When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In particular, the authors of [1] consider the Young tableaux of the permutation and define a semi-ordering of such permutations based on their relative 'dominance'. To define this dominance ordering, the authors decompose a permutation on a set of $n$ elements into its constituent cycles. This decomposition is expressed as a partition of the cardinality $n$ of the set.

Definition (Dominance Ordering). Let $\lambda,\mu$ be partitions of $n$. Then $\lambda$ dominates $\mu$ if, for each $i$, $$\sum_{k=1}^i\lambda_k\geq \sum_{k=1}^i \mu_k.$$

Thus for a set of size $n$, the third-order partition $\lambda=(n-3,1,2)$ would dominate the fourth-order partition $\mu=(n-4,1,1,1)$. We think of the representations corresponding to $\lambda$ as being of 'higher frequency' than those corresponding to $\mu$. To clarify, it associates each irreducible representation of $S_n$ to a dominance ranking. This (partial) ordering establishes a foundation upon which we can band-limit a function, etc.

It seems natural that we could construct a basis of irreducible representations of $SO(3)$ in a similar manner. Has an analogous definition of the 'frequency' of Fourier basis representations been established for $SO(3)$?

For further reading, I suggest [1] Fourier Theoretic Probabilistic Inference over Permutations, Huang et. al.

Thank you!

When generalizing the basic tenets of Fourier Theory to the symmetric group $S_n$, we can define a notion of the frequency of a basis function (i.e. an irreducible representation of $S_n$). In particular, the authors of [1] consider the Young tableaux of the permutation and define a semi-ordering of such permutations based on their relative 'dominance'. To define this dominance ordering, the authors decompose a permutation on a set of $n$ elements into its constituent cycles. This decomposition is expressed as a partition of the cardinality $n$ of the set.

Definition (Dominance Ordering). Let $\lambda,\mu$ be partitions of $n$. Then $\lambda$ dominates $\mu$ if, for each $i$, $$\sum_{k=1}^i\lambda_k\geq \sum_{k=1}^i \mu_k.$$

Thus for a set of size $n$, the third-order partition $\lambda=(n-3,1,2)$ would dominate the fourth-order partition $\mu=(n-4,1,1,1)$. We think of the representations corresponding to $\lambda$ as being of 'higher frequency' than those corresponding to $\mu$. To clarify, it associates each irreducible representation of $S_n$ to a dominance ranking. This (partial) ordering establishes a foundation upon which we can band-limit a function, etc.

It seems natural that we could construct a basis of irreducible representations of $SO(3)$ in a similar manner. **Has an analogous definition of the 'frequency' of Fourier basis representations been established for $SO(3) $?

Thank you!