Invariants of partitions and representative elements

TL;DR

This paper explores the relationship between symbol and fingerprint invariants of rigid semisimple operators, providing a method to construct representative elements with matching invariants and establishing a connection between these invariants.

Contribution

It introduces a construction for representative elements sharing the same symbol invariant and relates it to the fingerprint invariant, linking two key invariants in the theory.

Findings

Constructed representative elements with the same symbol invariant.

Derived the fingerprint of these representative elements.

Established a mapping between symbol and fingerprint invariants.

Abstract

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 described by pairs of partitions $(\lambda^{'}, \lambda^{"})$. We construct a nice representative element of the rigid semisimple operators with the same symbol invariant. The fingerprint of the representative element can be obtained immediately. We also discuss the representative element of rigid semisimple operator with the same fingerprint invariant. Our construction can be regarded as the maps between these two invariants.