Abstract
Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Aragona, L. Campioni and R. Civino, The number of maximal unrefinable partitions, https://arxiv.org/abs/2206.04261.
R. Aragona, L. Campioni, R. Civino and M. Lauria, On the maximal part in unrefinable partitions of triangular numbers, Aequationes Mathematicae 96 (2022), 1339–1363.
R. Aragona, L. Campioni, R. Civino and M. Lauria, Verification and generation of unrefinable partitions, Information Processing Letters 181 (2023), Article no. 106361.
R. Aragona, R. Civino, N. Gavioli and C. M. Scoppola, A chain of normalizers in the Sylow 2-subgroups of the symmetric group on 2nletters, Indian Journal of Pure and Applied Mathematics 52 (2021), 735–746.
R. Aragona, R. Civino, N. Gavioli and C. M. Scoppola, Rigid commutators and a normalizer chain, Monatshefte für Mathematik 196 (2021), 431–455.
R. Aragona, R. Civino, N. Gavioli and C. M. Scoppola, Unrefinable partitions into distinct parts in a normalizer chain, Discrete Mathematics Letters 8 (2022), 72–77.
G. E. Andrews and D. Newman, Partitions and the minimal excludant, Annals of Combinatorics 23 (2019), 249–254.
C. Ballantine and M. Merca, The minimal excludant and colored partitions, Séminaire Lotharingien de Combinatoire 84B (2020), Article no. 23.
J. Q. Du and D. Tang, On a conjecture for a refinement of the sum of minimal excludants, Bulletin of the Australian Mathematical Society https://doi.org/10.1017/S0004972723000709.
A. S. Fraenkel and U. Peled, Harnessing the unwieldy MEX function, in Games of No Chance 4, Mathematical Sciences Research Institute Publications, Vol. 63, Cambridge University Press, New York, 2015, pp. 77–94.
V. Gurvich, Further generalizations of the Wythoff game and the minimum excludant, Discrete Applied Mathematics 160 (2012), 941–947.
B. Hopkins, J. A. Sellers and D. Stanton, Dyson’s crank and the mex of integer partitions, Journal of Combinatorial Theory. Series A 185 (2022), Article no. 105523.
OEIS Foundation, The On-Line Encyclopedia of Integer Sequences, https://oeis.org.
V. I. Sushchansky and N. V. Netreba, Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups, Algebra and Discrete Mathematics 1 (2005), 122–132.
H. Strade, Simple Lie Algebras Over Fields of Positive Characteristic. Vol. 1, De Gruyter Expositions in Mathematics, Vol. 38, De Gruyter, Berlin, 2017.
Funding
Open access funding provided by Università degli Studi dell’Aquila within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
All the authors are members of INdAM-GNSAGA (Italy).
R. Civino is funded by the Centre of excellence ExEMERGE at University of L’Aquila, with which also the other authors collaborate.
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution and reproduction in any medium, provided the appropriate credit is given to the original authors and the source, and a link is provided to the Creative Commons license, indicating if changes were made (https://creativecommons.org/licenses/by/4.0/).
About this article
Cite this article
Aragona, R., Civino, R. & Gavioli, N. A modular idealizer chain and unrefinability of partitions with repeated parts. Isr. J. Math. 260, 441–461 (2024). https://doi.org/10.1007/s11856-023-2559-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-023-2559-8