INVOLUTIVE YANG–BAXTER: CABLING, DECOMPOSABILITY, DEHORNOY CLASS

. We develop new machinery for producing decomposability tests for involutive solutions to the Yang–Baxter equation. It is based on the seminal decomposability theorem of Rump, and on “cabling” operations on solutions and their eﬀect on the diagonal map T . Our machinery yields an elementary proof of a recent decomposability theorem of Camp–Mora and Sastriques, as well as original decomposability results. It also provides a conceptual interpretation (using the language of braces) of the Dehornoy class, a combinatorial invariant naturally appearing in the Garside-theoretic approach to involutive solutions.


Introduction
A finite non-degenerate involutive set-theoretic solution to the Yang-Baxter equation, simply called solution in this paper, is a non-empty finite set X endowed with an involutive map r : X ˆX Ñ X ˆX, rpx, yq " pσ x pyq, τ y pxqq, satisfying r 1 r 2 r 1 " r 2 r 1 r 2 , where the maps r i : X 3 Ñ X 3 are defined by r 1 " r ˆId X and r 2 " Id X ˆr, and the maps σ x and τ x are bijective for all x P X.Throughout the paper, we will assume that n " |X| ą 1.The origins, applications and recent results on solutions can be found in the extensive literature which followed [16,19].
A solution is called decomposable if the set X decomposes into two nonempty disjoint parts X " Y \ Z, with rpY ˆY q " Y ˆY and rpZ ˆZq " Z ˆZ.This is equivalent to asking the permutation group GpX, rq of pX, rq, which is the group of permutations on X generated by all the σ x , to be nontransitive [16].A natural approach to the (as for now unattainable) problem of classifying all solutions consists in constructing all indecomposable solutions [29,5,26,27,7,12,13,6,4,28], and then understanding how these building blocks can cement together [25,1,2,8,9].
The first and most famous result on decomposability is the 1996 conjecture of Gateva-Ivanova, proved by Rump in 2005 [23]: a square-free solution (i.e., satisfying rpx, xq " px, xq for all x) is decomposable.In general, a solution needs not be square-free; however, the diagonal map x pxq always defines a permutation on X, satisfying rpT pxq, xq " pT pxq, xq.
The permutation T splits X into orbits.This induces a partition of n " |X|, which we call the T -partition of pX, rq and denote by P T " P T pX, rq.Two recent papers [22,3] revealed that this simple numerical datum may suffice to determine the (in)decomposability of a solution: ‚ P T " p1, . . ., 1q ñ pX, rq decomp.( In this paper, we give a short and elementary proof of the CMS theorem (which was originally proved using advanced group theory), and present several original decomposability results.To explain them, we need the structure group of our solution pX, rq [16].It is defined by the following presentation: GpX, rq " xX | xy " σ x pyqτ y pxq for all x, y P Xy.
It carries a second, commutative operation `, satisfying the following compatibility relation: apb `cq " ab ´a `ac.
Such ring-like structures are called braces.They are extensively used in order to bring ring-theoretic tools into the study of the YBE: see [24] and references thereto.In the present work, we will employ braces in a very different way.Namely, given a positive integer k, consider the map Theorem A. Take a solution pX, rq and a positive integer k.The map ι pkq above is injective.Its image is a sub-solution of GpX, rq.
Here the solution r is extended from X to GpX, rq in the usual way.A push-back through ι pkq then defines a new solution on X, called the k-cabled solution r pkq .Equation (2.2) describes it explicitly.Some relations between a solution and its cablings are given in Theorem B. Take a solution pX, rq and a positive integer k.
(1) The diagonal map of pX, r pkq q is T k .
(2) Let x P X lie in a GpX, rq-orbit of size m, and in a GpX, r pkq q-orbit of size m 1 .Then m 1 is a multiple of the maximal divisor m k of m which is coprime to k: m k | m and gcdpm k , kq " 1.
