bibkey: bardakov2024simplex authors: V. G. Bardakov; B. B. Chuzhinov; I. A. Emelyanenkov; M. E. Ivanov; T. A. Kozlovskaya; V. E. Leshkov year: 2024 title: “Set-Theoretical Solutions of the n-Simplex Equation” doi: 10.1134/S1055134424010012 url: https://arxiv.org/abs/2206.08906v1 claim: “The n-simplex equation R_1 R_2 … R_(n+1) = R_(n+1) … R_2 R_1 for a map R of X^n acts on X^N, N = n(n+1)/2, with the multi-indices given by the rows of the matrix MI_n; a solution is simple when it has the form (x_1, …, x_n) -> (x_s(1), …, x_s(n)). Question 2 (Question 4.22 of the journal version) asks for which n > 2 there are non-identity permutations without fixed points that give solutions of the n-simplex equation. Section 9.2 defines the first tetrahedral 4-groupoid (T1-groupoid) and the reduced T1-groupoid and states that it is not clear whether some T1-groupoid yields, after forgetting the operations ◁ and ▷, a system that is not a reduced T1-groupoid.” strata_touched:

• D5/S3/StatisticalMechanics/VertexModels/SimplexFixedPointFreePermutations
• D5/S3/StatisticalMechanics/VertexModels/TetrahedralGroupoidReductRefutation license: citation-only triage: anchor

Set-Theoretical Solutions of the n-Simplex Equation

V. Bardakov, B. Chuzinov, I. Emel’yanenkov, M. Ivanov, T. Kozlovskaya and V. Leshkov (arXiv spelling), arXiv:2206.08906v1 [nlin.SI, cross-listed to math-ph and math.GR] (2022); Мат. труды 27(1) (2024) 5–72; Siberian Adv. Math. 34(1) (2024) 1–40. Quotations are from the arXiv source.

The equation (Section 3.1):

In general case, the left side and the right side of the {\SE} are words of length $n+1$. … $R_{\overline{1}}{R}_{\overline{2}}\cdots R_{\overline{n+1}}=R_{\overline{n+1}}\cdots R_{\overline{2}}{R}_{\overline{1}}$, where $\overline{k}=(k_1,k_2,\dots ,k_{n+1})\in {\mathbb{N}}^{n+1}$ is a multi-index.

The multi-indices in the {\SE} can be regarded as rows of the multi-indices matrix $M{I}_n$ that is a $(n+1)\times n$ matrix satisfying the recurrence relation

For $n=3$ the equation is $R_{123}{R}_{145}{R}_{246}{R}_{356}=R_{356}{R}_{246}{R}_{145}{R}_{123}$. Simple solutions (Section 4.2 in both versions):

A solution $T:X^n\to X^n$ of the {\SE} is said to be simple if $T(x_1,\dots ,x_n)=(x_{s(1)},\dots ,x_{s(n)})$, where $s:\{1,\dots ,n\}\to \{1,\dots ,n\}$ is a map (not necessary injective).

The question:

1. Is there a indecomposable simple solution of the {\SE} for some $n$ different from five solutions \{ \id_X, P, Pr^2_1, Pr^2_2, Pr^3_2 \} ? 2) We know that the permutation $P_{12}$ is a solution of the YBE. For which $n>2$ there are non-identity permutations without fixed points that gives solutions of the \SE? Of course, using Proposition \ref{invsymm} it is not difficult to find some transpositions which are solutions of \SE.

The journal version (Мат. труды 27(1), p. 29) states the same question as Вопрос 4.22: “Для каких $n>2$ есть перестановки без неподвижных точек, являющиеся решениями n-SE?”

Elementary 1-solutions of the tetrahedron equation (Section 9.2 of the arXiv source; Section 9.2, pp. 51–53 of Мат. труды 27(1) (2024), with the same axioms):

For studying elementary 1-solutions of the TE let us introduce an algebraic system (X, \os, \oc, \oL, \oR) with four binary operations which satisfy axioms: x \oc y = (x \oL z) \oc (y \oL z); (x \oc y) \os (z \oc w) = (x \os z) \oc (y \os w); (x \oR y) \oR z = (x \oR z) \oR (y \os z); (x \os y) \oL z = x \oR (y \oc z). We call it first tetrahedral 4-groupoid and denote it by $T_1$-groupoid.

There are 2-groupoids, that give elementary 1-solutions. Consider for example a system (X, \os, \oc) with the following axioms: x \oc y = (x \oc z) \oc (y \oc z); (x \os y) \os z = (x \os z) \os (y \os z); (x \oc y) \os (z \oc w) = (x \os z) \oc (y \os w). We will call it a reduced first tetrahedral 4-groupoid or simply reduced $T_1$-groupoid.

It is generally true that if in $T_1$-groupoid x \os y = x \oR y then forgetting about operations x \oL y and x \oR y yields a reduced $T_1$-groupoid. It is not clear if there is a $T_1$-groupoid such that forgetting about x \oL y and x \oR y will not yield a reduced $T_1$-groupoid.

The journal version (p. 53) keeps both sentences: «В общем случае верно, что если в $T_1$-группоиде $x\overline{\ast }y=x\overline{▹}y$, то при отбрасывании операций $x\overline{◃}y$ и $x\overline{▹}y$ получается редуцированный $T_1$-группоид. Не ясно существуют ли такие $T_1$-группоиды, что отбрасывание $x\overline{◃}y$ и $x\overline{▹}y$ не даст редуцированный $T_1$-группоид.»

Verified locator

• DOI: https://doi.org/10.1134/S1055134424010012 (the English journal version; the Crossref record gives the six authors, volume 34, issue 1, pages 1–40; the question was read in the Russian original, Мат. труды 27(1) (2024), p. 29, Вопрос 4.22).
• URL: https://arxiv.org/abs/2206.08906v1 (source ArXiv--Tetrahedral_equation.tex retrieved 2026-10-01): the construction of the equation and the matrix $M{I}_n$ (Section 3.1), the definition of simple solutions and the question (Section 4.2), and the $T_1$-groupoids with the question about their reducts (Section 9.2, lines 2135–2255); the Russian original was read for Section 9.2 (pp. 51–53).