Coxeter Group Actions on Interacting Particle Systems

The Coxeter relations and KP map for non-commuting symbols

We give an action of the symmetric group on non-commuting indeterminates in terms of series in the corresponding Mal'cev-Newmann division ring. The action is constructed from the non-Abelian Hirota-Miwa (discrete KP) system. The corresponding companion map, which gives generators of the action, is discussed in the generic case and the corresponding explicit formulas have been found in the periodic reduction. We discuss also briefly connection of the companion to the KP map with context-free languages.

Chiral Polyhedra Derived From Coxeter Diagrams and Quaternions

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their duals the Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups $W(A_1 \oplus A_1 \oplus A_1)$, $W(A_3)$, $W(B_3)$ and $W(H_3)$ to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosah…

Finite orbits of the braid group action on sets of reflections

The finite orbits of the braid group action on Stokes matrices are studied and are shown to be the orbits on ordered sets of reflections, generating finite groups. All invariants of a reflection arrangement are determined. Determination of the orbits of the braid group on non-redundant generaing reflections in finite groups is done in a new way. The original idea is to introduce universal sets of generators in each group. These sets are found for all finite Coxeter groups. The orbits are classified for non-degenerate arrangement matrices.

从仿射A₄到仿射H₂：五重镶嵌的群论分析

From Affine $A_4$ to Affine $H_2$: Group Theoretical Analysis of Five-fold Tilings

本文通过群论分析，探讨了从仿射A₄到仿射H₂的转换，揭示了五重镶嵌的结构与对称性。

Action-angle variables for dihedral systems on the circle

A nonrelativistic particle on a circle and subject to a cos^{-2}(k phi) potential is related to the two-dimensional (dihedral) Coxeter system I_2(k), for k in N. For such `dihedral systems' we construct the action-angle variables and establish a local equivalence with a free particle on the circle. We perform the quantization of these systems in the action-angle variables and discuss the supersymmetric extension of this procedure. By allowing radial motion one obtains related two-dimensional systems, including A_2, BC_2 and G_2 three-particle rational Calogero models on R, which we also analyz…

变分相互作用粒子系统与Vlasov方程

Variational interacting particle systems and Vlasov equations

本文研究了相互作用粒子系统的优化问题，证明了临界点满足Vlasov方程，并将极小解与Hamilton-Jacobi-Bellman方程联系起来。