Frequency of a representation of SO(3)

Question

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!

Answer 1

Though your interests are partly separate from the purely mathematical framework here, the basic theory is well developed. Notation and terminology vary, of course: e.g., your "semi-ordering" is usually called a "partial ordering". In the case of the symmetric group $S_n$, a convenient modern treatment is given by Gordon James in The Representation Theory of Symmetric Groups, Lect. Notes in Math. 682, Springer, 1978. Here the dominance ordering of partitions already comes into play in the early sections, where he constructs explicit models over the integers of the irreducible representations as "Specht modules" starting with a family of natural induced representations ("permutation modules").

Note however that in his later sections James is mostly interested in how all of this machinery over the integers behaves under reduction modulo a prime. Here the dominance ordering plays an even more crucial role in organizing the submodule structures.

In any case, the dominance ordering for $S_n$ is natural in many situations. Also, it generalizes to all finite (real) reflection groups as the Chevalley-Bruhat ordering, which makes sense for any Coxeter group. This kind of partial ordering played a key role, for example, in Chevalley's older treatment of Schubert varieties for special linear groups, where $S_n$ occurs as the Weyl group.

Turning to the compact real Lie groups $SO(3)$, there is again a purely mathematical development of the finite dimensional representations in an algebraic framework. Among many textbook treatments, see for instance section II.5 of Representations of Compact Lie Groups by Brocker and tom Dieck, Grad. Texts in Math. 98, Springer, 1985. Here the representations are naturally labelled by highest weights, identifiable with ordinary non-negative even integers: the representation $\rho_n$ has dimension $n+1$ for $n=0,2,4, \dots$. The twofold cover $SU(2)$ then has representations parametrized by all non-negative integers. Here the parametrization by integers actually gives a total ordering of highest weights, though for higher rank compact Lie groups there is only a partial ordering.

Such an ordering isn't immediately essential in the $SO(3)$ setting, but as in the work of James and others for $S_n$ it plays a much larger role when the representations are constructed algebraically over the integers and then reduced modulo a prime. (They tend to lose irreducibility, which creates a hierarchy of interesting submodules.)

Aside from the language involved, your notion of "frequency" does correlate with the standard notion of "highest weight" in the representation theory of compact Lie groups. Here as in the classical theory of finite group representations, Fourier analysis is often used as a tool in manipulating the resulting characters. But probably the most direct connection between the partial orderings of partitions and of highest weights comes in the relationship of Weyl groups to compact Lie groups of Lie type $A$ and arbitrary rank (not just $SO(3)$).