---
_id: '60491'
abstract:
- lang: eng
  text: We investigate generalisations of 1-factorisations and hyperfactorisations
    of the complete graph $K_{2n}$. We show that they are special subsets of the association
    scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and
    extends results by Cameron (1976) and gives rise to new existence and non-existence
    results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$
    and using the representation theory of $S_{2n}$.
author:
- first_name: Lukas-André Dominik
  full_name: Klawuhn, Lukas-André Dominik
  id: '91965'
  last_name: Klawuhn
  orcid: 0009-0009-7736-4885
- first_name: John
  full_name: Bamberg, John
  last_name: Bamberg
citation:
  ama: Klawuhn L-AD, Bamberg J. On the association scheme of perfect matchings and
    their designs. Published online 2025.
  apa: Klawuhn, L.-A. D., &#38; Bamberg, J. (2025). <i>On the association scheme of
    perfect matchings and their designs</i>.
  bibtex: '@article{Klawuhn_Bamberg_2025, title={On the association scheme of perfect
    matchings and their designs}, author={Klawuhn, Lukas-André Dominik and Bamberg,
    John}, year={2025} }'
  chicago: Klawuhn, Lukas-André Dominik, and John Bamberg. “On the Association Scheme
    of Perfect Matchings and Their Designs,” 2025.
  ieee: L.-A. D. Klawuhn and J. Bamberg, “On the association scheme of perfect matchings
    and their designs.” 2025.
  mla: Klawuhn, Lukas-André Dominik, and John Bamberg. <i>On the Association Scheme
    of Perfect Matchings and Their Designs</i>. 2025.
  short: L.-A.D. Klawuhn, J. Bamberg, (2025).
date_created: 2025-07-02T07:37:23Z
date_updated: 2025-07-02T07:47:09Z
department:
- _id: '100'
external_id:
  arxiv:
  - '2507.00813'
language:
- iso: eng
status: public
title: On the association scheme of perfect matchings and their designs
type: preprint
user_id: '91965'
year: '2025'
...
---
_id: '61759'
abstract:
- lang: eng
  text: 'Intersection distribution and non-hitting index are concepts introduced recently
    by Li and Pott as a new way to view the behaviour of a collection of finite field
    polynomials. With both an algebraic interpretation via the intersection of a polynomial
    with a set of lines, and a geometric interpretation via a (q+1)-set possessing
    an internal nucleus, the concepts have proved their usefulness as a new way to
    view various long-standing problems, and have applications in areas such as Kakeya
    sets. In this paper, by exploiting connections with diverse areas including the
    theory of algebraic curves, cyclotomy and the enumeration of irreducible polynomials,
    we establish new results and resolve various Open Problems of Li and Pott. We
    prove geometric results which shed new light on the relationship between intersection
    distribution and projective equivalence of polynomials, and algebraic results
    which describe and characterise the degree of Sf - the index of the largest non-zero
    entry in the intersection distribution of f. We provide new insights into the
    non-hitting spectrum, and show the limitations of the non-hitting index as a tool
    for characterisation. Finally, the benefits provided by the connections to other
    areas are evidenced in two short new proofs of the cubic case. '
author:
- first_name: Lukas-André Dominik
  full_name: Klawuhn, Lukas-André Dominik
  id: '91965'
  last_name: Klawuhn
  orcid: 0009-0009-7736-4885
- first_name: Sophie
  full_name: Huczynska, Sophie
  last_name: Huczynska
- first_name: Maura
  full_name: Paterson, Maura
  last_name: Paterson
citation:
  ama: 'Klawuhn L-AD, Huczynska S, Paterson M. The Intersection Distribution: New
    Results and Perspectives. Published online 2025.'
  apa: 'Klawuhn, L.-A. D., Huczynska, S., &#38; Paterson, M. (2025). <i>The Intersection
    Distribution: New Results and Perspectives</i>.'
  bibtex: '@article{Klawuhn_Huczynska_Paterson_2025, title={The Intersection Distribution:
    New Results and Perspectives}, author={Klawuhn, Lukas-André Dominik and Huczynska,
    Sophie and Paterson, Maura}, year={2025} }'
  chicago: 'Klawuhn, Lukas-André Dominik, Sophie Huczynska, and Maura Paterson. “The
    Intersection Distribution: New Results and Perspectives,” 2025.'
  ieee: 'L.-A. D. Klawuhn, S. Huczynska, and M. Paterson, “The Intersection Distribution:
    New Results and Perspectives.” 2025.'
  mla: 'Klawuhn, Lukas-André Dominik, et al. <i>The Intersection Distribution: New
    Results and Perspectives</i>. 2025.'
  short: L.-A.D. Klawuhn, S. Huczynska, M. Paterson, (2025).
date_created: 2025-10-08T14:52:20Z
date_updated: 2025-12-19T11:23:10Z
department:
- _id: '100'
external_id:
  arxiv:
  - '2510.04675'
language:
- iso: eng
page: '36'
status: public
title: 'The Intersection Distribution: New Results and Perspectives'
type: preprint
user_id: '91965'
year: '2025'
...
---
_id: '63384'
abstract:
- lang: eng
  text: "Two fundamental ways to represent a group are as permutations and as matrices.