In particular, if the solution pX, rq is indecomposable and gcdp|X|, kq " 1, then pX, r pkq q remains indecomposable (since |X| k " |X|).Taking k " |T |, we then reduce the CMS theorem to Rump's result.On the other hand, taking k " p, k " a, and k " 2 respectively, we obtain new decomposability theorems: Theorem C. Take an indecomposable solution pX, rq of size pq, where p ‰ q are prime.Then its T -partition cannot contain a term s satisfying pp´1qq ă s ă pq and gcdps, pq " 1.
In parallel with its decomposition into T -cycles, a solution carries several other relevant decompositions: into imprimitivity blocks, and into GpX, r pkq qorbits (for well chosen k).Comparing them, and using the recent classification of primitive solutions from [11], we obtain Theorem F. Take an indecomposable solution pX, rq of size pq, with p ă q prime.Then its T -partition contains either only multiples of q, or at least one multiple of p.
In particular, for an indecomposable solution of size n " 2q with an odd prime q, the only possible T -partition with only odd terms is pq, qq.
More generally, Theorem B allows one to considerably reduce the list of possible T -partitions for indecomposable solutions.This has the potential to speed up algorithms constructing all indecomposable solutions of small size.
In another vein, cabling can produce new indecomposable solutions out of old ones: see Example 4.2.
In the final part of the paper, cabling and brace ideas are used to explore an important invariant of a solution pX, rq, which we propose to call its Dehornoy class.It is the smallest positive integer m such that (1.2) @x P X, σ T m´1 pxq ¨¨¨σ T pxq σ x " Id.
Such an m always exists, and is ă pn 2 q!.The elements mx, x P X, then generate a normal free abelian subgroup of GpX, rq of finite index.The corresponding finite quotient plays the same role as Coxeter groups play for Artin groups.In particular it suffices for reconstructing the Garside structure on the whole GpX, rq.For details, see [14,15].A partial generalisation to non-involutive solutions is proposed in [21].Now, the permutation group GpX, rq inherits the brace structure from GpX, rq [10].We then give a new conceptual interpretation of the Dehornoy class in terms of the abelian group pGpX, rq, `q: Theorem G.The Dehornoy class m of a solution pX, rq is the least common multiple of the orders of the generators σ x , x P X, of the group pGpX, rq, `q.If the solution is indecomposable, m is the order of any σ x .
Another type of problems where cabling can be useful is the structural study of braces.Since these questions are out of the focus of the present work, we simply illustrate this approach with a quick proof of two important properties of finite braces at the end of Section 3.

Cabling a solution
In this section we prove Theorem A. Recall that, with respect to the operation `, GpX, rq is a free abelian group, and the elements x P X yield its basis [24].Therefore the map ι pkq is injective.
To get the second assertion of the theorem, we will prove an explicit formula for the extension R of r to GpX, rq: Rpkx, lyq " plσ kx pyq, kT k´1 τ ly T ´k`1 pxqq, where k and l are positive integers.This yields (2.2) r pkq px, yq " pσ kx pyq, T k´1 τ ky T ´k`1 pxqq, and finishes the proof of Theorem A.
Recall that the operation `on GpX, rq is a natural extension of the law x `y " xσ ´1 x pyq, x, y P X.
In particular, x pxq is the inverse of the diagonal map T .One recognises the frozen words from [14] (for k " 2), and the twisted powers x rks from [15] (for general k).
Let us look at (2.4) r k,l p xU pxqU 2 pxq ¨¨¨U k´1 pxq , yU pyqU 2 pyq ¨¨¨U l´1 pyq q " pu, wq, where the tuples px 1 , . . ., x s q are denoted by x 1 ¨¨¨x s for simplicity, and the solution r is extended, this time, to the powers of X: r k,l " pr l ¨¨¨r 1 q ¨¨¨pr k`l´2 ¨¨¨r k´1 qpr k`l´1 ¨¨¨r k q : X k ˆXl Ñ X l ˆXk .
These maps induce the solution R on GpX, rq, as explained in [18] with an inductive argument, and in [20] with a graphical argument.Both entries in (2.4) are frozen tuples, that is, they remain unchanged when r is applied to any neighbouring positions, since rpz, U pzqq " pz, U pzqq for all z P X.But the YBE for r guarantees that Here r i is the solution r applied at the positions i and i `1 of a tuple.As a result, u and w are also frozen: u " y 1 U py 1 q ¨¨¨U l´1 py 1 q " ly 1 , w " x 1 U px 1 q ¨¨¨U k´1 px 1 q " kx 1 .

