Finite orbits of the pure braid group on the monodromy of the 2-variable Garnier system

Journal of Integrable Systems (2018) 3, 1–35 doi: 10.1093/integr/xyy005 Finite orbits of the pure braid group on the monodromy of the 2-variable Garnier system P. Calligaris Department of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK and M. Mazzocco† School of Mathematics, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK †Corresponding author. Email: [email protected] Communicated by: Nalini Joshi [Received on 26 January 2018; editorial decision on 28 April 2018; accepted on 23 May 2018] In this article, we realize the SL2(C)character variety of he Riemann sphere 5with five boundary components as a 5-parameter family of affine varieties of dimension 4. We show that the action of the mapping class group corresponds to certain action of the braid group on this family of affine varieties and classify exceptional finite orbits. This action represents the nonlinear monodromy of the 2 variable Garnier system and finite orbits correspond to its algebraic solutions. Keywords: Braid group; Garnier system; Character variety. 1. Introduction The Garnier system G2is the isomonodromy deformation of the following two-dimensional Fuchsian system: d dλ=4  k=1 Ak λ−ak,λ∈C, (1) a1,...,a4, being pairwise distinct complex numbers. The residue matrices Ajsatisfy the following conditions: eigen Aj=±θj 2and − n+2  k=1 Ak=A∞, where θj∈C,j=1, ..., 4 and we assume A∞:=1 2θ∞0 0−θ∞, with θ∞∈C\{0}. © The Author(s) 2018. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 2P. CALLIGARIS AND M. MAZZOCCO Fig. 1. The basis of loops for π1(5). The Riemann–Hilbert correspondence associates to each Fuchsian system (1) its monodromy representation class, or in other words, a point in the moduli space of rank two linear monodromy representations over the two-dimensional sphere 5with five boundary components: MG2:=Hom(π1(5),SL 2(C))/SL2(C), also called SL2(C)character variety of 5. After fixing a basis of oriented loops γ1,...,γ4,γ∞for π1(5)such that γ−1 ∞=γ1···γ4, as in Fig. 1, an equivalence class of an homomorphism in the character variety MG2is determined by the five matrices M1,...,M4,M∞∈SL2(C), that are images of γ1,...,γ4,γ∞. These matrices must satisfy the relation: M∞M4M3M2M1=I.(2) In this article, we assume that M∞is diagonalizable: eigen(M∞)=e±πiθ∞. As a consequence the character variety MG2is identified with the quotient space ˆ MG2, defined as: ˆ MG2:=(M1,...,M4)∈SL2(C)|eigen(M4M3M2M1)=e±πiθ∞/∼, (3) where ∼is equivalence up to simultaneous conjugation of M1,...,M4by a matrix in SL2(C). As the pole positions a1,...,a4in (1) vary in the configuration space of 4 points, the monodromy matrices M1,...,M4of (1) remain constant if and only if (see [1]) the residue matrices A1,...,A4are solutions of the Schlesinger equations [2] which in the 2 ×2 case reduce to the Garnier system G2[3,4]. The structure of analytic continuation of the solutions of the Garnier system is described by a certain action of the pure braid group P4[5] (see also [6]) that can be deduced from the following action of the braid group B4: B4׈ MG2−→ ˆ MG2,(4) defined in terms of the following generators: σ1:(M1,M2,M3,M4)→ (M2,M2M1M−1 2,M3,M4), σ2:(M1,M2,M3,M4)→ (M1,M3,M3M2M−1 3,M4), σ3:(M1,M2,M3,M4)→ (M1,M2,M4,M4M3M−1 4), (5) so that M∞is preserved. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 3 Our aim in this article is to classify the finite orbits of this action. In our classification, we exclude the case when the monodromy group M1,...,M4is reducible because in this case the Garnier system for which algebraic solutions are classified in [7] (indeed in this case the Garnier system can be solved in terms of Lauricella hypergeometric functions [8]), and the case in which one of the monodromy matrices is a root of the identity because in this case the Garnier system reduces to the sixth Painlev´ e equation [8] for which all algebraic solutions are classified in [9]. Therefore, we restrict to the following open set: U=(M1,...,M4)∈ˆ MG2|M1,...,M4irreducible, (6) Mi=±I,∀i=1, ...,4,∞/∼, To explain our classification result, we firstly identify the open set Uwith an affine variety (see Section 2): Lemma 1.1 Let the functions pi,pij,pijk be defined as: pi=Tr Mi,i=1, ...,4, pij =Tr MiMj,i,j=1, ...,4, i>j, pijk =Tr MiMjMk,i,j,k=1, ...,4, i>j>k, p∞=Tr M4M3M2M1, (7) then, for every choice of p1,...,p4,p∞, the open set of monodromy matrices Uis isomorphic to a four dimensional affine variety: A:=C[p21,p31,p32,p41,p42,p43,p321,p432,p431,p421]/I, (8) where Iis the ideal generated by the algebraically dependent polynomials f1,...,f15 defined in (47)–(61). Therefore, we think of p1,...,p4,p∞as a set of parameters and of pij,pijk as an over-determined system of coordinates on U, and we express the action (4) in terms of pi,pij,pijk as follows (see Section 3): Lemma 1.2 The following maps σi:A−→ A,i=1, 2, 3, acting on the coordinates p:=(p1,p2,p3,p4,p∞,p21,p31,p32,p41,p42,p43,p321,p432,p431,p421)∈C15, (9) as follows: σ1:p→ (p2,p1,p3,p4,p∞,p21,p32,p1p3−p31 −p21p32 +p2p321,p42, p1p4−p41 −p21p42 +p2p421,p43,p321,p1p43 −p431 −p21p432 +p2p∞, p432,p421), σ2:p→ (p1,p3,p2,p4,p∞,p31,p1p2−p21 −p31p32 +p3p321,p32,p41,p43, p2p4−p42 −p32p43 +p3p432,p321,p432,p2p41 −p421 −p32p431 +p3p∞, p431), σ3:p→ (p1,p2,p4,p3,p∞,p21,p41,p42,p1p3−p31 −p41p43 +p4p431, p2p3−p32 −p42p43 +p4p432,p43,p421,p432,p431, p21p3−p321 −p421p43 +p4p∞), (10) define an action of the braid group B4on A. