In this paper, we propose a conjectural multiplicity formula for general spherical varieties. For all the cases where a multiplicity formula has been proved, including Whittaker model, Gan-Gross-Prasad model, Ginzburg-Rallis model, Galois model and Shalika model, we show that the multiplicity formula in our conjecture matches the multiplicity formula that has been proved. We also give a proof of this multiplicity formula in two new cases.

We give a finite presentation for the braid twist group of a decorated surface. If the decorated surface arises from a triangulated marked surface without punctures, we obtain a finite presentation for the spherical twist group of the associated 3-Calabi–Yau triangulated category. The motivation/application is that the result will be used to show that the (principal component of) space of stability conditions on the 3-Calabi–Yau category is simply connected in the sequel [King and Qiu, Invent. Math., to appear].

We study the cluster algebras arising from cluster tubes with rank bigger than 1. Cluster tubes are 2−Calabi-Yau triangulated categories which contain no cluster tilting objects, but maximal rigid objects. Fix a certain maximal rigid object T in the cluster tube C_n of rank n. For any indecomposable rigid object M in C_n, we define an analogous X_M of Caldero-Chapton's formula (or Palu's cluster character formula) by using the geometric information of M. We show that X_M,X_{M′} satisfy the mutation formula when M,M′ form an exchange pair, and that X_? : M↦X_M gives a bijection from the set of indecomposable rigid objects in C_n to the set of cluster variables of cluster algebra of type C_{n−1}, which induces a bijection between the set of basic maximal rigid objects in C_n and the set of clusters. This strengths a surprising result proved recently by Buan-Marsh-Vatne that the combinatorics of maximal rigid objects in the cluster tube Cn encode the combinatorics of the cluster algebra of type B_{n−1} since the combinatorics of cluster algebras of type B_{n−1} or of type C_{n−1} are the same by a result of Fomin and Zelevinsky. As a consequence, we give a categorification of cluster algebras of type C.

Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.

In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.