Noncommutative probability and group theory

There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group Image may be NSFW.
Clik here to view.. The kind which is covariantly functorial is some variation on the group algebra Image may be NSFW.
Clik here to view. k[G] , which is the free Image may be NSFW.
Clik here to view.-module on Image may be NSFW.
Clik here to view. with multiplication inherited from the multiplication on Image may be NSFW.
Clik here to view.. The kind which is contravariantly functorial is some variation on the algebra Image may be NSFW.
Clik here to view. k^G of functions Image may be NSFW.
Clik here to view. $G \to k$ with pointwise multiplication.

When Image may be NSFW.
Clik here to view. $k = \mathbb{C}$ and when Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ , the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of Image may be NSFW.
Clik here to view., while in the case of Image may be NSFW.
Clik here to view. $\mathbb{C}^G$ , the corresponding state is by definition integration with respect to normalized Haar measure on Image may be NSFW.
Clik here to view..

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 Image may be NSFW.
Clik here to view. it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on Image may be NSFW.
Clik here to view. using representation theory. This construction will in some sense explain why the category Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ of (finite-dimensional continuous unitary) representations of Image may be NSFW.
Clik here to view. behaves like an inner product space (with Image may be NSFW.
Clik here to view. $\text{Hom}(V, W)$ being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on Image may be NSFW.
Clik here to view..

Discrete groups and Plancherel measure

Let Image may be NSFW.
Clik here to view. be a group and consider the Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ with involution given by extending Image may be NSFW.
Clik here to view. $g^{\dagger} = g^{-1}$ . When Image may be NSFW.
Clik here to view. is finite, Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ is finite-dimensional, so states on it are completely described by the results in the previous post. In general, we may construct states on Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ from unitary representations of Image may be NSFW.
Clik here to view. on inner product spaces by choosing unit vectors in them and considering pure states. If Image may be NSFW.
Clik here to view. is a finite-dimensional unitary representation of Image may be NSFW.
Clik here to view., then the normalized character

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(g) = \frac{1}{\dim V} \chi_V(g)$

extends to a state on Image may be NSFW.
Clik here to view.. The distribution of a random variable with respect to this state is given by the uniform distribution on its eigenvalues as an operator on Image may be NSFW.
Clik here to view., as can be seen by comparing moments.

There is also a distinguished state given by setting

Image may be NSFW.
Clik here to view. $\mathbb{E}(g) = 0$

for all Image may be NSFW.
Clik here to view. $g \neq 1$ which generalizes the normalized trace on Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ when Image may be NSFW.
Clik here to view. is finite. With respect to the corresponding inner product, the elements Image may be NSFW.
Clik here to view. $g \in G$ are orthonormal. The moments of a random variable with respect to this distinguished state are related to random walks on Image may be NSFW.
Clik here to view.: for example, if Image may be NSFW.
Clik here to view. $a = g + h + g^{-1} + h^{-1}$ , then Image may be NSFW.
Clik here to view. $\mathbb{E}(a^n)$ counts the number of closed walks of length Image may be NSFW.
Clik here to view. from the identity to itself on the Cayley graph of Image may be NSFW.
Clik here to view. with generating set Image may be NSFW.
Clik here to view. $\{ g, h, g^{-1}, h^{-1} \}$ .

When Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ is given the normalized trace, how should we interpret the corresponding noncommutative probability space Image may be NSFW.
Clik here to view. $\text{Spec } \mathbb{C}[G]$ ? When Image may be NSFW.
Clik here to view. is finite, we know that Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ is canonically a finite product

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{C}[G] \cong \prod_{i=1}^r (V_i \Rightarrow V_i)$

where Image may be NSFW.
Clik here to view. V_i runs through the irreducible representations of Image may be NSFW.
Clik here to view.. The corresponding noncommutative probability space is therefore a disjoint union of the spaces associated to each of the matrix Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebras Image may be NSFW.
Clik here to view. $V_i \Rightarrow V_i$ ; moreover, the system is in Image may be NSFW.
Clik here to view. $\text{Spec } (V_i \Rightarrow V_i)$ with probability given by the normalized trace of the idempotent corresponding to Image may be NSFW.
Clik here to view. $V_i \Rightarrow V_i$ above. The trace of an idempotent is the dimension of its image, so we conclude that the system is in Image may be NSFW.
Clik here to view. $\text{Spec } V_i \Rightarrow V_i$ with probability

Image may be NSFW.
Clik here to view. $\displaystyle p_i = \frac{1}{|G|} (\dim V_i)^2$ .

