Existence of simple involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation of every cardinality $m^2 n$

For finite involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation, the simple solutions are the indecomposable building blocks from which all others are assembled by extensions, within Drinfeld's 1992 program of classifying all such solutions. It is known that no simple solution can have composite square-free cardinality, while simple solutions exist for every prime cardinality, every square $n^2$, every prime power $p^n$ with $n>1$, and every cardinality $p^2 n$ for which the prime $p$ divides $q-1$ for each prime divisor $q$ of $n$, plus sporadic examples of cardinalities 12 and 27. Whether a simple solution exists for every cardinality of the form $m^2 n$ with integers $m,n>1$ — the generic non-square-free shape — was claimed by Cedó and Okniński in 2021, retracted in a 2024 corrigendum, and is explicitly recorded as open in their 2024/2026 work. In particular, concrete cardinalities such as 18, 24, and 45 lie outside all verified families.

Background

The Yang–Baxter equation (YBE) asks for a linear map $R:V\otimes V\to V\otimes V$ satisfying $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$; it originated in statistical mechanics and in the theory of quantum groups. Drinfeld (1992) proposed studying its set-theoretic solutions: pairs $(X,r)$ where $X$ is a set and $r:X\times X\to X\times X$ is a bijection satisfying the same braid relation $r_{12}r_{23}r_{12}=r_{23}r_{12}r_{23}$, where $r_{12},r_{23}$ act as $r$ on the indicated pair of factors of $X^3$. Writing $r(x,y)=(\sigma_x(y),\gamma_y(x))$, a solution is called non-degenerate if all maps $\sigma_x,\gamma_y:X\to X$ are bijections, and involutive if $r^2=\mathrm{id}$. The finite involutive non-degenerate solutions form the most developed branch of the theory, initiated by Etingof–Schedler–Soloviev (1999), who introduced the structure group $G(X,r)$ (generated by the elements of $X$ subject to the relations $xy=\sigma_x(y)\gamma_y(x)$) and proved, in particular, that up to isomorphism there is a unique indecomposable solution of each prime cardinality.

Several structural notions organize the classification. A solution is indecomposable if $G(X,r)$ acts transitively on $X$. The retract identifies $x,y\in X$ whenever $\sigma_x=\sigma_y$, producing a quotient solution; a solution equal to its own retract is irretractable, and one whose iterated retracts eventually reach cardinality 1 is a multipermutation solution. Following Vendramin and Castelli, a solution $(X,r)$ with $|X|>1$ is simple if every epimorphism of solutions $f:(X,r)\to(Y,s)$ is either an isomorphism or satisfies $|Y|=1$. Simple solutions matter because every finite involutive non-degenerate solution is built from simple ones by iterated dynamical extensions, so they are the atoms of Drinfeld's classification problem. A basic lemma (Cedó–Okniński): a finite simple solution is indecomposable if $|X|>2$ and irretractable if $|X|$ is not prime. Rump (2007) introduced left braces — sets $B$ carrying an abelian group operation $+$ and a second group operation $\circ$ linked by $a\circ(b+c)=a\circ b-a+a\circ c$ — which encode involutive non-degenerate solutions; the permutation group $\mathcal{G}(X,r)$ (generated by the $\sigma_x$) is naturally a left brace whose ideals control the congruences of $(X,r)$, and this is the main technical tool for proving simplicity.

The cardinality problem asks which finite cardinalities occur as sizes of simple solutions. The known obstruction is sharp: no indecomposable irretractable solution has square-free cardinality (Cedó–Okniński, Adv. Math. 430 (2023)); since a finite simple solution of composite cardinality is indecomposable and irretractable by the basic lemma above, no simple solution has composite square-free cardinality, while prime cardinalities are exempt and occur uniquely. So $m^2n$ is precisely the generic cardinality shape not excluded. On the positive side: each prime cardinality occurs (uniquely, by Etingof–Schedler–Soloviev); every square $n^2$ occurs (Cedó–Okniński 2021, explicit solutions on $(\mathbb{Z}/(n))^2$); every prime power $p^n$ with $n>1$ occurs (Cedó–Okniński 2024/2026); cardinality $p^2n$ occurs under the congruence condition that $p$ divides $q-1$ for every prime divisor $q$ of $n$ (Cedó–Okniński, J. Pure Appl. Algebra 229 (2025)); cardinality $p^2$ is completely classified (Dietzel–Properzi–Trappeniers, Comm. Algebra 53 (2025): indecomposable plus irretractable already forces simplicity); and Castelli (Proc. Amer. Math. Soc. 151 (2023)) gave sporadic examples of cardinalities 12 and 27. In 2021 Cedó and Okniński claimed the general case — an explicit simple solution of cardinality $m^2n$ for all $m,n>1$ (Theorem 4.12 of their Advances in Mathematics paper) — but the proof was incorrect and the claim was retracted in a 2024 corrigendum, which does however give one example of a simple solution of non-square cardinality. Both of their subsequent papers state the general existence question as open, and it is the natural completion of the cardinality picture.

