Symbol Invariant of Partition and the Construction

TL;DR

This paper introduces a unified, easy-to-use method for constructing symbols of partitions across classical groups, simplifying the understanding of Springer correspondence and related results.

Contribution

It provides a new, uniform definition and construction rules for symbols of partitions in classical groups, enhancing clarity and ease of application.

Findings

Unified construction rules for symbols across Bn, Cn, Dn theories

Simplified proof of symbol construction using formal partition operations

Closed-form formulas for symbols in different theories

Abstract

The symbol is used to describe the Springer correspondence for the classical groups. We propose equivalent definitions of symbols for rigid partitions in the $B_n$, $C_n$, and $D_n$ theories uniformly. Analysing the new definition of symbol in detail, we give rules to construct symbol of a partition, which are easy to remember and to operate on. We introduce formal operations of a partition, which reduce the difficulties in the proof of the construction rules. According these rules, we give a closed formula of symbols for different theories uniformly. As applications, previous results can be illustrated more clearly by the construction rules of symbol.