For a generic $n imes n$ complex matrix, we have the LU decomposition (obtained from Gaussian elimination method) and the Jordan canonical form associated to it. In this talk, we will discuss their generalization to loop groups: the Bruhat decomposition and the decomposition into $\sigma$-conjugacy classes. We will describe the relation between these two decompositions, via the comparision between the ($\sigma$-)conjugacy classes of loop groups and associated affine Weyl groups. In the end, we will discuss some applications in arithmetic geometry: the Goertz-Haines-Kottwtiz-Reuman conjecture on the dimensions of affine Deligne-Lusztig varieties, the (generalized) Grothendieck conjecture on the closure relations between the $\sigma$-conjugacy classes, and the Kottwitz-Rapoport conjecture on the isocrystals with additional structures.