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 4P. CALLIGARIS AND M. MAZZOCCO Therefore, our problem is to find all points p∈Asuch that their orbit under the action of the pure braid group P4induced by the action (10) of the braid group B4is finite. Incidentally, the action (10) can also be interpreted as the Mapping Class Group action on the character variety MG2. Our approach is based on the simple observation that given p∈Asuch that it generates a finite orbit under the action of the pure braid group P4, then for any subgroup H⊂P4the action of Hover p∈A must also produce a finite orbit (this is a well-known fact, see for example [7]). We select four subgroups H1,H2,H3,H4⊂P4such that the restricted action is isomorphic to the action of the pure braid group P3 on the SL2(C)character variety of the Riemann sphere with four boundary components MPVI that can be identified with: ˆ MPVI :={(N1,N2,N3)∈SL2(C)|N∞N3N2N1=I, N∞=exp(iπθ∞σ3),θ∞∈C}/∼. (11) In other words, we show that in order for a point p∈Ato belong to a finite orbit of the pure braid group P4, it must have four projections on points q=(q1,q2,q3,q∞,q21,q31,q32)that have a finite orbit under the pure braid group P3. We then invert this way of thinking: since all finite orbits of the pure braid group P3on q= (q1,q2,q3,q∞,q21,q31,q32)have been classified in Lisovyy and Tykhyy’s work [9], we start from their list and reconstruct candidate points p ∈Athat satisfy the necessary conditions to belong to a finite orbit. We then classify all candidate points that indeed produce finite orbits. In order to avoid redundant solutions to this classification problem, we introduce the symmetry group Gof the affine variety (8) and factorize our classification modulo the action of G. The action of the symmetry group Gon Ais calculated in the Appendix using known results about B¨ acklund transformations of Schlesinger equations [10]. In order to produce our candidate points we use the classification result in [9] that shows that there are four types of finite orbits of the braid group B3: (1) Fixed points corresponding to Okamoto’s Riccati solutions [11]. (2) Dubrovin–Kitaev orbits, corresponding to algebraic solutions of type II, III and IV in [9]. (3) Picard orbits, corresponding to algebraic solutions obtained in terms of the Weierstrass elliptic function (see [12,13]). (4) 45 exceptional finite orbits [9]. Remark 1.1 Orbits of type II, III and IV in [9] where first obtained by Dubrovin in [14]. Later Kitaev showed that these solutions satisfy parametric families of the sixth Painlev´ e equations and re-obtained them by the pull-back of the hypergeometric equation (see [15,16]). To keep down the number of pages and of technical lemmata, we restrict our classification to exceptional orbits, namely orbits for which the corresponding monodromy group is not reducible, none of the monodromy matrices is a multiple of the identity and at most one projection giving either a Dubrovin– Kitaev or a Picard orbit is allowed. Therefore, our classification does not include the solutions found by Tsuda [17] by calculating fixed points of bi-rational canonical transformations, nor the ones found by Diarra in [18] using the method of pull-back introduced in [15,16], nor the ones found in [7] as they correspond to reducible monodromy groups, nor the families of algebraic solutions obtained by Girand in [19] by restricting a logarithmic flat connection defined on the complement of a quintic curve on P2 Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 5 on generic lines of the projective plane—indeed these algebraic solutions have at least two projections giving Dubrovin–Kitaev orbits. Our final classification result consists in a list of 54 exceptional finite orbits of the action (10) obtained up to the action of the group of symmetries G(see Table 2). One orbit (element 25 in Table 2) corresponds to an infinite monodromy group despite the fact that all of its projections to points corresponding to PVI generate finite monodromy groups. The other 53 of these orbits correspond to finite monodromy groups.1 We believe that these 53 orbits are also interesting because even if it is obvious that for finite monodromy groups the braid group orbits must be finite, the problem of classifying the representations of the SL2(C) character variety of the Riemann sphere with five boundary components on finite groups is not trivial. From the monodromy data M1,...,M4, it is in principle possible to recover the explicit formulation of the associated solution of G2using the method developed by Lisovyy and Gavrylenko in [20]of Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems of the form (1). However, the shortest finite orbit classified in our paper has length 36, for this reason the associated algebraic solution of G2has eventually 36 branches, and we doubt that the expression of this solution can have a nice and compact form. All the algorithms necessary to produce this classification can be found in [21]. 2. Co-adjoint coordinates on MG2 As explained in the Section 1, we identify the character variety MG2with the quotient space ˆ MG2 defined in (3). Following [22,23], the first step to endow ˆ MG2with a system of co-adjoint coordinates is to introduce a parameterization of the monodromy matrices in terms of their traces and traces of their products. The following result is a generalization of a result proved by Iwasaki for the case of the sixth Painlev´ e equation [24]:2 Theorem 2.1 Let (M1,...,M4)∈U,p∈Aas in Lemma 1.1 and g(x,y,z):=x2+y2+z2−xyz −4, then in the open set: U(0) jk :=ˆ MG2∩{(p2 jk −4)g(pjk,pl,pjkl)= 0}, (12) there exists a global conjugation P∈SL2(C)such that the matrices M1,...,M4can be parametrized as follows (up to conjugation by P): Ml=⎛ ⎜ ⎝ pjkl−plλ− jk rjk −g(pjk,pl,pjkl) r2 jk 1−pjkl−plλ+ jk rjk ⎞ ⎟ ⎠,Mk=⎛ ⎜ ⎝−pj−pkλ+ jk rjk −ykl−yjlλ− jk r2 jk ykl−yjlλ+ jk g(pjk,pl,pjkl) pj−pkλ− jk rjk ⎞ ⎟ ⎠, 1We are grateful to Gael Cousin for asking us this question. 2We thank the referee for pointing out that some of these results should be known to experts in geometric invariant theory, indeed the recent paper [25] provides an algorithm, implemented in Mathematica, SageMath and in Python, that takes a finite presentation for a finitely presentable discrete group Fand produces a finite presentation of the coordinate ring of the G-character variety of Fwhere Gis a rank 1 complex affine algebraic group. We did not try to use this algorithm as we had already obtained our coordinates by hands. