Math 202: Class Algebras

For nonabelian groups G, the convolution algebra \mathcal{C}(G) is noncommutative and we cannot hope for a Fourier basis. However, it is reasonable to hope that we can implement a Fourier transform on well-chosen commutative subalgebras of \mathcal{C}(G).

The first candidate to look at is the center \mathcal{Z}(G) of \mathcal{C}(G). Despite the fact that \mathcal{Z}(G) is by definition a subalgebra of \mathcal{C}(G), it is useful to think of the class algebra as a standalone algebra associated to every finite group. The starting point here is the fact that we have an explicit description of \mathcal{Z}(G) which is more workable than its initial definition as the center of \mathcal{C}(G).

Theorem Z1. The center \mathcal{Z}(G) of the convolution algebra \mathcal{C}(G) of a finite group G consists of functions satisfying A(g)=A(hgh^{-1}) for all g,h \in G. Equivalently, central functions are those satisfying A(gh)=A(hg) for all g,h \in G.

As a consequence of Theorem Z1, indicator functions of conjugacy classes in G form a basis of \mathcal{Z}(G), and this basis is orthogonal relative to the scalar product on \mathcal{C}(G) in which the group basis is orthonormal. More precisely, let \Omega be a set indexing the conjugacy classes of G, and for each \alpha \in \Omega let K_\alpha \subseteq G be the corresponding conjugacy class. The indicator function of K_\alpha is then

|K_\alpha\rangle=\sum\limits_{g \in K_\alpha} |g\rangle,

and because distinct conjugacy classes are disjoint we have

\langle K_\alpha | K_\beta \rangle = \delta_{\alpha\beta}|K_\alpha|, \quad \alpha,\beta \in \Omega.

Theorem Z2. The class basis \{|K_\alpha\rangle \colon \alpha \in \Omega\} is an orthogonal basis of \mathcal{Z}(G).

The problem we pose is to determine whether the class algebra \mathcal{Z}(G) contains a hidden Fourier basis. The answer is yes, though it requires substantial work to see why. Ultimately, we have the following generalization of the Fourier isomorphism on the convolution algebra of a finite abelian group.

Theorem Z3. The class algebra \mathcal{Z}(G) is isomorphic to the function algebra \mathcal{F}(\Lambda) of a finite set \Lambda equicardinal with \Omega.

The proof of Theorem Z3 is based on a higher-dimensional generalization of the multiplicative characters of G. A unitary representation of G is a pair (V,\varphi) consisting of a nonzero Hilbert space V together with a group homomorphism

\varphi \colon G \longrightarrow \mathbb{U}(V),

where \mathbb{U}(V) is the group of unitary elements in the endomorphism algebra \mathrm{End}(V). Observe that if \dim V=1, we can identify \varphi with a group homorphism G \to \mathbb{U}_1, i.e. with a multiplicative character. To associate a function on G to a general unitary representation (V,\varphi) of G, we use the unique (up to scaling) trace on \mathrm{End}(V) and define the character

\chi^V(g) = \mathrm{Tr} \varphi(g), \quad g \in G.

Then, by construction, \chi^V \in \mathcal{Z}(G), and by analogy with the Fourier transform on the convolution algebra \mathcal{C}(G) we hope that developing the theory of characters will lead to a Fourier basis of the class algebra \mathcal{Z}(G) of an arbitrary finite group, leading to a proof of Theorem Z3. As a first step, we seek conditions under which the characters \chi^V and \chi^W of two unitary representations (V,\varphi) and (W,\psi) of G will be orthogonal in \mathcal{Z}(G), i.e. conditions on these unitary representations which will force

\langle \chi^V,\chi^W \rangle = \sum\limits_{g \in G} \overline{\chi^V(g)}\chi^W(g)=0.

This is where the key notion of G-equivariant linear transformations enters: we say that a linear transformation T \in \mathrm{Hom}(V,W) intertwines the unitary representation (V,\varphi) and (W,\psi) if

T \circ \varphi(g) = \psi(g) \circ T, \quad \text{for all }g \in G.

The set \mathrm{Hom}_G(V,W) of all intertwining transformations is a subspace of \mathrm{Hom}(V,W) whose dimension is a key quantity.

Theorem Z4. We have \langle \chi^V,\chi^W\rangle = |G|\dim\mathrm{Hom}_G(V,W).

Thus our character orthogonality question becomes the question of when the space of intertwiners between two given unitary representations is zero-dimensional. To understand when this occurs, the key notion is that of a subrepresentation. A subrepresentation of a unitary representation (V,\varphi) of G is a nonzero subspace V_1 \leq V such that each of the unitary operators \varphi(g) maps V_1 \to V_1. In this case, we may define \varphi_1(g) to be the restriction of \varphi(g) to V_1, giving a unitary representation (V_1,\varphi_1) which is said to be a subrepresentation of (V,\varphi).

Let us take a moment to note that \mathrm{Hom}_G(V,W) is a space of linear maps between two possibly different Hilbert spaces, so the Singular Value Decomposition is a naturally associated tool. First of all, associated to any intertwiner T \in \mathrm{Hom}_G(V,W) are its kernel \mathrm{Ker}(T) and image \mathrm{Im}(T), and it is straightforward to check that these are subrepresentations of V and W, respectively. The SVD gives us positive numbers \sigma_1>\dots>\sigma_r>0 and orthogonal decompositions

V=V_1 \oplus \dots \oplus V_r \oplus \mathrm{Ker}(T)\quad\text{and}\quad W=W_1\oplus\dots\oplus W_r \oplus \mathrm{Im}(T)^\perp

such that the restriction of T to V_i has the form \sigma_iU_i with U_i \in \mathrm{Hom}(V_i,W_i) an isometric isomorphism of Hilbert spaces. It is an important fact that the SVD is fully compatible with representation theory.

