Math 202C: Lecture 12

*** Problems in this lecture due 05/03/2026 at 23:59 ***

Let \mathcal{A} be an algebra. In Week I of Math 202B, we saw that subalgebras of \mathcal{A} always come in pairs: associated to each subalgebra \mathcal{B} \leq \mathcal{A} is its centralizer Z(\mathcal{B},\mathcal{A}), the set of elements in A which commute with every element in \mathcal{B}. We used this fact to characterize maximal abelian subalgebras of \mathcal{A} as precisely those commutative subalgebras \mathcal{B} \leq \mathcal{A} which are equal to their own centralizer, \mathcal{B}=Z(\mathcal{B},\mathcal{A}).

A problem we might have already posed back then is to characterize those subalgebras \mathcal{B} \leq \mathcal{A} for which Z(\mathcal{B},\mathcal{A}) is commutative. Today, we will address this question in the case where \mathcal{A}=\mathcal{C}(G) is the convolution algebra of a finite group G and \mathcal{B}=\mathcal{C}(H) is the convolution algebra of a subgroup H \leq G, viewed as the subalgebra of \mathcal{C}(G) spanned by the orthogonal set \{|h\rangle \colon h \in H\}.

For any element |A\rangle \in \mathcal{C}(G), let A \in \mathrm{End}\mathcal{C}(G) denote its image in the regular representation of \mathcal{C}(G). Furthermore, let \Lambda(G) be a set indexing isomorphism classes of irreducible unitary representations of G, and for each \lambda \in \Lambda(G) let A^\lambda \in \mathrm{End}(V^\lambda) denote the image of |A\rangle \in \mathcal{C}(G) in a representative V^\lambda of the isomorphism class indexed by \lambda \in \Lambda(G).

Problem 12.1. Prove that |A\rangle \in Z(\mathcal{C}(H),\mathcal{C}(G)) if and only if A^\lambda B^\lambda = B^\lambda A^\lambda for each |B\rangle \in \mathcal{C}(H) and every \lambda \in \Lambda(G).

Let \lambda \in \Lambda(G) be arbitrary but fixed. Thanks to Problem 12.1, understanding when Z(\mathcal{C}(H),\mathcal{C}(G)) is commutative reduces to understanding when the algebra \mathrm{End}_H(V^\lambda) is commutative. If it were the case that V^\lambda was an irreducible representation of \mathcal{C}(H) this would be extremely easy, since then Schur’s Lemma would force \mathrm{End}_H(V^\lambda) to be one-dimensional. However, V^\lambda is an irreducible representation of \mathcal{C}(G), and there is no reason why it should be irreducible as a representation of the subalgebra \mathcal{C}(H). So, we will have to understand what conditions would cause \mathrm{End}_H(V^\lambda) to be commutative by analyzing the behavior of V^\lambda under restriction from G to H.

Let \Lambda(H) be a set parameterizing isomorphism classes of irreducible unitary representations of H (or equivalently irreducible linear representations of \mathcal{C}(H)). Consider the isotypic decomposition of V^\lambda as a representation of H, which has the form

V^\lambda =\bigoplus\limits_{\mu \in \Lambda(H)} V^\lambda[\mu],

where two summands V^\lambda[\mu] and V^\lambda[\nu] are isomorphic unitary representations of H if and only if \mu = \nu. In finer detail, each V^\lambda[\mu] further decomposes as

V^\lambda[\mu]=W^\mu_1 \oplus \dots \oplus W^\mu_{m_{\lambda\mu}},

where W^\mu_i and W^\mu_j are isomorphic irreducible unitary representations of H and m_{\lambda\mu}=\mathrm{Mult}(W^\mu,V^\lambda) is a positive integer.

Problem 12.2. Prove that if T \in \mathrm{End}_H(V^\lambda), then T maps each H-isotypic component V^\lambda[\mu] of V^\lambda into itself.

Problem 12.2 allows us to focus on intertwiners T \in \mathrm{End}_H(V^\lambda[\mu]) for a fixed \mu \in \Lambda(H). Each such operator may be visualized as a block matrix

T = \begin{bmatrix} {} & \vdots & {} \\ \dots & T_{ji}^\lambda & \dots \\ {} & \vdots & {} \end{bmatrix}_{i,j=1}^{m_{\lambda\mu}},

where T_{ij}^\lambda \in \mathrm{End}_H(W^\mu_j,W^\mu_i). Now, since W^\mu_j and W^\mu_i are irreducible unitary representations of H, Schur’s Lemma tells us that \mathrm{End}_H(W^\mu_j,W^\mu_i) is one-dimensional, so in fact each block T_{ij}^\lambda can be identified with a scalar t_{ji}^\lambda and T itself can be identified with an m_{\lambda\mu} \times m_{\lambda\mu} matrix of complex numbers. We thus have an algebra isomorphism

\mathrm{End}_H(V^\lambda) \simeq \bigoplus\limits_{\mu \in \Lambda(H)} \mathrm{End}(\mathbb{C}^{m_{\lambda\mu}}),

which is important as a standalone fact, but also tells us that \mathrm{End}_H(V^\lambda) is commutative if and only if m_{\lambda\mu}=1 for each \mu \in \Lambda(H). We have thus established the following theorem.

Theorem 12.1. The centralizer Z(\mathcal{C}(H),\mathcal{C}(G)) is a commutative subalgebra of \mathcal{C}(G) if and only if the restriction of each irreducible unitary representation of G to a unitary representation of H is multiplicity free: for every \lambda \in \Lambda(G) we have

V^\lambda = \bigoplus\limits_{\mu \in \Lambda(H)} m_{\lambda\mu}W^\mu, \quad m_{\lambda\mu} \in \{0,1\}.

Published by Jonathan Novak

Mathematician at UC San Diego.

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