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 6P. CALLIGARIS AND M. MAZZOCCO Mj=⎛ ⎜ ⎝−pk−pjλ+ jk rjk −yjl−yklλ+ jk r2 jk yjl−yklλ− jk g(pjk,pl,pjkl) pk−pjλ− jk rjk ⎞ ⎟ ⎠,Mi=⎛ ⎜ ⎝ pijk−piλ− jk rjk −yil+yijklλ+ jk r2 jk yil+yijklλ− jk g(pjk,pl,pjkl)−pijk −piλ+ jk rjk ⎞ ⎟ ⎠. (13) Alternatively on the open set: U(1) jk :=ˆ MG2∩{(p2 jk −4)g(pj,pk,pjk)= 0}, (14) the matrices M1,...,M4can be parametrized as follows (up to conjugation by P): Ml=⎛ ⎜ ⎝ pjkl−plλ− jk rjk −ykl−yjlλ+ jk r2 jk ykl−yjlλ− jk g(pjk,pj,pk)−pjkl−plλ+ jk rjk ⎞ ⎟ ⎠,Mk=⎛ ⎜ ⎝−pj−pkλ+ jk rjk −g(pjk,pj,pk) r2 jk 1pj−pkλ− jk rjk ⎞ ⎟ ⎠, Mj=⎛ ⎜ ⎝−pk−pjλ+ jk rjk g(pjk,pj,pk)λ+ jk r2 jk −λ− jk pk−pjλ− jk rjk ⎞ ⎟ ⎠,Mi=⎛ ⎜ ⎝ pijk−piλ− jk rjk −yik−yijλ+ jk r2 jk yik−yijλ− jk g(pjk,pj,pk)−pijk −piλ+ jk rjk ⎞ ⎟ ⎠. (15) Finally, on the open set: U(2) jk :=ˆ MG2∩{(p2 jk −4)g(pjk,pi,pijk )= 0}, (16) the matrices M1,...,M4can be parametrized as follows (up to conjugation by P): Ml=⎛ ⎜ ⎝ pjkl−plλ− jk rjk −yil+yijklλ− jk r2 jk yil+yijklλ+ jk g(pjk,pi,pijk )−pjkl−plλ+ jk rjk ⎞ ⎟ ⎠,Mk=⎛ ⎜ ⎝−pj−pkλ+ jk rjk −yik−yijλ− jk r2 jk yik−yijλ+ jk g(pjk,pi,pijk ) pj−pkλ− jk rjk ⎞ ⎟ ⎠, Mj=⎛ ⎜ ⎝−pk−pjλ+ jk rjk −yij−yikλ+ jk r2 jk yij−yikλ− jk g(pjk,pi,pijk ) pk−pjλ− jk rjk ⎞ ⎟ ⎠,Mi=⎛ ⎜ ⎝ pijk−piλ− jk rjk −g(pjk,pi,pijk ) r2 jk 1−pijk−piλ+ jk rjk ⎞ ⎟ ⎠, (17) where: rjk :=p2 jk −4, λ± jk =pjk ±rjk 2, (18) ykl :=2pkl +pjkpjl −pjpjkl −pkpl, yjl :=2pjl +pjkpkl −pkpjkl −pjpl, (19) yik :=2pik +pijpjk −pjpijk −pipk, (20) yij :=2pij +pikpjk −pkpijk −pipj, (21) yil :=2pil +pijkpjkl −pjk pijkl −pipl, yijkl :=2pijkl −pilpjk −pipjkl −pijkpl+pipjk pl. (22) Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 7 Proof. Consider (M1,...,M4)∈U. We only prove the statement for the open subset U(0) jk . For the parametrizations on the open subsets U(1) jk and U(2) jk a similar proof applies. Under the hypothesis that pjk =±2, there exists a matrix P∈SL2(C)such that the product matrix MjMkcan be brought into diagonal form: jk :=P(MjMk)P−1=diag{λ+ jk ,λ− jk }, (23) where the eigenvalues λ± jk are given in (18), where the positive branch of the square root is chosen. Consequently, we conjugate by Pthe matrices Ml,Mk,Mj,Mias follows: P(Ml,Mk,Mj,Mi)P−1=(U,V,W,T). (24) Since, W=jkV−1, we only need to produce the parametrization of the matrices U,V,T. Solving the equations Tr U=pl,Trjk U=pjkl and Tr V=pk,TrjkV−1=pjand Tr T=piand Tr TWV = Tr Tjk =pijk we obtain the diagonal elements of U,Vand T, respectively: u11 =pjkl −plλ− jk rjk ,u22 =−pjkl −plλ+ jk rjk , (25) v11 =−pj−pkλ+ jk rjk ,v22 =pj−pkλ− jk rjk , (26) t11 =pijk −piλ− jk rjk ,t22 =−pijk −piλ+ jk rjk . (27) We now calculate the off-diagonal elements. Since det U=1, then the following identity holds: u12u21 =−g(pjk,pl,pjkl) r2 jk , (28) and in U(0) jk g(pjk,pl,pjkl)= 0. Since Pis unique up to left multiplication by a diagonal matrix D∈SL2(C), we are allowed to fix u21 =1. Then equation (28) gives us the element u12. The system of equations Tr VU =pkl and Tr jkV−1U=pjl gives us a parametrization for the off-diagonal elements of V: v12 =−yik −yijλ− jk r2 jk ,v21 =yik −yijλ+ jk g(pjk,pi,pijk ), (29) where yik and yij are defined in (20) and (21), respectively. Finally, consider the system of equations Tr TU =pil and Tr TWVU =Tr TjkU=pijkl, then we have the following parametrization for t12 and t21: t12 =−yil +yijklλ+ jk r2 jk ,t21 =yil +yijklλ− jk g(pjk,pl,pjkl), (30) where yil and yijkl are defined in (19) and (22), respectively. This concludes the proof.  Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 8P. CALLIGARIS AND M. MAZZOCCO Theorem 2.1 shows that (p1,...,p4,p21,...,p43,p321,...,p421)parameterize the following open subset of U:  j>k U(0) jk ∪U(1) jk ∪U(2) jk . (31) We now show that it is possible to parameterize the monodromy matrices in terms of p∈Aalso outside of this open subset. Lemma 2.2 Let (M1,...,M4)∈Uand p∈A. Assume that pjk =±2 for at least one choice of j= k, j,k=1, ..., 4 and g(pjk,pl,pjkl)=g(pj,pk,pjk )=g(pjk,pi,pijk )=0, (32) where g(x,y,z):=x2+y2+z2−xyz −4, then there exists at least an index lfor which plk = λlλk+1 λlλk and a global conjugation P∈SL2(C)such that: PMkP−1=λk1 01 λk,PMjP−1=λj−λjλk 01 λj, (33) PMlP−1=λl0 plk −λlλk−1 λlλk 1 λl, (34) PMiP−1=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ λi0 pik −λiλk−1 λiλk 1 λi, for pil =λiλl+1 λiλl, ⎛ ⎝λi pil−λiλl−1 λiλl plk−λlλk−1 λlλk 01 λi ⎞ ⎠, for pil = λiλl+1 λiλl, (35) where λs+1 λs=ps,∀s=1, ...,4. Proof. Proceeding as in the proof of Theorem 2.1, we bring the product matrix MjMkinto the diagonal form. Condition (32) implies that the following equations must be satisfied (we have absorbed the global conjugation Pin the matrices M1,...,M4, here): (M1)12(M1)21 =(M2)12(M2)21 =(M3)12(M3)21 =(M4)12(M4)21 =0. By global conjugation by a permutation matrix, we can assume that (Mk)12 = 0 and then by global diagonal conjugation we can put Mkin Jordan normal form. Then, since Mj=jkM−1 kwe immediately obtain (33). Since the monodromy group must be irreducible, one of the two remaining matrices, call it Ml, must have non-zero 21 entry. Then since Tr(MlMk)=plk , we obtain (Ml)21 =plk −λlλk−1 λlλk= 0, and therefore (34). Now if the last matrix is also lower triangular, by imposing Tr MiMk=pik, we obtain the first formula in (35), and it is immediate to check that then pil =λiλl+1 λiλl. Otherwise, if Miis upper triangular, by imposing Tr MiMl=pil, we obtain the second formula (35), and it is immediate to check that then pil = λiλl+1 λiλl. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 9 Proposition 2.3 Let (M1,...,M4)∈Uand p∈A. Assume that pjk =2jk for all j,k=1, ..., 4, where jk =±1. Then, if that at least one matrix Miis diagonalizable there exists a choice of the ordering of the indices i,j,k,l∈{1, 2, 3, 4}such that the following parameterization holds true: PMiP−1=λi0 01 λi,λi=±1, λi+1 λi=pi, (36) PMkP−1=⎛ ⎝−pk−2kiλi λ2 i−1−(pkλi−ki(λ2 i+1))2 (λ2 i−1)2 1λi(pkλi−2ki) λ2 i−1⎞ ⎠, (37) and for pk= kipi: PMjP−1=⎛ ⎝−pj−2jiλi λ2 i−1 (λ2 i−1)(2kj−pikjλi)+(pkλi−2ki)(pjλi−2ji) (pkλi−ki(λ2 i+1))2 −λ2 i(pkλi−2ki)(pjλi−2ji)+λi(λ2 i−1)(pikj−2kjλi) (λ2 i−1)2 λi(pjλi−2ji) λ2 i−1⎞ ⎠(38) PMlP−1=⎛ ⎝−pl−2liλi λ2 i−1 (λ2 i−1)(2kl−piklλi)+(pkλi−2ki)(plλi−2li) (pkλi−ki(λ2 i+1))2 −λ2 i(pkλi−2ki)(plλi−2li)+λi(λ2 i−1)(pikl−2klλi) (λ2 i−1)2 λi(plλi−2li) λ2 i−1⎞ ⎠(39) and if pk=kipi, then pikj(λ2 i+1)= 2λi(kiji +kj)and pikl(λ2 i+1)= 2λi(kili +kl)and PMjP−1=⎛ ⎜ ⎝ λi(pikjλi−2kj) ki(λ2 i−1) λ4 i(pikjλi−2kj)2−2kj jiλ2 i(pikjλi−2kj)(λ2 i−1)+(λ2 i−1)2 (λ2 i−1)2λi(2λi(kiji+kj)−pikj(λ2 i+1)2) λi(pikj(λ2 i+1)−2λi(kiji +kj)) 2kijiλi(λ2 i−1)−λ3 i(pikjλi−2kj) ki(λ2 i−1) ⎞ ⎟ ⎠(40) PMlP−1=⎛ ⎜ ⎝ λi(piklλi−2kl) ki(λ2 i−1) λ4 i(piklλi−2kl)2−2kl liλ2 i(piklλi−2kl)(λ2 i−1)+(λ2 i−1)2 (λ2 i−1)2λi(2λi(kili+kl)−pikl(λ2 i+1)2) λi(pikl(λ2 i+1)−2λi(kili +kl)) 2kiliλi(λ2 i−1)−λ3 i(piklλi−2kl) ki(λ2 i−1) ⎞ ⎟ ⎠(41) If none of the monodromy matrices is diagonalizable, then there exists a choice of the ordering of the indices i,j,k,l∈{1, 2, 3, 4}such that the following parameterization holds true: PMiP−1=i1 0i,PMjP−1=j0 4ij j, (42) PMkP−1=⎛ ⎝ pijk−2ik j−2jki+2ijk 4ij jk−jk 2ij 2(ik −ik)2ikj+2jk i+8ijk−2ijk−pijk 4ij ⎞ ⎠(43) PMlP−1=⎛ ⎝ pijl−2ilj−2jl i+2ijl 4ij jl−jl 2ij 2(il −il)2ilj+2jli+8ij l−2ijl−pijl 4ij ⎞ ⎠(44) Proof. First let us assume that at least one matrix Miis diagonal and work in the basis in which Mi assumes the form (36) with λi=±1. