Supersymmetry at the Linear Collider

If supersymmetry (SUSY) is realized at the electroweak scale, its underlying structure and breaking mechanism may be explored with great precision by a future linear $e^+ e^-$ collider (LC) with a clean environment, tunable collision energy, high luminosity polarized beams, and additional $e^-e^-$, $e\gamma$ and $\gamma\gamma$ modes. In this report we summarize four papers submitted to the ICHEP04 conference about the precise measurements of the top squark parameters and $\tan\beta$, the impacts of the CP phases on the search for top/bottom squarks, the Majorana nature and CP violation in the neutralino system, the implications of the SUSY dark matter scenario for the LC experiments, and the characteristics of the neutralino sector of the next--to--minimal supersymmetric standard model at the LCComment: 5 pages, 3 figures, contribution to the proceedings of the 32nd International Conference on High-Energy Physics (ICHEP 04), Beijing, China, 16-22 Aug 200

The definability criterions for convex projective polyhedral reflection groups

Following Vinberg, we find the criterions for a subgroup generated by reflections \Gamma \subset \SL^{\pm}(n+1,\mathbb{R}) and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply the criterions for groups generated by reflections that act cocompactly on irreducible properly convex open subdomains of the $n$-dimensional projective sphere. This gives a method for constructing injective group homomorphisms from such Coxeter groups to \SL^{\pm}(n+1,\mathbb{Z}). Finally we provide some examples of \SL^{\pm}(n+1,\mathbb{Z})-representations of such Coxeter groups. In particular, we consider simplicial reflection groups that are isomorphic to hyperbolic simplicial groups and classify all the conjugacy classes of the reflection subgroups in \SL^{\pm}(n+1,\mathbb{R}) that are definable over $\mathbb{Z}$. These were known by Goldman, Benoist, and so on previously.Comment: 31 pages, 8 figure