Rigid Surface Operator and Symbol Invariant of Partitions

The symbol is used to describe the Springer correspondence for the classical groups by Lusztig. We refine the explanation that the S-duality maps of the rigid surface operators are symbol preserving maps. And we find that the maps \(X_S\) $X_S$ and \(Y_S\) $Y_S$ used in the construction of S-duality maps are essentially the same. We clear up cause of the mismatch problem of the total number of the rigid surface operators between the \(B_n\) $B_n$ and \(C_n\) $C_n$ theories. And we construct all the \(B_n/C_n\) $B_n/C_n$ rigid surface operators which can not have a dual. A classification of the problematic surface operators is made.