Related papers: Planar networks and simple Lie groups beyond type …

We characterize finite-dimensional thick representations over ${\Bbb C}$ of connected complex semi-simple Lie groups by irreducible representations which are weight multiplicity-free and whose weight posets are totally ordered sets.…

Network models, which abstractly are given by lax symmetric monoidal functors, are used to construct operads for modeling and designing complex networks. Many common types of networks can be modeled with simple graphs with edges weighted by…

CGL extensions, named after G. Cauchon, K. Goodearl, and E. Letzter, are a special class of noncommutative algebras that are iterated Ore extensions of associative algebras with compatible torus actions. Examples of CGL extensions include…

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a…

In the prequel of this paper, we have associated a family of cluster X-varieties to the dual Poisson-Lie group(G*,\pi_*) of (G,\pi_G) when (G,\pi_G) is a complex semi-simple Lie group of adjoint type, given with the standard Poisson…

Let $C$ be a symmetrizable generalized Cartan matrix. We introduce four different versions of double Bott-Samelson cells for every pair of positive braids in the generalized braid group associated to $C$. We prove that the decorated double…

A tensor network is a product of tensors associated with vertices of some graph $G$ such that every edge of $G$ represents a summation (contraction) over a matching pair of indexes. It was shown recently by Valiant, Cai, and Choudhary that…

Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G. We show that the action on triples by orientation preserving…

Fock and Goncharov introduced a quantization of higher Teichm\"uller theory using cluster Poisson varieties and their noncommutative deformations, associating to a complex semisimple Lie group $G$ and a marked surface $S$ a quantum algebra…

Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti-Guido-Longo (BGL)…

Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$ on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant multiplicities via the…

The inclusion of link weights into the analysis of network properties allows a deeper insight into the (often overlapping) modular structure of real-world webs. We introduce a clustering algorithm (CPMw, Clique Percolation Method with…

We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to…

Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on…

The recent high level of interest in weighted complex networks gives rise to a need to develop new measures and to generalize existing ones to take the weights of links into account. Here we focus on various generalizations of the…

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group $D$. The double of a group $G$ has a pointwise decomposition $D\sim G\times G^*$, where $G$ and…

This paper is devoted to the representation theory of quantum coordinate algebra $\mathbb{C}_q[G]$, for a semisimple Lie group $G$ and a generic parameter $q$. By inspecting the actions of normal elements on tensor modules, we generalize a…

We derive an efficient method to perform clustering of nodes in Gaussian graphical models directly from sample data. Nodes are clustered based on the similarity of their network neighborhoods, with edge weights defined by partial…

We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial)…

We show the existence of cluster $\mathcal{A}$-structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig's coordinates. Several…