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We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$and $Ma{t}_n$that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group $\mathcal{G}$corresponds to a cluster structure in $\mathcal{O}\left(\mathcal{G}\right)$. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of $A_n$type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $S{L}_n$compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $S{L}_n$equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications. 

As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot– Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators. 

We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the A_n-groupoid of upper-triangular matrices to express roots of the characteristic equation det(A−λA^T)=0, with A∈A_n, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichmüller spaces for the algebra sl_n. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of A_n-groupoid to a A_{Sp_2m}-groupoid. 

As was pointed out by S. Karp, Theorem B of paper [V. Sedykh and B. Shapiro, On two conjectures concerning convex curves, Int. J. Math. 16(10) (2005) 1157–1173] is wrong. Its claim is based on an erroneous example obtained by multiplication of three concrete totally positive 4 × 4 upper-triangular matrices, but the order of multiplication of matrices used to produce this example was not the correct one. Below we present a right statement which claims the opposite to that of Theorem B. Its proof can be essentially found in a recent paper [N. Arkani-Hamed, T. Lam and M. Spradlin, Non- perturbative geometries for planar N = 4 SYM amplitudes, J. High Energy Phys. 2021 (2021) 65]. 

Using Fock--Goncharov higher Teichmüller space variables we derive Darboux coordinate representation for entries of general symplectic leaves of the A_n groupoid of upper-triangular matrices and, in a more general setting, of higher-dimensional symplectic leaves for algebras governed by the reflection equation with the trigonometric R-matrix. The obtained results are in a perfect agreement with the previously obtained Poisson and quantum representations of groupoid variables for A_3 and A_4 in terms of geodesic functions for Riemann surfaces with holes. We represent braid-group transformations for A_n via sequences of cluster mutations in the special A_n-quiver. We prove the groupoid relations for quantum transport matrices and, as a byproduct, obtain the Goldman bracket in the semiclassical limit. 

We continue the study of multiple cluster structures in the rings of regular functions on $$GL_n$$, $$SL_n$$ and $$\operatorname{Mat}_n$$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $$\mathcal G$$ corresponds to a cluster structure in $$\mathcal O(\mathcal G)$$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $$A_n$$ type, which includes all the previously known examples. Namely, we subdivide all possible $$A_n$$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $$SL_n$$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $$SL_n$$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications. 

We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿𝑛 constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties. 

In this paper we settle a special case of the Grassmann convexity conjecture formulated by the second and the third authors about a decade ago. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders 4 and 5. 

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.