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 16 P. CALLIGARIS AND M. MAZZOCCO 5.1 The classification result by Lisovyy and Tykhyy In order to expand Lisovyy and Tykhyy list of 45 finite orbits (see Table 5 in [9]) it is best to introduce the following quantities: ω1:=q1q∞+q3q2,ω2:=q2q∞+q3q1,ω3:=q3q∞+q2q1, ω4:=q2 3+q2 2+q2 1+q2 ∞+q3q2q1q∞.(75) The group F4of Okamoto transformations of the sixth Painlev´ e equation acts as K4S3on (ω1,...,ω4) [9]. Extending this action to the qijs, namely acting on (ω1,...,ω4,q21,q31,q32)it is straightforward to prove the following: Proposition 5.1 The group F4of the Okamoto transformations of the sixth Painlev´ e equation is generated by the following transformations that act on (ω1,...,ω4,q21,q31,q32)as follows: si(q21,q31,q32,ω1,ω2,ω3,ω4)=(q21,q31,q32,ω1,ω2,ω3,ω4),i=1, 2, 3, ∞,δ, r1(q21,q31,q32,ω1,ω2,ω3,ω4)=(−q21,−q31,q32,ω1,−ω2,−ω3,ω4), r2(q21,q31,q32,ω1,ω2,ω3,ω4)=(−q21,q31,−q32,−ω1,ω2,−ω3,ω4), r3(q21,q31,q32,ω1,ω2,ω3,ω4)=(q21,−q31,−q32,−ω1,−ω2,ω3,ω4), P13(q21,q31,q32,ω1,ω2,ω3,ω4)=(q32,ω2−q31 −q21q32,q21,ω3,ω2,ω1,ω4), P23(q21,q31,q32,ω1,ω2,ω3,ω4)=(ω2−q31 −q21q32,q21,q32,ω1,ω3,ω2,ω4). Proof. The proof of this is a consequence of the results of [10,27].  In particular we observe that P13 and P23 are elements of the braid group B3—since we act only on points that have finite orbits under the action of the braid group, the action of the whole group F4 produces a finite set of values. All these values will be in the form (ω1,...,ω4,q21,q31,q32); in order to extract q1,q2,q3and q∞we use the fact that we can consider the relations (75) as a system of equations in q1,q2,q3and q∞and that each qihas the form: qi=2 cos πθi,i=1, 2, 3, ∞. One particular solution of equations (75) is listed in [9] in terms of θ1,θ2,θ3,θ∞for each point in the Table5in[9]. We can then compute all other solutions q1,q2,q3and q∞by using the following: Lemma 5.2 Suppose ω1,ω2,ω3,ω4are given and consider system (75) in the variables q1,q2,q3,q∞, then this system admits at most 24 solutions. Any two such solutions are related by the following elements of F4: id,α,β,γ,α·β,α·γ,β·γ, α·β·γsδ,α·sδ,β·sδ,γ·sδ,α·β·sδ, α·γ·sδ,β·γ·sδ,α·β·γ·sδ,sδs1,α·sδ·s1, β·sδ·s1,γ·sδ·s1α·β·sδ·s1,α·γ·sδ·s1, β·γ·sδ·s1,α·β·γ·sδ·s1. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 17 where α,β,γ,sδ,s1act as follows on the parameters θi: α(θ1,θ2,θ3,θ∞)=(1+θ1,1+θ2,1+θ3,1+θ∞) β(θ1,θ2,θ3,θ∞)=(θ2,θ1,θ∞−2, θ3),γ(θ 1,θ2,θ3,θ∞)=(θ3,θ∞−2, θ1,θ2) sδ(θ1,θ2,θ3,θ∞)=(θ1−δ,θ2−δ,θ3−δ,θ∞−δ),δ=θ1+θ2+θ3+θ∞ 2, s1(θ1,θ2,θ3,θ∞)=(−θ1,θ2,θ3,θ∞). Proof. It is an immediate consequence of Proposition 10 in [9].  This lemma allows us to calculate all the solutions of the system (75) in terms of the given ω1,ω2,ω3,ω4 starting from only one solution q1,q2,q3and q∞. We are therefore able to set up our expansion algorithm: Algorithm 1 For every line of Table 5 in [9], take the values (ω1,...,ω4,q21,q31,q32)and the corresponding (q1,q2,q3,q∞)givenin[9]. (1) Apply to (ω1,...,ω4,q21,q31,q32)all 48 transformations of the group K4S3. For each new set of values (ω 1,...,ω 4,q 21,q 31,q 32)obtained in this way, compute the corresponding (q 1,...,q ∞)as the result of the same transformation on (q1,q2,q3,q∞). (2) For every element (ω 1,...,ω 4,q 21,q 31,q 32)obtained in step 1, generate their orbit under the action of the braid group B3. For each new set of values (ω 1,...,ω 4,q 21,q 31,q 32)obtained in this way, compute the corresponding (q 1,...,q ∞)as the result of the same braid on (q 1,q 2,q 3,q ∞). (3) For every element (ω 1,...,ω 4,q 21,q 31,q 32)and (q 1,...,q ∞)obtained in step 2, find all other solutions (q 1,q 2,q 3,q ∞)of the system (75) for (ω 1,...,ω 4)by applying the transformations in Lemma 5.2 to (q 1,...,q ∞). (4) Merge (q 1,q 2,q 3,q ∞)and (q 21,q 31,q 32)into: q =(q 1,q 2,q 3,q ∞,q 21,q 31,q 32). (5) Generate the P3-orbit of q and save the result in the set E45 Once this algorithm ends, the set E45 will contain only a finite number of orbits. This set contains 86,768 points. 6. Matching procedure In this section, we propose a procedure to construct all candidate points p∈A: Definition 6.1 A point psuch that its four projections ˆq,ˇq,¯q,˜q, defined in (74), generate finite orbits under the action of P3and such that at most one projections is a Picard or Dubrovin–Kitaev orbit, is said tobeacandidate point. Note that, to generate a candidate point p, it is not necessary to know all four projections ˆq,ˇq,¯q,˜q. Indeed, looking at Table 1, we see that if we give three projections, then only the value of p∞and one Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 18 P. CALLIGARIS AND M. MAZZOCCO value pijk will be undetermined, but we can calculate these values from (51) and by choosing appropriately one of the four relations f1,...,f4, defined in (47)–(50), respectively. So, in order to obtain the set Cof all candidate points, we can set up three matching procedures, each of them based on the knowledge of only three projections. We denote by ˜ C,ˆ C,ˇ Cand ¯ Cthe sets obtained by matching three projections and missing ˜q,ˆq,ˇqor ˜q, respectively. In order to construct the set C, the union of all the above four sets ˜ C,ˆ C,ˇ C,¯ Cmust be taken: C=˜ C∪ˆ C∪ˇ C∪¯ C. (76) As we are going to show in the next Lemma, it is enough to know only one of the sets ˜ C,ˆ C,ˇ C,¯ Cto generate the whole set C: Lemma 6.1 Consider m∈Uand the permutation π(1234)that acts on the co-adjoint coordinates of mas follows: π(1234)(p)=(p4,p1,p2,p3,p∞,p41,p42,p21,p43,p31,p32,p421,p321,p432,p431), then: π(1234)(˜ C)=ˇ C,π(1234)(ˇ C)=¯ C,π(1234)(¯ C)=ˆ C,π(1234)(ˆ C)=˜ C. (77) Proof. We only prove the first of (77), the other relations can be proved in a similar