Properties of cabled solutions
In this section we first prove Theorem B and then turn to other properties of cabled solutions.
Proof of Theorem B. For all positive integer k and x P X, the tuple xU pxqU 2 pxq ¨¨¨U 2k´1 pxq P X 2k is frozen.Since applying the solution r k,k to a 2k-tuple boils down to applying r repeatedly at different positions, one gets In other words, Rpkx, kU k pxqq " pkx, kU k pxqq, that is, Since T is the inverse of U , this yields r pkq pT k pxq, xq " pT k pxq, xq.Therefore T k is the diagonal map for r pkq .Now, let x P X lie in the GpX, rq-orbit of size m and in the GpX, r pkq qorbit of size m 1 .Denote by GpX, rq x and GpX, r pkq q x its stabilisers in the two groups.One has |GpX, rq| " m|GpX, rq x |, |GpX, r pkq q| " m 1 |GpX, r pkq q x |.
The permutation groups GpX, rq and GpX, r pkq q inherit brace structures from the corresponding structure groups GpX, rq and GpX, r pkq q (see [10]).Moreover, the abelian group pGpX, r pkq q, `q is obtained from the abelian group pGpX, rq, `q by multiplying each of its generators σ x , x P X, by k.Thus its size is the size of pGpX, rq, `q divided by some product p d 1 1 ¨¨¨p d l l of powers of prime divisors of k.Also, since the permutation group GpX, r pkq q is the subgroup of GpX, rq generated by the permutations σ kx , x P X, the stabiliser GpX, r pkq q x is a subgroup of GpX, rq x .Hence |GpX, r pkq q x | " |GpX, rq x | { t for some positive integer t.Summarising, we obtain Since this fraction is an integer, it is a multiple of an integer of the form m p Recalling that the p i are prime divisors of k, we see that is a multiple of the maximal divisor m k of m which is coprime to k, as announced.
Proposition 3.1.The iteration of cablings remains a cabling.More precisely, given a solution pX, rq and positive integers k and k 1 , one has pr pkq q pk 1 q " r pkk 1 q .
Recall the relation x T connecting σ's and τ 's (see for instance [16,Proposition 2.2]).It implies that the σ-component uniquely determines a solution.We are done.
Recall that a solution pX, rq is called retractable if for some x ‰ x 1 P X, one has σ x " σ x 1 (and hence τ x " τ x 1 ).Identifying all such x and x 1 , one gets the retraction RetpX, rq of pX, rq; it is a solution again, as explained in [16].This is an important property of solutions: see [17] and references thereto.
Proposition 3.2.If a solution pX, rq is retractable, then so are all its cablings.More precisely, RetpX, r pkq q is a quotient of RetpX, rq pkq for all positive integers k.
Proof.Using the brace structure GpX, rq inherits from GpX, rq, we can write σ kx " kσ x .Thus the relation σ x " σ x 1 implies σ kx " σ kx 1 .From (2.2), one then concludes that elements x and x 1 identified in RetpX, rq are necessarily identified in RetpX, r pkq q as well.
Until now, all connections between solutions and braces that we used went through the brace structures on the structure and permutation groups of a solution.But one can go the other way round, and define a solution on any brace [24].This gives one the intuition on how to cable a brace.Concretely, take a brace pB, `, ˝q and a positive integer k.The elements ka, a P B, form a sub-brace B pkq of B, called its k-cabling.Indeed, we have ka `kb " kpa `bq, ka ˝kb " kppkaq ˝b ´pk ´1qaq, as follows from the commutativity of `and from relation (1.1) respectively.The additive structure of B pkq is obtained from pB, `q by multiplication by k.One can thus easily determine its size.The multiplicative group pB, ˝q then has a subgroup of the same size.Here are two direct applications: (1) A quick proof of the solvability of the multiplicative group of a finite brace (first established in [16,Theorem 2.15]).Indeed, let pB, `, ˝q be a brace of size ab with gcdpa, bq " 1.Looking at the additive structure, one sees that B paq is of size b.Therefore pB paq , ˝q is a b-Hall subgroup of pB, ˝q.Thus pB, ˝q is solvable.