Problem Statement

For arbitrary integers $m,n>1$, does there exist a finite simple involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang–Baxter equation with $|X|=m^2n$? Here 'simple' means: $|X|>1$ and every epimorphism of solutions $f:(X,r)\to(Y,s)$ is either an isomorphism or satisfies $|Y|=1$ (Definition 1.1 of Cedó–Okniński, Publ. Mat. 70 (2026)), a notion that coincides with Vendramin's simplicity for finite indecomposable solutions. This is the open problem recorded explicitly by Cedó and Okniński after the retraction of their 2021 Theorem 4.12, which had claimed such solutions for all $m,n>1$. Because simple solutions of every prime, prime-power, and square cardinality already exist, an affirmative answer would establish that every non-square-free cardinality — equivalently, every cardinality not excluded by the square-free obstruction — admits a simple solution, completing the determination of which finite cardinalities occur; a negative answer for some $m^2n$ would reveal the first cardinality obstruction beyond square-freeness. Concretely, no construction from the verified infinite families covers cardinalities such as $18=3^2\cdot 2$ or $45=3^2\cdot 5$, which fail the congruence condition of the strongest partial result (J. Pure Appl. Algebra 229 (2025), Theorem 5.3).

References

1. F. Cedó and J. Okniński, Simple solutions of the Yang–Baxter equation of cardinality $p^n$, Publicacions Matemàtiques 70 (2026), no. 2, 303–313. DOI: 10.5565/PUBLMAT7022601 (preprint arXiv:2407.07907). https://doi.org/10.5565/PUBLMAT7022601
2. F. Cedó and J. Okniński, New simple solutions of the Yang–Baxter equation and their permutation groups, Journal of Pure and Applied Algebra 229 (2025), no. 9, Article 108025. DOI: 10.1016/j.jpaa.2025.108025 (preprint arXiv:2401.12904). https://doi.org/10.1016/j.jpaa.2025.108025
3. F. Cedó and J. Okniński, Constructing finite simple solutions of the Yang–Baxter equation, Advances in Mathematics 391 (2021), 107968. DOI: 10.1016/j.aim.2021.107968 (preprint arXiv:2012.08400). https://doi.org/10.1016/j.aim.2021.107968
4. F. Cedó and J. Okniński, Corrigendum to "Constructing finite simple solutions of the Yang-Baxter equation" [Adv. Math. 391 (2021) 107968], Advances in Mathematics 451 (2024), 109807. DOI: 10.1016/j.aim.2024.109807. https://doi.org/10.1016/j.aim.2024.109807
5. M. Castelli, A characterization of finite simple set-theoretic solutions of the Yang–Baxter equation, Proceedings of the American Mathematical Society 151 (2023), 5047–5057. DOI: 10.1090/proc/16329 (preprint arXiv:2203.16693). https://doi.org/10.1090/proc/16329
6. C. Dietzel, S. Properzi and S. Trappeniers, Indecomposable involutive set-theoretical solutions to the Yang–Baxter equation of size $p^2$, Communications in Algebra 53 (2025), no. 3, 1238–1256. DOI: 10.1080/00927872.2024.2405024 (preprint arXiv:2403.18653). https://doi.org/10.1080/00927872.2024.2405024
7. M. Bonatto and M. Castelli, Involutive (simple) latin solutions of the Yang–Baxter equation and related (left) quasigroups, preprint (2025), arXiv:2501.03660. https://arxiv.org/abs/2501.03660
8. F. Cedó and J. Okniński, Indecomposable solutions of the Yang–Baxter equation of square-free cardinality, Advances in Mathematics 430 (2023), 109221. DOI: 10.1016/j.aim.2023.109221. https://doi.org/10.1016/j.aim.2023.109221
9. V. G. Drinfeld, On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics 1510, Springer, Berlin, 1992, pp. 1–8. DOI: 10.1007/BFb0101175. https://doi.org/10.1007/BFb0101175
10. P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Mathematical Journal 100 (1999), 169–209. DOI: 10.1215/S0012-7094-99-10007-X. https://doi.org/10.1215/S0012-7094-99-10007-X
11. W. Rump, Braces, radical rings, and the quantum Yang–Baxter equation, Journal of Algebra 307 (2007), 153–170. DOI: 10.1016/j.jalgebra.2006.03.040. https://doi.org/10.1016/j.jalgebra.2006.03.040
12. M. Castelli, On the indecomposable involutive solutions of the Yang–Baxter equation of finite primitive level, Publicacions Matemàtiques 69 (2025), no. 2, 429–444. DOI: 10.5565/PUBLMAT6922509 (preprint arXiv:2311.01114). https://doi.org/10.5565/PUBLMAT6922509
13. I. Colazzo, E. Jespers, Ł. Kubat and A. Van Antwerpen, Simple solutions of the Yang-Baxter equation, preprint (2023–2025), arXiv:2312.09687. https://arxiv.org/abs/2312.09687