way. Thanks to Theorem 2.1, a point p∈˜ Cparameterizes a quadruple mof monodromy matrices m:=(M1,M2,M3,M4) up to global diagonal conjugation. Analogously, the three projections ˆq,ˇq,¯q∈ˆ MPVI parameterize three triples of monodromy matrices ˆn,ˇn,¯n∈ˆ MPVI , such that, up to global diagonal conjugation: ˆ N1=M2,ˆ N2=M3,ˆ N3=M4,ˆ N∞=(M4M3M2)−1, ¯ N1=M1,¯ N2=M3,¯ N3=M4,¯ N∞=(M4M3M1)−1, ˇ N1=M1,ˇ N2=M2,ˇ N3=M4,ˇ N∞=(M4M2M1)−1. Now take the point p=π(1234)(p), this parameterizes the triple m=π(1234)(m)up to global diagonal conjugation. Consider now the three projections ˆq,˜q,¯q∈ˆ MPVI of p. They parameterize three triples of monodromy matrices ˆn,˜n,¯n∈ˆ MPVI , such that, up to global diagonal conjugation: ˆ N 1=M 2=M1,ˆ N 2=M 3=M2,ˆ N 3=M 4=M3, ˆ N ∞=(M 4M 3M 2)−1=(M3M2M1)−1, ¯ N 1=M 1=M4,¯ N 2=M 3=M2,¯ N 3=M 4=M3, ¯ N ∞=(M 4M 3M 1)−1=(M3M2M4)−1, ˜ N 1=M 1=M4,˜ N 2=M 2=M1,˜ N 3=M 3=M2, ˜ N ∞=(M 3M 2M 1)−1=(M2M1M4)−1. These relations show that ˆn=˜n,¯n=π(123)ˆn,˜n=π(123)ˇn, Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 19 where π(123)(q)=(q3,q1,q2,q∞,q32,q21,q31), Now since ˜n,π(123)ˆn,π(123)ˇn∈ˆ MPVI , this shows that p∈ˇ C. Viceversa, we can prove in a similar way that given p∈ˇ C, then p=π−1 (1234)p∈˜ C. This concludes the proof.  We are now ready to describe how to implement the matching algorithmically. 6.1 Matching with the PVI 45 exceptional algebraic solutions In this section, we give an algorithm that produces the finite set CE45×E45×E45 of all candidate points psuch that three over four projections ˆq,ˇq,¯q,˜q, defined in (74), are in the set E45. Algorithm 2 (1) Consider (ˆq,ˇq,¯q)∈E45 ×E45 ×E45. (2) Check if ˆq,ˇq,¯qsatisfy relations given by the columns of Table 1, then go to the next step, otherwise go to Step 1. (3) Determine the two roots p(i) 321, for i=1, 2, using equation (47). For each i=1, 2: (4) Calculate the values of p(i) ∞using equation (51). (5) Use Table 1to determine all the other components of p(i). (6) If p(i)satisfies equations (52)–(61) then go to the next Step, otherwise go to Step 1. (7) Save p(i)in the set ˜ CE45×E45×E45 , eliminate (ˆq,ˇq,¯q)from E45 ×E45 ×E45 and go to Step 1. Since E45 is a finite set, this algorithm terminates and produces a finite set ˜ CE45×E45×E45 . Finally the big set CE45×E45×E45 can be generated by Lemma 6.1 as follows: CE45×E45×E45 =˜ CE45×E45×E45 3  i=1 πi (1234)(˜ CE45×E45×E45 ). The Algorithm 2together with the action of the permutations producing the set CE45×E45×E45 can be found [21]. This set contains all candidate points p∈Asuch that three projections (74) are in the set E45 and consists of 3,355,200 points. 6.2 Matching with Okamoto’s Riccati solutions We call Okamoto-type solutions the algebraic solutions of the PVI equation belonging to Okamoto’s Riccati solutions. The set O of all finite orbits corresponding to Okamoto-type solutions is an infinite set, therefore to construct candidate points with projections in this set is not a straightforward adaptation of Algorithm 2. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 20 P. CALLIGARIS AND M. MAZZOCCO Definition 6.2 A point pis called not relevant if the associated monodromy group is reducible or there exists an index i=1, ...,4,∞such that Mi=±I. A point pis called relevant otherwise. In this subsection, we are going to prove a few lemmata that show that in order for pto be relevant, the number of projections corresponding to solutions of Okamoto type is limited. We will then characterize these projections and formulate algorithms that exploit these characterizations to classify candidate points with projections of Okamoto type. Proposition 6.2 If a point p∈Ais such that any three of its four projections ˆq,ˇq,¯q,˜qare in the set O of all finite orbits corresponding to algebraic solutions of Okamoto type then the point pis not relevant. Consequently, all points psatisfying hypotheses of Proposition 6.2 will be irrelevant to our classification (and then excluded from it). Before proving this result, we will need the following two definitions: Definition 6.3 The set OID is the set of all the q∈O such that the associated triple of monodromy matrices n∈ˆ MPVI admits one matrix equals to ±I. Definition 6.4 The set ORED is the set of all the q∈O such that if we consider the associated triple of monodromy matrices n∈ˆ MPVI then the monodromy group N1,N2,N3is reducible. Proof of Proposition 6.2:In order to prove the statement, we distinguish three cases: (i) Assume phas three projections in OID. It is enough to consider m∈ˆ MG2and the following three projections: ˜n=(M1,M2,M3),ˆn=(M2,M3,M4),ˇn=(M1,M2,M4), (78) because all other cases differ from this case only by a permutation of the matrices Mi, see Lemma 6.1.If any of Mi=±I, then we conclude. If not, we are left with the following case: ˜ N∞=M3M2M1=˜I,ˆ N∞=M4M3M2=ˆI,ˇ N∞=M4M2M1=ˇI, where ˜,ˆ,ˇ=±1. Combining these relations we obtain: M4=˜ˆM1,M3=˜ˇM4, and therefore M3=ˆˇM1,M2=˜ˆˇM−2 1, so that finally m=(M1,˜ˆˇM−2 1,ˆˇM1,˜ˆM1)which is reducible. Therefore pis not relevant. (ii) Suppose pis such that three projections over four are in the set ORED. Again it is enough to consider the three projections (78). Since the three monodromy groups defined by the triples ˜n,ˆn,ˇnare reducible, these triples have each a common eigenvector, let us denote them ˜v,ˆvand ˇv, respectively. Now the matrix M2that appears in all the three projections, has three eigenvectors ˜v,ˆvand ˇv, which implies that one of the following identities must hold: ˜v=ˆvor ˜v=ˇvor ˆv=ˇv. Therefore the monodromy group is reducible and the point pis not relevant. (iii) When there are three projections in O, not all of the same type, we apply Lemma 6.3. This concludes the proof. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 21 Lemma 6.3 If a point p∈Ais such that one of its four projections ˆq,ˇq,¯q,˜qis in the set OID and another one projection is in the set ORED, then such point pis not relevant. Proof. Consider m∈ˆ MG2and the following two distinct generic projections: (Mi,Mj,Mk)∈OID,i>j>k,i,j,k=1, ..., 4, (79) (Mi,Mj,Mk)∈ORED,i>j>k,i,j,k=1, ..., 4. (80) If either Mi,Mj,Mkis equal to ±I, then we conclude. Otherwise suppose: MiMjMk=±I. (81) Moreover, suppose the monodromy group associated to the triple (Mi,Mj,Mk)is reducible, then the matrices Mi,Mj,Mkhave a common eigenvector v.In(79) and in (80), at least two indices i,j,kthat are equal to two indices i,j,k, without loss of generality, suppose i= i,j=jand k=k, then equation (81) implies Mi=±(MjMk)−1, which shows that vis also an eigenvector for Mjand therefore the monodromy group Mi,Mi,Mj,Mkis reducible as we wanted to prove.  