(2) Let B be a finite brace with cyclic additive group, and d a divisor of its size |B|.Then pB, ˝q contains a subgroup of size d.Indeed, looking at the additive structure and using the cyclicity of pB, `q, one sees that B p|B|{dq is of size d.

Applications: (in)decomposability results
We now turn to applications of Theorem B. Its assertion is particularly transparent when the solution pX, rq is indecomposable, and the cabling parameter k is coprime to its size |X|, which is now the size of the only GpX, rq-orbit.Since |X| k " |X|, the theorem implies that the solution pX, r pkq q remains indecomposable, with diagonal map T k .Here are some interesting particular cases.
(1) If gcdp|T |, |X|q " 1, then pX, r p|T |q q has to be indecomposable, with diagonal map Id, which is impossible by Rump's theorem.We thus recover the Camp-Mora-Sastriques (CMS) theorem.(2) If the cycle decompositions of T and T k are different, we get a new indecomposable solution on the same set X.For instance, if pX, rq is the indecomposable solution with T -partition p2, 6q (cf.[22, Table 3.2]), we have |X| " 2 `6 " 8, which is comprime with k " 3. Then pX, r p3q q is an indecomposable solution with T -partition p2, 2, 2, 2q, and hence not isomorphic to pX, rq.
To treat other cases, we need the following elementary observation.
Lemma 4.1.Given a solution pX, rq with diagonal map T , any T -orbit in X lies entirely within a single GpX, rq-orbit.
Proof.Take an element x P X from a GpX, rq-orbit Y .By [16], r restricts to Y ˆY and defines a solution on Y .The diagonal map of this restricted solution has to be the restriction of T to Y .Thus the T -orbit of x lies entirely within Y .Now, take an indecomposable solution pX, rq and a cabling parameter k which is not coprime to |X|, but which makes |X| k big enough.Then the sizes of all GpX, r pkq q-orbits are multiples of |X| k .On the other hand, by the above lemma, all the T k -orbits lie entirely inside these GpX, r pkq q-orbits.In several cases, for numerical reasons, this can happen only when there is only one GpX, r pkq q-orbit.The solution pX, r pkq q is then indecomposable, which imposes some constraints on the sizes of the T k -orbits, for instance by the CMS theorem.This leads to a contradiction in various cases which are not themselves covered by the CMS theorem.Here are some of them.
(1) Take an indecomposable solution pX, rq of size pq, where p ‰ q are primes.Assume that a T -orbit is of size pp ´1qq ă s ă pq, with gcdps, pq " 1.We will show that this is impossible, and thus prove Theorem C. For any t P N, the diagonal map T p t of pX, r pp t q q inherits this orbit, since gcdps, pq " 1.Thus this T p t -orbit of size s lies entirely within a GpX, r pp t q q-orbit, whose size is a multiple of |X| p t " q.Since pp ´1qq ă s ă pq, this GpX, r pp t q q-orbit has to be the whole set X.In other words, the p t -cabled solution pX, r pp t q q is indecomposable.But, for t big enough, the sizes of all T p t -orbits are coprime to p.But they are also coprime to q since there is one orbit of size pp ´1qq ă s ă pq and several smaller orbits of total size pq ´s ă q.As a consequence, gcdp|X|, |T p t |q " gcdppq, |T p t |q " 1.
By the CMS theorem, the solution pX, r pp t q q is then decomposable, contradiction.
