Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the 4-dimensional semi-regular polytope snub 24-cell and its symmetry group ${(W(D_{4})\mathord{/{\vphantom {(W(D_{4}) C_{2}}}. \kern-\nulldelimiterspace} C_{2}}):S_{3} $ of order 576 are obtained from the quaternionic representation of the Coxeter-Weyl group \textbf{$W(D_{4}).$}The symmetry group is an extension of the proper subgroup of the Coxeter-Weyl group \textbf{$W(D_{4})$}by the permutation symmetry of the Coxeter-Dynkin diagram \textbf{$D_{4} .$} The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector \textbf{$Λ=(τ, 1, τ, τ)$}or\textbf{}on the vector\textbf{$Λ=(σ, 1, σ, σ)$}in the Dynkin basis with\textbf{$τ=\frac{1+\sqrt{5}}{2} {\rm and}σ=\frac{1-\sqrt{5}}{2} {\rm .}$} The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasi regular 4D polytope invariant under the Coxeter-Weyl group \textbf{$W(F_{4}).$}Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24 cell is also constructed. Relevance of these structures to the Coxeter groups \textbf{$W(H_{4}){\rm and}W(E_{8})$}has been pointed out.
Radii of the E8 Gosset Circles as the Mass Excitations in the Ising Model
Zamolodchikov猜想暗示了特殊李群E8的存在，这一猜想似乎被Coldea等人对一维伊辛模型的量子相变实验所证实。由仿射Toda场论得出的E8模型预测了8个束缚态，其质量关系为递增阶m1，m2= tau m1，m3，m4，m5，m6=tau m3，m7= tau m4，m8= tau m5，其中tau=（1+ \sqrt（5））/2表示黄金比例。Coxeter群W（H4）是Coxeter-Weyl群W（E8）的极大子群。这些质量原来是成比例的半径戈塞特圆的考克斯特平面上获得的正交投影的根系E8。我们还注意到，质量m1，m3，m4和m5对应于通过投影600胞腔的顶点而获得的圆的半径，600胞腔是非晶体学Coxeter群W（H4）的4D多面体。简单根在考克斯特平面上的特殊非正交投影导致束缚态质量的数值精确为0.4745，0.7678，0.9438，1.141，1.403，1.527，1.846和2.270。我们注意到的前两个数值的前两个质量的束缚态所确定的Coldea等惊人的平等。al.
The Saito determinant for extended affine Weyl discriminant strata
University of Sheffield（谢菲尔德大学）；CNRS, DR 13, Montpellier（法国国家科学研究中心，DR 13，蒙彼利埃）；Università di Milano-Bicocca（米兰比可卡大学）；INFN sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）
