We prove the quantum version - for Hecke algebras
$H(A_n)$ of type $A$ at roots of unity - of Kleshchev's modular branching rule for symmetric groups. This
result describes the socle of the restriction of an irreducible $H(A_n)$-module to the subalgebra
$H(A_{n-1})$. As a consequence, we describe the quantum version of the Mullineux involution describing the
irreducible module obtained on twisting an irreducible module with the sign representation.
1991
Mathematics Subject Classification: 20C05, 20G05.