This defines Plancherel measure on the irreducible representations of Image may be NSFW.
Clik here to view.. The corresponding commutative probability space can be constructed as follows. Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ has a distinguished commutative subalgebra given by its center Image may be NSFW.
Clik here to view. $Z(\mathbb{C}[G])$ . When Image may be NSFW.
Clik here to view. is finite, Image may be NSFW.
Clik here to view. $Z(\mathbb{C}[G])$ is canonically a finite product

Image may be NSFW.
Clik here to view. $\displaystyle Z(\mathbb{C}[G]) \cong \prod_{i=1}^r Z(V_i \Rightarrow V_i) \cong \mathbb{C}^r$

where Image may be NSFW.
Clik here to view. is the number of irreducible representations of Image may be NSFW.
Clik here to view.. Consequently, Image may be NSFW.
Clik here to view. $\text{Spec } Z(\mathbb{C}[G])$ can be canonically identified with the set of irreducible representations of Image may be NSFW.
Clik here to view., and the normalized trace on Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ descends to the state on Image may be NSFW.
Clik here to view. $Z(\mathbb{C}[G])$ describing Plancherel measure on the irreducible representations.

It is plausible that similar results hold when Image may be NSFW.
Clik here to view. is infinite, although to actually obtain them it would be sensible to complete Image may be NSFW.
Clik here to view. $\mathbb{C}[G]$ to get more analytic structure.

Compact groups and Haar measure

Let Image may be NSFW.
Clik here to view. be a compact Hausdorff group and consider the C*-algebra Image may be NSFW.
Clik here to view. C(G) of continuous functions Image may be NSFW.
Clik here to view. $G \to \mathbb{C}$ with pointwise conjugation and pointwise multiplication. By the Riesz representation theorem, a state on Image may be NSFW.
Clik here to view. C(G) is precisely a Radon probability measure on Image may be NSFW.
Clik here to view.. A distinguished such state is given by integration against normalized Haar measure Image may be NSFW.
Clik here to view. $\mu$ :

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(f) = \int_G f \, d \mu$ .

By the uniqueness of Haar measure, this is the unique state which is invariant under translation by elements of Image may be NSFW.
Clik here to view.. The corresponding probability space is of course just Image may be NSFW.
Clik here to view. equipped with normalized Haar measure.

Image may be NSFW.
Clik here to view. C(G) has a natural closed subalgebra Image may be NSFW.
Clik here to view. $C_{\text{cl}}(G)$ 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 Image may be NSFW.
Clik here to view., and the restriction of the state above to Image may be NSFW.
Clik here to view. $C_{\text{cl}}(G)$ describes the pushforward of Haar measure to the space of conjugacy classes of Image may be NSFW.
Clik here to view..

Integration against Haar measure on the conjugacy classes of Image may be NSFW.
Clik here to view. is completely determined by the representation theory of Image may be NSFW.
Clik here to view. in the following sense. Inside Image may be NSFW.
Clik here to view. $C_{\text{cl}}(G)$ is a natural subspace Image may be NSFW.
Clik here to view. $\text{Char}(G)$ spanned by the characters Image may be NSFW.
Clik here to view. $\chi_V$ of finite-dimensional continuous unitary representations Image may be NSFW.
Clik here to view. of Image may be NSFW.
Clik here to view.. This subspace is closed under multiplication by taking tensor products and closed under conjugation by taking duals, so it is a Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebra. By the Peter-Weyl theorem, Image may be NSFW.
Clik here to view. $\text{Char}(G)$ is in fact a dense Image may be NSFW.
Clik here to view. $^{\dagger}$ -subalgebra, so Image may be NSFW.
Clik here to view. $\mathbb{E}$ is completely determined by its values on characters, but by Schur’s lemma we know that the integral