Theorem Z5. If T \in \mathrm{Hom}_G(V,W), then the spaces V_i are subrepresentations of V, the spaces W_i are subrepresentations of W, and the isometric isomorphisms U_i \in \mathrm{Hom}(V_i,W_i) are G-equivariant.

A unitary representation (V,\varphi) of G is called irreducible if it admits no proper subrepresentation. From the above, we have Schur’s Lemma.

Theorem Z6. A nonzero G-equivariant linear map out of an irreducible unitary representation is injective, a nonzero G-equivariant linear map into an irreducible unitary representation is surjective, a nonzero G-equivariant linear map between irreducible unitary representations is bijective.

A consequence of this, which is often also called Schur’s Lemma, is the following. Note: Theorem Z6 does not require the Fundamental Theorem of Algebra, but Theorem Z7 does.

Theorem Z7. If (V,\varphi) is an irreducible representation of G, then \mathrm{End}_G(V)=\mathbb{C}I. Moreover, if (W,\psi) is a second irreducible unitary representations of G which is isomorphic to (V,\varphi), then \dim \mathrm{Hom}_G(V,W) is one-dimensional.

There is an important consequence of Theorem Z7, which is the third and final result in the trinity of theorems collectively known as “Schur’s Lemma.” To state it, note first that there is a bijection between unitary representations of G and linear representations of \mathcal{C}(G). Namely, given a unitary representation (V,\varphi) of G there is a corresponding linear representation (V,\Phi) of \mathcal{C}(G) whose action is defined by

Theorem Z8. If |A\rangle \in \mathcal{C}(G) belongs to \mathcal{Z}(G), then the image \Phi|A\rangle of |A\rangle in any irreducible linear representation (V,\Phi) of \mathcal{C}(G) is a scalar operator.

Combining Theorems Z4,Z6, and Z7 we reach the following conclusion: characters of non-isomorphic irreducible unitary representations are orthogonal.

Theorem Z9. If (V,\varphi) and (W,\psi) are irreducible unitary representations of G, then \langle \chi^V,\chi^W\rangle is equal to |G| if they are isomorphic and equal to 0 if they are not isomorphic.

Let \Lambda be a set parameterizing isomorphism classes of irreducible unitary representations of G, and for each \lambda \in \Lambda let (V^\lambda,\varphi^\lambda) be a representative of the corresponding isomorphism class. Theorem Z8 tells us that \{\chi^\lambda \colon \lambda \in \Lambda\} is a linearly independent set in the class algebra \mathcal{Z}(G). From this it follows that the number of isomorphism classes of irreducible unitary representations of G is bounded by the number of conjugacy classes in G.

The fact that the orthogonal set \{\chi^\lambda \colon \lambda \in \Lambda\} spans \mathcal{Z}(G) is deduced by combining isotypic decompositions with the Wedderburn transform on the full convolution algebra \mathcal{C}(G). First, there is a bijection between unitary representations of G and linear representations of \mathcal{C}(G). Namely, given a unitary representation (V,\varphi) of G there is a corresponding linear representation (V,\Phi) of \mathcal{C}(G) whose action is defined by

\Phi|A\rangle = \sum\limits_{g \in G} A(g) \varphi(g).

Conversely, given a linear representation (V,\Phi) of \mathcal{C}(G) we get a unitary representation (V,\varphi) of G given by

\varphi(g)=\Phi|g\rangle, \quad g \in G.

Theorem Z10. As a unitary representation of G, we have \mathrm{Mult}(V^\lambda,\mathcal{C}(G)) = \dim V^\lambda for every \lambda \in \Lambda.

Theorem Z10 is a key result. To prove it, one first shows that for any unitary representation V of G we have

\mathrm{Mult}(V^\lambda,V) = \frac{1}{|G|}\langle \chi^\lambda,\chi^V\rangle.

Make sure you know how to do this. Then, one computes the character of the regular representation,

\chi^{\mathcal{C}(G)}(g) = \delta_{g,e}|G|,

and combines this with the above multiplicity formula. With Theorem Z9 in hand, the next result is easily obtainable. For each \lambda \in \Lambda, define |C^\lambda\rangle by

|C^\lambda\rangle = \sum\limits_{\alpha \in \Omega} \chi^\lambda_\alpha|K_\alpha\rangle,

where \chi^\lambda_\alpha=\chi^\lambda(g) for any g \in K_\alpha.

Theorem Z11. The orthogonal complement of \mathrm{Span}\{|C^\lambda\rangle\colon \lambda \in \Lambda\} in \mathcal{Z}(G) is the zero space. Equivalently, \mathrm{Span}\{|C^\lambda\colon \lambda \in \Lambda\} is a basis of \mathcal{Z}(G).

The proof of Theorem Z11 uses Theorem Z9 together with the third part of Schur’s Lemma, Theorem Z8.

This completes Phase I of the construction of the Fourier transform on the class algebra \mathcal{Z}(G) of an arbitrary finite group: we have built an orthogonal basis of \mathcal{Z}(G) from characters of irreducible unitary representations of G. This mirrors the abelian theory, though it is much more involved. Phase II consists of showing that the character basis of \mathcal{Z}(G), suitably normalized, is a Fourier basis.

Published by Jonathan Novak

Mathematician at UC San Diego.

Leave a Reply