Lemma 6.4 Let pbe a relevant point such that one of its projections qis in the set OID, then qsatisfies: q21 =±q3,q31 =±q2,q32 =±q1,q∞=±2. (82) Proof. Consider the triple of matrices n=(N1,N2,N3)determined by q∈OID. If any of the Niis equal to ±I, by the matching procedure, we end up with a point pthat is not relevant, therefore, we avoid this case. Otherwise, assume N∞=N3N2N1=±I, then: N1=±(N3N2)−1,N2=±(N1N3)−1,N3=±(N2N1)−1. (83) By taking the traces we obtain (82). This concludes the proof.  Lemma 6.5 Let qbe the co-adjoint coordinates on ˆ MPVI .Ifqis in the set ORED, then qsatisfies: qij =1 2(qiqj−ijsisj),i>j,i,j=1, 2, 3, q∞=1 4(q1q2q3−12s1s2q3−13s1s3q2−23s2s3q1)(84) where sk=4−q2 kfor some choice of the signs k=±1 for k=1, 2, 3. Proof. Consider the triple of matrices n=(N1,N2,N3)determined by q∈ORED, they define a reducible monodromy group. Therefore, we can choose a basis in which they are all upper triangular. Then their diagonal elements are given by their eigenvalues eigenv(Ni)=exp (lπθi), where l=±1, so that: Tr(Ni)=2 cos πθi,i=1, 2, 3, ∞, (85) Tr(NiNj)=2 cos(π(iθi+jθj)),i,j=1, 2, 3, i>j, (86) Tr(N3N2N1)=2 cos(π(1θ1+2θ2+3θ3)). (87) Applying trigonometric identities we obtain relations (84). This concludes the proof.  Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 22 P. CALLIGARIS AND M. MAZZOCCO An obvious consequence of this result is: Lemma 6.6 Suppose p∈Ais a relevant point such that any two of its four projections ˆq,ˇq,¯q,˜q, defined in (74), are in the set ORED. Denote by qone of the remaining projections, then there exists a couple of indices (i,j),(i,j)with one index in (i,j)equal to one index in (i,j)such that: q2 ij +q2 i+q2 j−qijqiqj−4=0, i>j,i,j=1, 2, 3, q2 ij+q2 i+q2 j−qijqiqj−4=0, i>j,i,j=1, 2, 3. (88) Lemmata 6.4,6.5 and 6.6 lead to the development of additional matching algorithms in order to complete our classification for the cases when these points are included. Thanks to Lemma 6.3, in order to complete our classification of candidate points, we need to construct only the following four sets: CE45×OID×OID , the set of all candidate points with at least two projections in OID and one in E45, the set CE45×ORED×ORED , the set of all candidate points with at least two projections in ORED and one in E45, CE45×E45×OID , the set of all candidate points with at least two projections in E45 and one in OID, and CE45×E45×ORED , the set of all candidate points with at least two projections in E45 and one in ORED. The set CE45×ORED×ORED turns out to be empty. To construct the set CE45×OID×OID , we proceed as follows: firstly we construct the set ˜ CE45×OID×OID , where one over the three projections ˆq,ˇq,¯qis in the set E45 and two of the remaining projections are in the set OID, then, applying Lemma 6.1 we generate the whole set CE45×OID×OID . The set ˜ CE45×OID×OID is the union of the following three sets of candidate points p: (A2.1) ¯ ˜ CE45×OID×OID :candidate points pwith ˆq,ˇq∈OID,¯q∈E45. (A2.2) ˇ ˜ CE45×OID×OID :candidate points pwith ˆq,¯q∈OID,ˇq∈E45. (A2.3) ˆ ˜ CE45×OID×OID :candidate points pwith ¯q,ˇq∈OID,ˆq∈E45. Here, we state only the algorithm that generates the subset (A2.1), the other algorithms for the subsets (A2.2) and (A2.3) can be derived in a similar way. The algorithm is based on the following result, which is an obvious consequence of Lemma 6.4: Lemma 6.7 If a point p∈A, is such that ˆq,ˇq∈OID, then ¯qmust satisfy: ¯q2=ˆˇ¯q1,¯q32 =ˆˇ¯q31, (89) and pis such that: p1=¯q1,p2=ˇ¯q31,p3=ˆˇ¯q1,p4=¯q3,p21 =ˇ¯q3,p31 =¯q21,p32 =ˆ¯q3, p41 =¯q31,p42 =ˇ¯q1,p43 =ˆˇ¯q31,p432 =ˆ2, p431 =¯q∞,p421 =ˇ2. (90) Algorithm 3 (1) Take ¯q∈E45. (2) Check if ¯qsatisfies: ¯q2=ˆˇ¯q1, and ¯q32 =ˆˇ¯q31, then go to the next Step, otherwise go to Step 1. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 23 (3) Determine the components of pinvolved in identities (90). (4) Determine the values p(i) 321, for i=1, 2, using equation (47). For each i=1, 2: (5) Calculate the values of p(i) ∞using equation (51). (6) Use identities given by the columns of Table 1in order to determine the other components of p(i). (7) If p(i)satisfies equations (52)–(61) then go to the next Step, otherwise Step 1. (8) Save p(i)in the set ¯ ˜ CE45×OID×OID , and go to Step 1. When Algorithm 3and the algorithms for subsets (A2.2) and (A2.3) end, the following set is obtained: ˜ CE45×OID×OID =¯ ˜ CE45×OID×OID ∪ˇ ˜ CE45×OID×OID ∪ˆ ˜ CE45×OID×OID , then, by Lemma 6.1, we generate the set CE45×OID×OID as: CE45×OID×OID =˜ CE45×OID×OID 3  i=1 πi (1234)(˜ CE45×OID×OID ), (91) where permutation π(1234)is defined in Lemma 6.1. This set contains 6,385 points and Algorithm 3can be found in [21]. We proceed in a similar way to construct the set CE45×E45×ORED of all candidate points p∈ˆ MG2such that one over the four projections ˆq,ˇq,¯q,˜qis in the set ORED and two of the remaining projections are in the set E45. We give here only the algorithm such that ˆq,ˇq∈E45,¯q∈ORED—all other cases can be derived in similar way. Algorithm 4 (1) Consider ˆq,ˇq∈E45 ×E45. (2) Check if ˆq,ˇqsatisfy relations given by the columns of the first and third rows of Table 1then go to the next step, otherwise go to Step 1. (3) Calculate p31 and p431 using Table 1and conditions (84). (4) Determine the values p(i) 321, for i=1, 2, using equation (47). For each i=1, 2: (5) Calculate the values of p(i) ∞using equation (51). (6) Use identities given by the columns of Table 1in order to determine the other components of p(i). (7) If p(i)satisfies equations (52)–(61) then go to the next step, otherwise Step 1. (8) Save p(i)in the set ¯ ˜ CE45×E45×ORED , and go to Step 1. When Algorithm 4and the analogous algorithms for ¯q,ˇq∈E45,ˆq∈ORED and for ¯q,ˆq∈E45, ˇq∈ORED, respectively end, the set ˜ CE45×E45×ORED is obtained. Then, as before, the set CE45×E45×ORED is Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 24 P. CALLIGARIS AND M. MAZZOCCO given