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(\chi_V) = \int_G \chi_V(g) \, d \mu = \dim \text{Inv}(V)$

is the dimension of the invariant subspace of Image may be NSFW.
Clik here to view.. In other words, Image may be NSFW.
Clik here to view. $\mathbb{E}$ is completely determined by its values on irreducible characters, and these are given by Image may be NSFW.
Clik here to view. for the trivial representation and Image may be NSFW.
Clik here to view. otherwise. In particular, the joint moments of a collection of Image may be NSFW.
Clik here to view. $\chi_V$ are given by the dimension of the invariant subspace of their tensor product, so understanding these dimensions is essentially equivalent to knowing Image may be NSFW.
Clik here to view. $\mathbb{E}$ .

Example. Let Image may be NSFW.
Clik here to view. $G = \text{SU}(2)$ and let Image may be NSFW.
Clik here to view. be the defining representation. The character Image may be NSFW.
Clik here to view. $\chi_V(g)$ is just the trace of Image may be NSFW.
Clik here to view. $g \in \text{SU}(2)$ regarded as a matrix, which completely determines its conjugacy class; consequently, Image may be NSFW.
Clik here to view. $\chi_V$ already separates points, and to understand Haar measure on the conjugacy classes of Image may be NSFW.
Clik here to view. $\text{SU}(2)$ it suffices to understand the moments of Image may be NSFW.
Clik here to view. $\chi_V$ . But these are just the dimensions Image may be NSFW.
Clik here to view. $\dim \text{Inv}(V^{\otimes n})$ of the invariant subspaces of the tensor powers of Image may be NSFW.
Clik here to view.. The explicit description

Image may be NSFW.
Clik here to view. $\displaystyle V \otimes S^n(V) \cong S^{n+1}(V) \oplus S^{n-1}(V)$

of the tensor product of Image may be NSFW.
Clik here to view. with the other irreducible representations of Image may be NSFW.
Clik here to view. $\text{SU}(2)$ can be used to compute by a combinatorial argument that

Image may be NSFW.
Clik here to view. $\displaystyle \dim \text{Inv}(V^{\otimes (2n-1)}) = 0, \dim \text{Inv}(V^{\otimes 2n}) = C_n$

where Image may be NSFW.
Clik here to view. C_n are the Catalan numbers. Consequently, the moments of Image may be NSFW.
Clik here to view. $\chi_V$ are the same as those of the Wigner semicircular distribution with Image may be NSFW.
Clik here to view. R = 2 , and this completely describes Haar measure on the conjugacy classes of Image may be NSFW.
Clik here to view. $\text{SU}(2)$ .

Example. Let Image may be NSFW.
Clik here to view. $G = \text{SU}(3)$ and let Image may be NSFW.
Clik here to view. be the defining representation. The character Image may be NSFW.
Clik here to view. $\chi_V(g)$ is again just the trace of Image may be NSFW.
Clik here to view. $g \in \text{SU}(3)$ regarded as a matrix, which completely determines its conjugacy class. This is less obvious than for Image may be NSFW.
Clik here to view. $\text{SU}(2)$ and it is false for higher Image may be NSFW.
Clik here to view. $\text{SU}(n)$ : it comes from the fact that the conjugacy class of an element of Image may be NSFW.
Clik here to view. $\text{SU}(n)$ 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 Image may be NSFW.
Clik here to view. $\Lambda^k(V)$ of Image may be NSFW.
Clik here to view., and when Image may be NSFW.
Clik here to view. n = 3 the representation Image may be NSFW.
Clik here to view. $\Lambda^3(V)$ is trivial and Image may be NSFW.
Clik here to view. $\Lambda^2(V)$ is dual to Image may be NSFW.
Clik here to view., hence the character of one determines the other.

