There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group . The kind which is covariantly functorial is some variation on the group algebra , which is the free -module on with multiplication inherited from the multiplication on . The kind which is contravariantly functorial is some variation on the algebra of functions with pointwise multiplication.
When and when is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of , the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of , while in the case of , the corresponding state is by definition integration with respect to normalized Haar measure on .
In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on using representation theory. This construction will in some sense explain why the category of (finite-dimensional continuous unitary) representations of behaves like an inner product space (with being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on .
Discrete groups and Plancherel measure
Let be a group and consider the -algebra with involution given by extending . When is finite, is finite-dimensional, so states on it are completely described by the results in the previous post. In general, we may construct states on from unitary representations of on inner product spaces by choosing unit vectors in them and considering pure states. If is a finite-dimensional unitary representation of , then the normalized character
extends to a state on . The distribution of a random variable with respect to this state is given by the uniform distribution on its eigenvalues as an operator on , as can be seen by comparing moments.
There is also a distinguished state given by setting
for all which generalizes the normalized trace on when is finite. With respect to the corresponding inner product, the elements are orthonormal. The moments of a random variable with respect to this distinguished state are related to random walks on : for example, if , then counts the number of closed walks of length from the identity to itself on the Cayley graph of with generating set .
When is given the normalized trace, how should we interpret the corresponding noncommutative probability space ? When is finite, we know that is canonically a finite product
where runs through the irreducible representations of . The corresponding noncommutative probability space is therefore a disjoint union of the spaces associated to each of the matrix -algebras ; moreover, the system is in with probability given by the normalized trace of the idempotent corresponding to above. The trace of an idempotent is the dimension of its image, so we conclude that the system is in with probability
This defines Plancherel measure on the irreducible representations of . The corresponding commutative probability space can be constructed as follows. has a distinguished commutative subalgebra given by its center . When is finite, is canonically a finite product
where is the number of irreducible representations of . Consequently, can be canonically identified with the set of irreducible representations of , and the normalized trace on descends to the state on describing Plancherel measure on the irreducible representations.
It is plausible that similar results hold when is infinite, although to actually obtain them it would be sensible to complete to get more analytic structure.
Compact groups and Haar measure
Let be a compact Hausdorff group and consider the C*-algebra of continuous functions with pointwise conjugation and pointwise multiplication. By the Riesz representation theorem, a state on is precisely a Radon probability measure on . A distinguished such state is given by integration against normalized Haar measure :
By the uniqueness of Haar measure, this is the unique state which is invariant under translation by elements of . The corresponding probability space is of course just equipped with normalized Haar measure.
has a natural closed subalgebra consisting of class functions (functions invariant under conjugation), which is therefore also a C*-algebra; its Gelfand spectrum can be identified with the space of conjugacy classes of , and the restriction of the state above to describes the pushforward of Haar measure to the space of conjugacy classes of .
Integration against Haar measure on the conjugacy classes of is completely determined by the representation theory of in the following sense. Inside is a natural subspace spanned by the characters of finite-dimensional continuous unitary representations of . This subspace is closed under multiplication by taking tensor products and closed under conjugation by taking duals, so it is a -subalgebra. By the Peter-Weyl theorem, is in fact a dense -subalgebra, so is completely determined by its values on characters, but by Schur’s lemma we know that the integral
is the dimension of the invariant subspace of . In other words, is completely determined by its values on irreducible characters, and these are given by for the trivial representation and otherwise. In particular, the joint moments of a collection of are given by the dimension of the invariant subspace of their tensor product, so understanding these dimensions is essentially equivalent to knowing .
Example. Let and let be the defining representation. The character is just the trace of regarded as a matrix, which completely determines its conjugacy class; consequently, already separates points, and to understand Haar measure on the conjugacy classes of it suffices to understand the moments of . But these are just the dimensions of the invariant subspaces of the tensor powers of . The explicit description
of the tensor product of with the other irreducible representations of can be used to compute by a combinatorial argument that
where are the Catalan numbers. Consequently, the moments of are the same as those of the Wigner semicircular distribution with , and this completely describes Haar measure on the conjugacy classes of .
Example. Let and let be the defining representation. The character is again just the trace of regarded as a matrix, which completely determines its conjugacy class. This is less obvious than for and it is false for higher : it comes from the fact that the conjugacy class of an element of is determined by its eigenvalues, which are in turn determined by symmetric functions of the eigenvalues. But these are determined by the characters of the exterior powers of , and when the representation is trivial and is dual to , hence the character of one determines the other.
As a consequence, to understand Haar measure on the conjugacy classes of it suffices to understand the joint moments of the real and imaginary part of . Computations of some of these moments using highest weight theory can be found at this MO question.
A strategy for extending this algebraic description of Haar measure on the conjugacy classes to Haar measure on the group itself can be found in David Speyer’s answer to this MO question.
Haar measure and representation theory
The category of finite-dimensional continuous unitary representations of a compact (Hausdorff) group bears a striking resemblance to an inner product space, mainly due to the properties of the Hom functor . The Hom functor is is bilinear in the sense that it respects finite direct sums in both arguments. We always have because of the identity morphism. The Hom functor is also contravariantly functorial in the first argument and covariantly functorial in the second, analogous to how the inner product (in the physicist’s convention) is conjugate-linear in the first argument and linear in the second. Schur’s lemma can be restated as saying that the irreducible representations of are an orthonormal basis with respect to . Finally, there is the formula
showing that naturally extends to an inner product on the space of class functions on .
Baez’s Higher-Dimensional Algebra II: 2-Hilbert Spaces uses this observation as motivation to categorify the notion of a Hilbert space. In this post I would prefer instead to hint at a categorification of the notion of a random algebra. The first step is to observe that
and to replace thinking directly about with thinking about tensor product, dual, and invariant subspace. These structures make seem less like an inner product space and more like a random algebra. The tensor product is multiplication, taking duals is the -operation, and taking invariant subspaces is the state.
In fact, all of this structure descends to the Grothendieck group of , which is the universal way to assign every object in an element of an abelian group in such a way that direct sum is taken to multiplication. More explicitly, the Grothendieck group is the free abelian group on symbols standing for the irreducible (finite-dimensional continuous unitary) representations of . Tensor product naturally descends to a multiplication on the Grothendieck group, so in this context it is sometimes called the Grothendieck ring or representation ring of . Explicitly, if
Dual naturally descends to a -involution given by extending , and taking invariant subspaces naturally descends to a map from the Grothendieck ring of to the Grothendieck ring of , which is just ; explicitly, is sent to if it is trivial and otherwise.
Tensoring with (and extending the -operation appropriately), we get precisely the random algebra above. In other words, can be thought of as a categorified random algebra whose decategorification is precisely . This is a more precise version of the statement that understanding the representation theory of is equivalent to understanding Haar measure on the conjugacy classes of and suggests a general strategy for finding interesting random algebras, which is to decategorify interesting monoidal categories with duals, such as the category of representations of a Hopf algebra.