by: CE45×E45×ORED =˜ CE45×E45×ORED 3  i=1 πi (1234)(˜ CE45×E45×ORED ). This set contains 342,368 points and Algorithm 4can be found in [21]. We now produce the algorithm that generates the set CE45×E45×OID of all candidate points p∈Asuch that one projection is in the set OID and two of the remaining three projections are in the set E45.Wegive here only the algorithm such that ˆq,ˇq∈E45,¯q∈OID - all other cases can be derived in similar way. This is a simple adaptation of Algorithm 4in which we substitute Steps (2) and (8): Algorithm 5 (1), (2), (4), (5), (6), (7) see Algorithm 4. (3) Calculate p31 and p431 using Table 1and conditions (82). (8) Save p(i)in the set ¯ ˜ CE45×E45×OID , and go to Step 1. When Algorithm 5and the analogous algorithms for ¯q,ˇq∈E45,ˆq∈OID and for ¯q,ˆq∈E45,ˇq∈OID, respectively end, we obtain ˜ CE45×E45×OID , then as before: CE45×E45×OID =˜ CE45×E45×OID 3  i=1 πi (1234)(˜ CE45×E45×OID ). This set contains 245,760 points, and Algorithm 5can be found in [21]. Finally the set of all candidate points is: C=CE45×E45×E45 ∪CE45×OID×OID ∪CE45×E45×ORED ∪CE45×E45×OID . (92) This is a finite set consisting of 3,461,273 points (duplicated points are erased). We re-define this set by throwing away all points that produce M∞=±I, so that the resulting set Chas 3,287,140 elements. 7. Extracting finite orbits Now, we need to determine which points in Clead to a finite orbit of the P4-action. The following result is fundamental to achieve this: Lemma 7.1 Let p∈Cacandidate point, then its orbit is finite if and only if β(p)∈Cfor every braid β∈P4. Proof. Suppose β(p)∈Cfor every β∈P4, then the orbit is finite since Cis finite too. Vice versa, suppose phas a finite P4-orbit, then for every β,β(p)must have a finite orbit. Hence, β(p)must be an element of C. Therefore, to select the finite orbits is equivalent to find the subset C0⊂Csuch that: C0={p∈C|β(p)∈C,β∈P4}. (93) Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 25 To construct the set C0, we use the following: Algorithm 6 (1) Consider p∈C. (2) Apply to it all the generators (65)ofP4. (3) If there exists an i=1, ..., 6 such that p(i)/∈Cthen delete pfrom the set Cand go to Step 1, otherwise save pin C0and go to Step 1. This algorithm is designed in such a way that points already considered are not considered again, or in other words, we order points in Cand proceed in order. This algorithm ends when in the set Cthere are no more elements to delete. The final set C0contains 1,270,050 points and Algorithm 6can be found in [21]. Note that C0contains only elements that generate finite orbits under the P4-action. In fact, assume by contradiction that p∈C0has an infinite orbit. Then there exists a braid βsuch that β(p)∈ C. Now every braid β∈P4can be thought as an ordered combination of generators βij: β=βij...β ij    n , (94) where nindicates the length of the word. Let us introduce the following notation: p(0)=p,p(1)=βij(p(0)),...,p(n)=β(p)=βij(p(n−1))=βij...β ij    n (p(0)). (95) Since we supposed p(n)/∈C, Algorithm 6deletes p(n−1)from the set C. In the next iteration it deletes p(n−2) and so on, till when p(0)=pis deleted from C, and therefore pis not in C0, contradicting our hypothesis. 8. Extracting non-equivalent orbits In this section, we quotient the set C0of all points pgiving rise to a finite orbit with respect to the action of the pure braid group, so that we select only one representative point for every finite orbit, and by the action of the symmetry group Gof ˆ MG2described in the next theorem proved in the Appendix. Theorem 8.1 The group G:=P13,P23,P34,P1∞, sign1,..., sign4,π(12)(34),π(1234)(96) where P13(p)=σ2σ−1 1σ−1 2(p), (97) P23(p)=σ2σ−1 1σ−1 2σ−1 1σ2σ−1 1σ−1 2(p), (98) Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 32 P. CALLIGARIS AND M. MAZZOCCO Funding EPSRC DTA allocation to the Mathematical Sciences Department at Loughborough University to P.C. A. The symmetry group Gof ˆ MG2 The general theory of the bi-rational transformations of the Garnier systems was developed in [28], where Kimura proved that the symmetric group S5acts as a group of bi-rational transformations on the Garnier system (see also [17,29,30]). These bi-rational transformations map algebraic solutions to algebraic solutions with the same number of branches. This means that the corresponding action on the co-adjoint coordinates maps finite orbits to finite orbits with the same number of points. To compute this action, we use the following result proved in [10]: Lemma A.1 The symmetric group S5giving rise to Kimura’s bi-rational transformations of the Garnier system acts on MG2as the group P13,P23,P34,P1∞where the transformations P13,P23,P34 act on the monodromy matrices as follows: P13 :(M1,M2,M3,M4)→ (M−1 1M−1 2M3M2M1,M2,M2M1M−1 2,M4), P23 :(M1,M2,M3,M4)→ ((M−1 2M3M2M1)−1M1M−1 2M3M2M1, (M2M1)−1M3M2M1,M2,M4), P34 :(M1,M2,M3,M4)→ (M∞M3M2M1(M∞M3M2)−1,M2, (M3M2M1M−1 2)−1M4(M3M2M1M−1 2),M3), (A.1) while transformation P1∞acts on the monodromy matrices as: P1∞:(M1,M2,M3,M4)→(−C1M∞C−1 1,C−1 1M2C1,C−1 1M3C1, C−1 1M4C1), (A.2) where C1is the diagonalizing matrix of M1. Corollary A.2 The group P13,P23,P34,P1∞acts on the co-adjoint coordinates as in (97)–(100). Proof. This is a straightforward computation relying on the definition of the co-adjoint coordinates and the skein relation.  We wish to extend the class of transformations satisfying this property by adding to P13,P23,P34,P1∞ the following set of transformations that also map finite orbits to finite orbits with the same number of points (see Theorem A.5): (i) Sign flips, or transformations that change signs to matrices Mifor i=1, ..., 4, corresponding to the so-called Schlesinger transformations introduced by Jimbo–Miwa in [31]: sign(1,2,3,4):(M1,M2,M3,M4,M∞)→ (1M1,2M2,3M3,4M4, 1234(M4M3M2M1)−1), (A.3) where i=±1 for i=1, ...,4. