Fingerprint Invariant of Partitions and Construction

The fingerprint invariant of partitions can be used to describe the Kazhdan-Lusztig map for the classical groups. We discuss the basic properties of fingerprint. We construct the fingerprints of rigid partitions in the $B_n$, $C_n$, and $D_n$ theories. To calculate the fingerprint of a rigid semisimple operator $({\lambda }^{{}^{\mathrm{\prime }}};{\lambda }^{\mathrm{"}})$, we decompose ${\lambda }^{{}^{\mathrm{\prime }}}+{\lambda }^{\mathrm{"}}$ into several blocks. We define operators to calculate the fingerprint for each block using the results of fingerprint of the unipotent operators.