Left Cells in the Affine Weyl Group of TypeF̃4 Original Research Article

By applying an algorithm designed before, we complete the description for all the left cells of the affine Weyl groupWaof typeF̃4by finding a representative set of its left cells together with all its left cell graphs (or with all the associated essential graphs) in each of its two-sided cells. The generalized τ-invariants of left cells ofWaare exhibited graphically. A group-theoretical interpretation is given on the numbers of left cells ofWain some two-sided cells. Thus so far the left cells of all the affine Weyl groups of ranks less than or equal to 4 have been known explicitly. Some techniques are developed in applying the algorithm. As a consequence, we complete the verification of a conjecture concerning the characterization of left cells of Weyl groups and affine Weyl groups.