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 33 (ii) Permutations of the matrices Mifor i=1, ..., 4 generated by: π(12)(34):(M1,M2,M3,M4,M∞)→ (M−1 2,M−1 1,M−1 4,M−1 3,M2M1M4M3), (A.4) π(1234):(M1,M2,M3,M4,M∞)→ (M4,M1,M2,M3,(M3M2M1M4)−1). (A.5) The following two results give the action of the sign flips and permutations on the co-adjoint coordinates and can be proved by straightforward computations: Proposition A.3 The sign flips are invertible maps generated by the four basic elements: sign1:=sign(−1,1,1,1), sign2:=sign(1,−1,1,1), (A.6) sign3:=sign(1,1,−1,1)sign4:=sign(1,1,1,−1) that act as follow on the co-adjoint coordinates (9)asin(101)–(104). Proposition A.4 The generators π(12)(34)and π(1234)act on the co-adjoint coordinates (9)asin(105) and (106). Finally, we characterize the group Gof symmetries of ˆ MG2: Definition A.1 A symmetry for ˆ MG2is an invertible map :ˆ MG2→ ˆ MG2such that given an element p∈ˆ MG2and its orbit O(p), the following is true: |O((p))|=|O(p)|. (A.7) Theorem A.5 The group G:=P13,P23,P34,P1∞, sign1,..., sign4,π(12)(34)π(1234)(A.8) is a group of symmetries for ˆ MG2. Proof. The statement is true for the subgroup P13,P23,P34,P1∞by construction. We now prove that each generator in sign1,..., sign4,π(12)(34)π(1234)satisfies (A.7). It is straightforward to prove the following relations: σ1sign2=sign1σ1,σ1sign3=sign3σ1,σ1sign4=sign4σ1, σ2sign1=sign1σ2,σ2sign2=sign3σ2,σ2sign3=sign2σ2, σ2sign4=sign4σ2,σ3sign1=sign1σ3,σ3sign2=sign2σ3, σ3sign3=sign4σ3,σ3sign4=sign3σ3. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 34 P. CALLIGARIS AND M. MAZZOCCO so that all sign flips are indeed symmetries. Regarding the permutations, it is straightforward to prove the following relations: σ2π(1234)=π(1234)σ1,σ1π(12)(34)=π(12)(34)σ−1 1, σ2π(12)(34)=π(12)(34)(1234)3σ2σ3,σ3π(12)(34)=π(12)(34)σ−1 3, σ1π(1234)=π(1234)π(1234)σ−1 2σ−1 1,σ2π(1234)=π(1234)σ1, σ3π(1234)=π(1234)σ2. This conclude the proof.  References 1. Bolibruch, A. (1997) On isomonodromic deformations of Fuchsian systems. J. Dynam. Control Syst., 3, 589–604. 2. Schlesinger, L. (1912) Uber eine Klasse von Differentsial System Beliebliger Ordnung mit Festen Kritischer Punkten. J. fur Math., 141, 96–145. 3. Garnier, R. (1912) Sur des ´ equations diff´ erentielles du troisi` eme ordre dont l’int´ egrale g´ en´ erale est uniforme et sur une classe d’´ equations nouvelles d’ordre sup´ erieur dont l’int´ egrale g´ en´ erale a ses points critiques fixes. Ann. Sci. ´ Ecole Norm.. Sup., 29, 1–126. 4. Garnier, R. (1926) Solution du probleme de Riemann pour les systemes diff´ erentielles lin´ eaires du second ordre. Ann. Sci. ´ Ecole Norm.. Sup., 43, 239–352. 5. Dubrovin, B. & Mazzocco, M. (2000) Monodromy of certain Painlev´ e VI transcendents and reflection groups. Invent. Math., 141, 55–147. 6. Cousin, G. (2017) Algebraic isomonodromic deformations of logarithmic connections on the Riemann sphere and finite braid group orbits on character varieties. Math. Ann., 367, 965–1005. 7. Cousin, G. & Moussard, D. (2016) Finite braid group orbits in Aff(C)-character varieties of the punctured sphere. ArXiv1604.04234. 8. Mazzocco, M. (2001) The geometry of the classical solutions of the Garnier systems. Int. Math. Res. Notices, 12, 613–646. 9. Lisovyy, O. & Tykhyy, Y. (2014) Algebraic solutions of the sixth Painlev´ e equation. J. Geom. Phys., 85, 124–163. 10. Dubrovin, B. & Mazzocco, M. (2007) Canonical structure and symmetries of the Schlesinger equations. Commun. Math. Phys., 271, 289–373. 11. Okamoto, K. (1987) Studies on the Painlev´ e equations I, sixth Painlev´ e equation. Ann. Mat. Pura Appl., 146, 337–381. 12. Mazzocco, M. (2001) Picard and Chazy solutions to the Painlev´ e VI equation. Math. Ann., 321, 157–195. 13. Picard, E. (1889) M´ emoire sur la th´ eorie des fonctions alg´ ebriques de deux variables. J. Math. Pures Appl., 5, 135–320. 14. Dubrovin, B. (1996) Geometry of 2D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993) (M. Francaviglia & S. Greco eds). Lecture Notes in Mathematics, vol. 1620, Fond. CIME/CIME Found. Subseries. Berlin: Springer, pp. 120–348. 15. Andreev, F. & Kitaev, A. (2002) Transformations RS2 4(3)of the ranks ≤4 and algebraic solutions of the sixth Painlev´ e equation. Commun. Math. Phys., 228, 151–176. 16. Doran, C. (2001) Algebraic and geometric isomonodromic deformations. J. Diff. Geom., 59, 33–85. 17. Tsuda, T. (2006) Toda equation and special polynomials associated with the Garnier system. Adv. Math., 206, 657–683. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025 FINITE ORBITS OF THE BRAID GROUP ON THE GARNIER SYSTEM 35 18. Diarra, K. (2013) Construction et classification de certaines solutions alg´ ebriques des syst` emes de Garnier. Bull. Braz. Math. Soc. (N.S.), 44, 129–154. 19. Girand,A. (2016) A new two-parameter family of isomonodromic deformations over the five punctured sphere. Bull. Soc. Math. France, 144, 339–368. 20. Gavrylenko, P. & Lisovyy, O. (2016) Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions. ArXiv 1608.00958. 21. Calligaris, P. & Mazzocco, M. Algorithms for finite orbits of the pure braid on the monodromy of the 2-variable Garnier system. Available at https://doi.org/10.17028/rd.lboro.4924181. 22. Procesi, C. (1976) The invariant theory of n×nmatrices. Adv. Math., 19, 306–381. 23. Procesi, C. (2007) Lie Groups, An Approach through Invariants and Representations, Universitext. New York: Springer. 24. Iwasaki, K. (2003) An Area-Preserving Action of the Modular Group on Cubic Surfaces and the Painlev´ eVI Equation. Commun. Math. Phys., 242, 185–219. 25. Ashley, C., Burelle, J. P. & Lawton, S. (2017) Rank 1 character varieties of finitely presented groups. arXiv1703.08241. 26. Grayson, D. & Stillman, M. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/. 27. Inaba, M., Iwasaki, K. & Saito, M. (2004) B¨ acklund transformations of the sixth Painlev´ e equation in terms of Riemann–Hilbert correspondence. Int. Math. Res. Not., 2004, 1–30. 28. Kimura, H. (1990) Symmetries of the Garnier system and of the associated polynomial Hamiltonian system. Proc. Jpn. Acad. Ser. A Math. Sci., 66, 176–178. 29. Iwasaki, K., Kimura, H., Shimomura, S. & Yoshida, M. (1991) From Gauss to Painlev´ e. Aspects of Mathematics, E16. Braunschweig: Friedr. Vieweg & Sohn, pp. xii–347. 30. Mazzocco, M. (2004) Irregular isomonodromic deformations for Garnier systems and Okamoto’s canonical transformations. J. Lond. Math. Soc. (2), 70, 405–419. 31. Jimbo, M. & Miwa, T. (1981) Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II. Phys. D, 2, 306–352. Downloaded from https://academic.oup.com/integrablesystems/article/3/1/xyy005/5032871 by UPC Btca Rector G Ferrate user on 16 January 2025