The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set tors A of torsion classes over a finitedimensional algebra A. We show that tors A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of tors A. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that tors A is completely congruence uniform. When I is a two-sided ideal of A, tors(A/I) is a lattice quotient of tors A which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of tors A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras Π, for which tors Π is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between tors kQ and the Cambrian lattice when Q is a Dynkin quiver. We also prove that, in type A, the algebraic quotients of tors Π are exactly its Hasse-regular lattice quotients.
ContentsDEMONET, IYAMA, READING, REITEN, AND THOMAS 4.2. Wide subcategories 29 4.3. Algebraic characterizations of the forcing order 33 5. Algebraic lattice congruences on torsion classes 38 5.1. General results on morphisms of algebras 38 5.2. Algebraic lattice quotients 42 6. The preprojective algebra and the weak order 46 6.1. Weak order on Weyl groups 47 6.2. Preprojective algebras and Weyl groups 49 6.3. The preprojective algebra and the weak order in type A 52 6.4. Combinatorial realizations 55 7. Cambrian and biCambrian lattices 56 7.1. A representation-theoretic interpretation of Cambrian lattices 56 7.2. The bipartite biCambrian congruence 62 Acknowledgments 63 References 63
A new construction of the associahedron was recently given by Arkani-Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply-laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the F -polynomials of the corresponding cluster algebras.
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