A generalization of Coxeter groups, root systems, and Matsumoto's theorem

融合范畴中的Coxeter箭图表示与Gabriel定理

Coxeter quiver representations in fusion categories and Gabriel's theorem

本文引入Coxeter箭图表示概念，利用$U_q(\mathfrak{s}\mathfrak{l}_2)$融合范畴推广了经典箭图表示理论，证明了涵盖所有Coxeter--Dynkin图的广义Gabriel定理，并揭示了不可分解表示与正根的双射关系。

Kac--Moody Borel包络代数的双有理等价

Birational Equivalences for Kac--Moody Borel Enveloping Algebras

本文证明有限秩Kac--Moody代数的Borel包络代数与剩余Coxeter中心化子张量仿射空间双有理等价，并推广到量子情形。

The adjoint group of a Coxeter quandle

We give explicit descriptions of the adjoint group of the Coxeter quandle $Q_W$ associated with an arbitrary Coxeter group $W$. The adjoint group of $Q_W$ turns out to be an intermediate group between $W$ and the corresponding Artin group $A_W$, and fits into a central extension of $W$ by a finitely generated free abelian group. We construct $2$-cocycles of $W$ corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of $Q_W$ is isomorphic to the commutator subgroup of $W$. Finally, the root system $Φ_W$ associated with a Coxeter group $W$…

Skew divided difference operators in the Nichols algebra associated to a finite Coxeter group

Let $(W,S)$ be a finite Coxeter system with root system $R$ and with set of positive roots $R^+$. For $α\in R$, $v,w\in W$, we denote by $\partial_α$, $\partial_w$ and $\partial_{w/v}$ the divided difference operators and skew divided difference operators acting on the coinvariant algebra of $W$. Generalizing the work of Liu, we prove that $\partial_{w/v}$ can be written as a polynomial with nonnegative coefficients in $\partial_α$ where $α\in R^+$. In fact, we prove the stronger and analogous statement in the Nichols-Woronowicz algebra model for Schubert calculus on $W$ after Bazlov. We draw …

Action of Coxeter groups on m-harmonic polynomials and KZ equations

The Matsuo-Cherednik correspondence is an isomorphism from solutions of Knizhnik-Zamolodchikov equations to eigenfunctions of generalized Calogero-Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials, recently introduced in math-ph/0105014. We compute the Poincare' polynomials for the space H_m and of its isotypical components corresponding to each irreducible representation of the group G.…

The $E_8$ geometry from a Clifford perspective

This paper considers the geometry of $E_8$ from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system $H_3$ gives rise to the largest (and therefore exceptional) non-crystallographic root system $H_4$. Arnold's trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and …