As a consequence, to understand Haar measure on the conjugacy classes of Image may be NSFW.
Clik here to view. $\text{SU}(3)$ it suffices to understand the joint moments of the real and imaginary part of Image may be NSFW.
Clik here to view. $\chi_V$ . 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 Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ of finite-dimensional continuous unitary representations of a compact (Hausdorff) group Image may be NSFW.
Clik here to view. bears a striking resemblance to an inner product space, mainly due to the properties of the Hom functor Image may be NSFW.
Clik here to view. $\text{Hom}(V, W)$ . The Hom functor is is bilinear in the sense that it respects finite direct sums in both arguments. We always have Image may be NSFW.
Clik here to view. $\dim \text{Hom}(V, V) > 0$ 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 Image may be NSFW.
Clik here to view. are an orthonormal basis with respect to Image may be NSFW.
Clik here to view. $\dim \text{Hom}(V, W)$ . Finally, there is the formula

Image may be NSFW.
Clik here to view. $\displaystyle \dim \text{Hom}(V, W) = \int_G \overline{\chi_V(g)} \chi_W(g) \, d \mu$

showing that Image may be NSFW.
Clik here to view. $\dim \text{Hom}(V, W)$ naturally extends to an inner product on the space of class functions on Image may be NSFW.
Clik here to view..

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

Image may be NSFW.
Clik here to view. $\displaystyle \text{Hom}(V, W) \cong \text{Inv}(V^{\ast} \otimes W)$

and to replace thinking directly about Image may be NSFW.
Clik here to view. $\text{Hom}(V, W)$ with thinking about tensor product, dual, and invariant subspace. These structures make Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ seem less like an inner product space and more like a random algebra. The tensor product is multiplication, taking duals is the Image may be NSFW.
Clik here to view. $^{\dagger}$ -operation, and taking invariant subspaces is the state.

In fact, all of this structure descends to the Grothendieck group of Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ , which is the universal way to assign every object in Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ 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 Image may be NSFW.
Clik here to view. v_i standing for the irreducible (finite-dimensional continuous unitary) representations Image may be NSFW.
Clik here to view. V_i of Image may be NSFW.
Clik here to view.. 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 Image may be NSFW.
Clik here to view.. Explicitly, if

Image may be NSFW.
Clik here to view. $\displaystyle V_i \otimes V_j \cong \bigoplus_k c_{ij}^k V_k$

then

Image may be NSFW.
Clik here to view. $\displaystyle v_i v_j = \sum_k c_{ij}^k v_k$ .

Dual naturally descends to a Image may be NSFW.
Clik here to view. $^{\dagger}$ -involution given by extending Image may be NSFW.
Clik here to view. $v_i \mapsto v_i^{\ast}$ , and taking invariant subspaces naturally descends to a map from the Grothendieck ring of Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ to the Grothendieck ring of Image may be NSFW.
Clik here to view. $\text{Rep}(1)$ , which is just Image may be NSFW.
Clik here to view. $\mathbb{Z}$ ; explicitly, Image may be NSFW.
Clik here to view. v_i is sent to Image may be NSFW.
Clik here to view. if it is trivial and Image may be NSFW.
Clik here to view. otherwise.

Tensoring with Image may be NSFW.
Clik here to view. $\mathbb{C}$ (and extending the Image may be NSFW.
Clik here to view. $^{\dagger}$ -operation appropriately), we get precisely the random algebra Image may be NSFW.
Clik here to view. $\text{Char}(G)$ above. In other words, Image may be NSFW.
Clik here to view. $\text{Rep}(G)$ can be thought of as a categorified random algebra whose decategorification is precisely Image may be NSFW.
Clik here to view. $\text{Char}(G)$ . This is a more precise version of the statement that understanding the representation theory of Image may be NSFW.
Clik here to view. is equivalent to understanding Haar measure on the conjugacy classes of Image may be NSFW.
Clik here to view. 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.

Noncommutative probability and group theory

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