Related papers: Explicit Forms of Cluster Variables on Double Bruh…

Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its…

For a semisimple adjoint algebraic group $G$ and a Borel subgroup $B$, consider the double classes $BwB$ in $G$ and their closures in the canonical compactification of $G$: we call these closures large Schubert varieties. We show that these…

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called…

The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $E_{8}$ or its highest weight is minuscule. In this paper, we prove…

Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational…

In this paper, we characterize the dynamical orders of minimal Morse components and, partially, of control sets on maximal compact subgroups of a semisimple Lie group as an algebraic order similar to the Bruhat order of the Weyl group. We…

In a recent paper Cohen, Liu and Yu introduce the Type $C$ Brauer algebra. We show that this algebra is an iterated inflation of hyperoctahedral groups, and that it is cellularly stratified. This gives an indexing set of the standard…

Motivated by the geometry of certain hyperplane arrangements, Manin and Schechtman defined for each positive integer n a hierarchy of finite partially ordered sets B(n, k), indexed by positive integers k, called the higher Bruhat orders.…

Let $\mathscr{C}$ be a $2$-Calabi-Yau triangulated category with two cluster tilting subcategories $\mathscr{T}$ and $\mathscr{U}$. Results by Demonet-Iyama-Jasso and J{\o}rgensen-Yakimov known as tropical duality says that the index with…

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition…

A monomial basis and a filtration of subalgebras for the universal enveloping algebra $U(g_l)$ of a complex simple Lie algebra $g_l$ of type $A_l$ is given in this note. In particular, a new multiplicity formula for the Weyl module…

We characterise the strictly closed left invariant C*-subalgebras of the C*-algebra C_b(G) of bounded continuous functions on a locally compact group G. On the dual side, we characterise the strictly closed invariant C*-subalgebras of the…

Let G be a split semi-simple adjoint group, and S a colored decorated surface, given by an oriented surface with punctures, special boundary points, and a specified collection of boundary intervals. We introduce a moduli space P(G,S)…

Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the…

Let $G$ be a reductive group with Borel $B$ and Weyl group $W$. Then $B$-double cosets in $G$ are indexed by the Weyl group, say $O(w)$ for $w\in W$. Then we prove the minimal $B$-double coset in the convolution $O(w_1)*O(w_2)$ is…

This is an expanded version of the notes for the two lectures at the 2004 International Mathematics Conference (Chonbuk National University, August 4-6, 2004). The first lecture discusses the origins of cluster algebras, with the focus on…

We describe a ring whose category of Cohen-Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of k-planes in n-space. More precisely, there is a…

Let $G$ be a connected reductive complex algebraic group with a maximal torus $T$. We denote by $\Lambda$ the cocharacter lattice of $(T,G)$. Let $\Lambda^+ \subset \Lambda$ be the submonoid of dominant coweights. For $\lambda \in…

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we…

The authors compute the support varieties of all irreducible modules for the small quantum group $u_\zeta(\mathfrak{g})$, where $\mathfrak{g}$ is a simple complex Lie algebra, and $\zeta$ is a primitive $\ell$-th root of unity with $\ell$…