Fingerprint Invariant of Partitions and Construction

TL;DR

This paper explores the properties of the fingerprint invariant of partitions, particularly for classical groups, and introduces a method to compute fingerprints of rigid partitions and semisimple operators using block decomposition.

Contribution

It provides a systematic construction of fingerprints for rigid partitions in Bn, Cn, Dn theories and defines operators for calculating fingerprints of semisimple operators.

Findings

Constructed fingerprints for rigid partitions in classical groups.

Developed operators for fingerprint calculation of semisimple operators.

Enhanced understanding of the fingerprint invariant's role in representation theory.

Abstract

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 $(\lambda^{'};\lambda^{"})$, we decompose $\lambda^{'}+\lambda^{"}$ into several blocks. We define operators to calculate the fingerprint for each block using the results of fingerprint of the unipotent operators.