Construction of the Symbol Invariant of Partition

Abstract

Symbol is used to describe the Springer correspondence for the classical groups. We prove two structure theorems of symbol. We propose a construction of the symbol of the rigid partitions in the $B_n$, $C_n$, and $D_n$ theories. This construction is natural and consists of two basic building blocks. Using this construction, we give closed formulas of symbols for the rigid partitions in the $B_n, C_n$, and $D_n$ theories. One part of the closed formula is universal and other parts are determined by the specific theory. A comparison of between this closed formula and the old one is made. Previous results can be illustrated more clearly by this closed formula.