Invariants of partitions

The symbol invariant is used to describe the Springer correspondence for the classical groups by Lusztig. And the fingerprint invariant can be used to describe the Kazhdan-Lusztig map. They are invariants of rigid semisimple operators labeled by pairs of partitions $(\lambda^{'}, \lambda^{"})$. It is conjectured that the symbol invariant is equivalent to the fingerprint invariant. We prove that the former imply the latter one. We make a classification of the maps preserving symbol, and then prove these maps preserve fingerprint. We also discuss the classification of the maps preserving fingerprint, which is found related to the conditions of the definition of the fingerprint. The constructions of the symbol and the fingerprint invariants in previous works play significant roles in the proof.