]

Serre, Linear Representations of Finite Groups , Springer, 1977

J.-P. Serre, Linear Representations of Finite Groups , Springer, 1977. Department of Mathematics, P aderborn University, W arburger Str. 100, 33098 P ader- born, Germany. Email address : klawuhn@math.upb.de

work page 1977