(2) Take an indecomposable solution pX, rq of size ab and T -partition pa, c, c 1 q, where b ą a `c, and the numbers a, b, c, c 1 are pairwise coprime, except for, possibly, c and c 1 .We will show that this is impossible, and thus prove Theorem D. The a-cabling of pX, rq has T -partition pc, c 1 , 1, . . ., 1q, with a ones.Since gcdp|X|, |T a |q divides gcdpab, cc 1 q " 1, the CMS theorem says that pX, r paq q is decomposable, and there are at least two GpX, r paq q-orbits.One of them does not contain the T a -orbit of size c 1 , hence its size is ď c `a ă b, which is impossible for a multiple of |X| a " |ab| a " b. (3) Take an indecomposable solution pX, rq of size 2d, with d odd, and T -partition p2a, b, cq, where gcdp2d, abcq " 1 and b ď c.We will show that it imposes heavy restrictions on a, b, c, and thus prove Theorem E. The 2-cabling of pX, rq has T -partition pa, a, b, cq, since b and c are odd.The sizes of its GpX, r p2q q-orbits are multiples of p2dq 2 " d, as d is odd.Since gcdp|X|, |T 2 |q divides gcdp2d, abcq " 1, the CMS theorem says that pX, r p2q q is decomposable, so there are precisely two GpX, r p2q q-orbits, each of size d.Each of the four T 2orbits lies entirely in one of these two GpX, r p2q q-orbits.Since the numbers a, b, c, d are all odd, this is possible only if d " 2a `b " c. (4) Assume that pX, rq is an indecomposable solution of size 30.We will show that its T -partition cannot be p21, 7, 1, 1q.Indeed, the 3cabled solution pX, r p3q q would then have T -partition p7, 7, 7, 7, 1, 1q, and be decomposable by the CMS theorem.On the other hand, its GpX, r p3q q-orbits are multiples of 30 3 " 10.But the only way to divide the multiset p7, 7, 7, 7, 1, 1q into parts whose total sums are all divisible by 10 is to take the whole multiset.Thus the solution pX, r p3q q is indecomposable, contradiction.As in the above situations, this example generalises to an infinite family.In another vein, cabling can produce new indecomposable solutions out of old ones.

Primitivity and further (in)decomposability results
A solution pX, rq is called imprimitive if the GpX, rq-action on X is so, and primitive otherwise.That is, an imprimitive solution X admits a non-trivial decomposition into blocks which is preserved by the GpX, rq-action.A recent result from [11] asserts that, up to isomorphism, the only primitive solutions are the permutation solutions pZ{pZ, rpa, bq " pb ´1, a `1qq, with p prime.By [16], these are the only indecomposable solutions of prime size.Thus, in the interesting case of non-prime size, an indecomposable solution can be split into imprimitivity blocks.Their interaction with T -cycles is quite intricate.We will now analyse this interaction in the particular settings of Theorem F, and deduce a proof of that theorem.
Consider an indecomposable solution pX, rq of size pq, with primes p ă q.Assume that its T -partition contains no multiples of p, and at least one term which is not a multiple of q.We will obtain a contradiction, proving Theorem F.
By Theorem B, one can choose a suitable k coprime with pq such that the solution pX, r pkq q is still indecomposable, has T -partition with all terms of the form p α q β , and permutation group GpX, r pkq q of size p a q b .(For the latter property, recall that the k-cabling multiplies all the elements of pGpX, rq, `q by k.) Since the cabling can only split T -orbits into equal parts, the Tpartition of pX, r pkq q still contains no multiples of p, and at least one term which is not a multiple of q.Thus it suffices to work with solutions having these properties.
Summarizing all the constraints on the T -partition we obtained, one sees that it has to be of the form pq, . . ., q, 1, . . ., 1q, with at least one term 1 and one term q (otherwise Rump's theorem applies).
Since pq is not prime, our solution is imprimitive.Thus X non-trivially decomposes into blocks preserved by the GpX, rq-action.Since pX, rq is indecomposable, GpX, rq permutes these blocks in a transitive manner, hence they are all of the same size.This leaves us with two possibilities.
