Related papers: Shard theory for $g$-fans

For a finite dimensional algebra $A$ over a field $k$, the 2-term silting complexes of $A$ gives a simplicial complex $\Delta(A)$ called the $g$-simplicial complex. We give tilting theoretic interpretations of the $h$-vectors and…

The $g$-fan $\Sigma(A)$ of a finite dimensional algebra $A$ is a non-singular fan in its real Grothendieck group, defined by tilting theory. If the union ${\rm P}(A)$ of the simplices associated with the cones of $\Sigma(A)$ is convex, we…

The $g$-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory. We give a classification of complete $g$-fans of rank 2. More explicitly, our first main result asserts that every complete…

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is…

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the $\tau$-cluster morphism category of a finite-dimensional algebra. This…

In the paper: Fans in the Theory of Real Semigroups. I. Algebraic Theory (submitted) we introduced the notion of fan in the categories of real semigoups and their dual abstract real spectra and developed the algebraic theory of these…

Motivated by the goal of studying cluster algebras in infinite type, we study the stability domains of modules for the preprojective algebra in the corresponding infinite types. Specifically, we study real bricks: those modules whose…

We study the $g$-fan associated with a skew-symmetrizable matrix in the sense of cluster algebras. We show that a skew-symmetrizable matrix is of finite type if and only if its $g$-fan is complete; equivalently (as we show), its support…

For an abelian length category $\mathcal{A}$ with only finitely many isoclasses of simple objects, we have the wall-chamber structure and the TF equivalence on the dual real Grothendieck group…

For a reductive group $G$, Harder-Narasimhan theory gives a structure theorem for principal $G$ bundles on a smooth projective curve $C$. A bundle is either semistable, or it admits a canonical parabolic reduction whose associated Levi…

For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called a quotient fan, and V. Pilaud and F.…

An arrangement of k-semilines in the Euclidean (projective) plane or on the 2-sphere is called a k-fan if all semilines start from the same point. A k-fan is an $\alpha$-partition for a probability measure $\mu$ if $\mu(\sigma_i)=\alpha_i$…

We establish a bijection between torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra A and pairs (Z, I) formed by a closed rigid set Z in the Ziegler spectrum of A and a set I of indecomposable…

Let $X_\Sigma$ be a smooth, not necessarily compact toric variety. We show that a certain complex, defined in terms of the fan $\Sigma$, computes the integral cohomology of $X_\Sigma$, including the module structure over the homology of the…

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…

We define the Z/2Z Hodge spaces H_{pq}(\Sigma) of a fan \Sigma. If \Sigma is the normal fan of a reflexive polytope \Delta then we use polyhedral duality to compute the Z/2Z Hodge Spaces of \Sigma. In particular, if the cones of dimension…

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact \'etale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated…

We study the $G$-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose…

We give an explicit formula for the Higgs and Coulomb branch Hilbert series for the class of 3d N=4 superconformal gauge theories T^{\sigma}_{\rho}(G) corresponding to a set of D3 branes ending on NS5 and D5-branes, with or without O3…

The goal of this note two-fold. First, I draw attention to the way that semistability (in the sense of King) gives us a geometrical picture of (some of) the extension-closed abelian subcategories of a finite-dimensional algebra. Second, I…