Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

We introduced new subclasses of separately radial symbols: symmetric separately radial (with corresponding group \(S_n\rtimes {\mathbb {T}}^n\) $S_n⋊{\mathbb{T}}^n$ ) and alternating separately radial (with corresponding group \(A_n\rtimes \mathbb {T}^n\) $A_n⋊{\mathbb{T}}^n$ ) symbols, as well as the associated Toeplitz operators on the weighted Bergman spaces on the unit ball on \(\mathbb {C}^n\) ${\mathbb{C}}^n$ . Using a purely representation theoretic approach, we study the symmetries of the corresponding sequence of eingenvalues. Furthermore, we show that the symmetric separately radial Toeplitz operators are more general than radial Toeplitz operators, i.e., every radial Toeplitz operator is symmetric separately radial.