Case 1: There are p blocks of size q.Since our solution is indecomposable, some map σ x permutes 1 ă p 1 ď p blocks in a cyclic manner.It thus has an orbit of size p 1 q 1 , with 1 ď q 1 ď q.Since this size is of the form p α q β (the group GpX, rq having the size of this form), and since p ă q are primes, one necessarily has p 1 " p.Thus σ x permutes all the p blocks in a cyclic manner.As a result, x and U pxq " σ ´1 x pxq lie in different blocks.Since U " T ´1, one obtains a T -cycle which does not entirely lie in a single block.Now, again by Theorem B, one can choose a suitable m such that the solution pX, r pp m q q is decomposable, with orbits whose sizes are multiples of q.The permutation group GpX, r pp m q q is a subgroup of the group GpX, rq of size p a q b .Its size, as well as the sizes of all the GpX, r pp m q q-orbits, are then of the same form.Being multiples of q, the sizes of the GpX, r pp m q q-orbits are then all precisely q.One of them has to entirely contain our T -cycle of size q (which is also a T pp m q -cycle).This GpX, r pp m q q-orbit then intersects several blocks.Since the subgroup GpX, r pp m q q of GpX, rq permutes these blocks, our GpX, r pp m q q-orbit has to be of size p 1 q 1 , with 1 ă p 1 ď p and 1 ď q 1 ď q.But q cannot be written in this way.
Case 2: There are q blocks of size p.The permutations τ x of X, for x P X, generate a transitive group, since so do the σ x , and the two are related by the conjugation by T (see relation (3.1)).Therefore some element f P X fixed by T is moved to an element c from a T -cycle of size q by some τ x .That is, c " τ x pf q.We will use the relation T pτ x pf qq " τ σ f pxq pf q from [22, Lemma 3.8].Applied k times, it yields As a result, the size of the σ f -orbit containing x is a multiple of the size of the T -orbit containing c, which is q.Since p ă q, it intersects q 1 ą 1 blocks of size p.On the other hand, since σ f fixes f , it fixes at least one block.Thus q 1 ă q.Summarizing, the size q of our orbit decomposes as p 1 q 1 , with 1 ď p 1 ď p and 1 ă q 1 ă q.But this is impossible.
Remark 5.1.Along the lines of the proof of [22,Lemma 3.8], one can establish the relation T k pτ x pyqq " τ σ ky pxq pT k pyqq, valid for all x, y P X (not necessarily T -fixed) and all k.

Applications: Dehornoy class
In this section, we will prove Theorem G.The main ingredient is Lemma 6.1.For all elements x from the same GpX, rq-orbit of a solution pX, rq, the order of σ x in the finite abelian group pGpX, rq, `q the same.
Thus lσ y vanishes if and only if lσ σxpyq does.As a consequence, σ y and σ σxpyq have the same order in pGpX, rq, `q for all x, y P X.
Remark 6.2.Relation (6.1) means that the cabling operation ι plq : x Þ Ñ lx is equivariant with respect to the left GpX, rq-actions induced by the maps σ x .It thus behaves better than the diagonal map T , which instead of the equivariance obeys the less tractable rule (3.1).
Proof of Theorem G. Relation (1.2) can be rewritten as @x P X, σ x σ U pxq ¨¨¨σ U m´1 pxq " Id, which, by (2.3), simply means mσ x " 0. This yields the first assertion of the theorem.The second then directly follows from Lemma 6.1.
We finish with the following observation, relating the Dehornoy class of a solution to its diagonal map: Proposition 6.3.Let pX, rq be a solution.Then the order |T | of its diagonal map divides its Dehornoy class m.
Proof.We need to prove the relation T m " Id, or, equivalently, U m " Id.Let us compute (6.2) r m,1 pxU pxqU 2 pxq ¨¨¨U m´1 pxq, U m pxqq in two ways.On the one hand, the definition of the Dehornoy class allows one to simplify (6.2) as r m,1 pmx, U m pxqq " pσ mx pU m pxqq, ¨q " ppmσ x qpU m pxqq, ¨q " pU m pxq, ¨q.
Hence U m pxq " x for all x P X.

Theorem D .
Take an indecomposable solution pX, rq of size ab and Tpartition pa, c, c 1 q, where the numbers a, b, c, c 1 are pairwise coprime, except for, possibly, c and c 1 .Then one cannot have b ą a `c.As a consequence, indecomposable solutions pX, rq of size 2b with odd b ě 5 cannot have T -partition p2, b ´4, b `2q.Theorem E. Take an indecomposable solution pX, rq of size 2d, with d odd, and T -partition p2a, b, cq, where gcdp2d, abcq " 1 and b ď c.Then 2a`b " c.