On the reductions and classical solutions of the Schlesinger equations

Abstract

Abstract. The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of "more complicated" Schlesinger equations S (n,m) to "simpler" S (n ′ ,m ′ ) having n ′ < n or m ′ < m.

Riemann–Hilbert problem for Hurwitz Frobenius manifolds: Regular singularities

“…• In another work [16] by Dubrovin and Mazzocco the idea of reducibility of Schlesinger systems was developed: a solution to a Schlesinger system is called reducible if it can be expressed in terms of solutions to Schlesinger systems with smaller number of singularities or lower matrix dimension. In particular, it was proved that if all monodromy matrices have the same blocktriangular structure, than the solution is reducible.…”

“…Now we note that the last d − 1 rows in the block S ∞ above the diagonal in the monodromy matrix M ∞ (A. 16) form an (d − 1) × (d − 1) matrix which coincides with the Cartan matrix for A d−1 . Therefore each row of the matrix M ∞ is a coordinate vector of a root of A d−1 with respect to a basis of weight vectors {v i } defined by…”

“…Furthermore, as can be seen from (A.7), (A.8), (A.11) and (A. 16), the only non-zero row in each of the blocks S k and S 2g+2n in the matrices (4.9) is the respective row from M ∞ , i.e., a row of a Cartan matrix for A d−1 .…”

“…A Schlesinger system corresponding to a block-diagonal structure of monodromy matrices as in (1.7), was called reducible in [16]; this means that its solution can in principle be expressed in terms of solutions of two Schlesinger systems of lower dimension ( (d − 1) × (d − 1) and (2g + m − 1) × (2g + m − 1) in our case).…”

Triangular Schlesinger systems and superelliptic curves

“…Note that triangular and, more generally, reducible, Schlesinger systems of arbitrary size p were already studied by B. Dubrovin, M. Mazzocco in [10], where the main question was the following: when are solutions of one Schlesinger system for N (p × p)-matrices expressed via solutions of some other "simpler" Schlesinger systems of smaller matrix size or involving less than N matrices? However, there was no restriction imposed on the exponents, and thus there was no discussion of the integration of such systems in an explicit, in particular algebro-geometric, form.…”