The Mukai conjecture is an inequality involving the second Betti number b2 and the index k of a Fano variety M, the index being the largest integer dividing the first Chern class of the tangent bundle, which is not zero in the Fano case. More precisely it asserts that

b2(k−1)<n+1 (∗),

where n denotes the complex dimension of M. In this talk I will explain a new approach to prove the Mukai conjecture for flag varieties and show how it relates with the Kostant game which is used to determine the system of roots of a Dynkin diagram.