    In this paper, we study linear representations of groups that intertwine with
    a permutation representation. Recently, D'Alconzo and Di Scala investigated how
    small the matrices in such a linear representation can be. The minimal dimension
    of such a representation is the \\emph{linear dimension of the group action} and
    this has applications in cryptography and cryptosystems.\r\n\r\nWe develop the
    idea of linear dimension from an algebraic point of view by using the theory of
    permutation modules. We give structural results about representations of minimal
    dimension and investigate the implications of faithfulness, transitivity and primitivity
    on the linear dimension. Furthermore, we compute the linear dimension of several
    classes of finite primitive permutation groups. We also study wreath products,
    allowing us to determine the linear dimension of imprimitive group actions. Finally,
    we give the linear dimension of almost simple finite $2$-transitive groups, some
    of which may be used for further applications in cryptography. Our results also
    open up many new questions about linear representations of group actions."
author:
- first_name: Alice
  full_name: Devillers, Alice
  last_name: Devillers
- first_name: Michael
  full_name: Giudici, Michael
  last_name: Giudici
- first_name: Daniel R.
  full_name: Hawtin, Daniel R.
  last_name: Hawtin
- first_name: Lukas-André Dominik
  full_name: Klawuhn, Lukas-André Dominik
  id: '91965'
  last_name: Klawuhn
  orcid: 0009-0009-7736-4885
- first_name: Luke
  full_name: Morgan, Luke
  last_name: Morgan
citation:
  ama: Devillers A, Giudici M, Hawtin DR, Klawuhn L-AD, Morgan L. Linear dimension
    of group actions. Published online 2025.
  apa: Devillers, A., Giudici, M., Hawtin, D. R., Klawuhn, L.-A. D., &#38; Morgan,
    L. (2025). <i>Linear dimension of group actions</i>.
  bibtex: '@article{Devillers_Giudici_Hawtin_Klawuhn_Morgan_2025, title={Linear dimension
    of group actions}, author={Devillers, Alice and Giudici, Michael and Hawtin, Daniel
    R. and Klawuhn, Lukas-André Dominik and Morgan, Luke}, year={2025} }'
  chicago: Devillers, Alice, Michael Giudici, Daniel R. Hawtin, Lukas-André Dominik
    Klawuhn, and Luke Morgan. “Linear Dimension of Group Actions,” 2025.
  ieee: A. Devillers, M. Giudici, D. R. Hawtin, L.-A. D. Klawuhn, and L. Morgan, “Linear
    dimension of group actions.” 2025.
  mla: Devillers, Alice, et al. <i>Linear Dimension of Group Actions</i>. 2025.
  short: A. Devillers, M. Giudici, D.R. Hawtin, L.-A.D. Klawuhn, L. Morgan, (2025).
date_created: 2025-12-19T11:20:46Z
date_updated: 2025-12-19T11:23:41Z
department:
- _id: '100'
external_id:
  arxiv:
  - '2512.16079'
language:
- iso: eng
status: public
title: Linear dimension of group actions
type: preprint
user_id: '91965'
year: '2025'
...
---
_id: '56429'
abstract:
- lang: eng
  text: "It is known that the notion of a transitive subgroup of a permutation group\r\n$P$
    extends naturally to the subsets of $P$. We study transitive subsets of the\r\nwreath
    product $G \\wr S_n$, where $G$ is a finite abelian group. This includes\r\nthe
    hyperoctahedral group for $G=C_2$. We give structural characterisations of\r\ntransitive
    subsets using the character theory of $G \\wr S_n$ and interpret such\r\nsubsets
    as designs in the conjugacy class association scheme of $G \\wr S_n$. In\r\nparticular,
    we prove a generalisation of the Livingstone-Wagner theorem and\r\ngive explicit
    constructions of transitive sets. Moreover, we establish\r\nconnections to orthogonal
    polynomials, namely the Charlier polynomials, and use\r\nthem to study codes and
    designs in $C_r \\wr S_n$. Many of our results extend\r\nresults about the symmetric
    group $S_n$."
author:
- first_name: Lukas-André Dominik
  full_name: Klawuhn, Lukas-André Dominik
  id: '91965'
  last_name: Klawuhn
  orcid: 0009-0009-7736-4885
- first_name: Kai-Uwe
  full_name: Schmidt, Kai-Uwe
  last_name: Schmidt
citation:
  ama: Klawuhn L-AD, Schmidt K-U. Transitivity in wreath products with symmetric groups.
    <i>arXiv:240920495</i>. Published online 2024.
  apa: Klawuhn, L.-A. D., &#38; Schmidt, K.-U. (2024). Transitivity in wreath products
    with symmetric groups. In <i>arXiv:2409.20495</i>.
  bibtex: '@article{Klawuhn_Schmidt_2024, title={Transitivity in wreath products with
    symmetric groups}, journal={arXiv:2409.20495}, author={Klawuhn, Lukas-André Dominik
    and Schmidt, Kai-Uwe}, year={2024} }'
  chicago: Klawuhn, Lukas-André Dominik, and Kai-Uwe Schmidt. “Transitivity in Wreath
    Products with Symmetric Groups.” <i>ArXiv:2409.20495</i>, 2024.
  ieee: L.-A. D. Klawuhn and K.-U. Schmidt, “Transitivity in wreath products with
    symmetric groups,” <i>arXiv:2409.20495</i>. 2024.
  mla: Klawuhn, Lukas-André Dominik, and Kai-Uwe Schmidt. “Transitivity in Wreath
    Products with Symmetric Groups.” <i>ArXiv:2409.20495</i>, 2024.
  short: L.-A.D. Klawuhn, K.-U. Schmidt, ArXiv:2409.20495 (2024).
date_created: 2024-10-08T13:14:45Z
date_updated: 2024-11-15T12:34:03Z
department:
- _id: '100'
external_id:
  arxiv:
  - '2409.20495'
language:
- iso: eng
page: '38'
publication: arXiv:2409.20495
status: public
title: Transitivity in wreath products with symmetric groups
type: preprint
user_id: '91965'
year: '2024'
...