Math 202C – Jonathan Novak

Math 202C: May 27 Lecture

*** Problems in this lecture due May 31 at 23:59 ***

Problem 1. Let $V$ be a one-dimensional Hilbert space and $A \in \mathrm{End}(V)$ a linear operator. Prove that there is $\alpha \in \mathbb{C}$ such that $Av = \alpha v$ for all $v \in V$ .

Yes, Problem 1 is trivial. Yes, I want you to write down a proof.

Problem 2. Let $V$ be a Hilbert space of finite dimension $n \in\mathbb{N}$ and let $A \in \mathrm{End}(V)$ be a linear operator. Prove that the function of a complex variable $z$ defined by $P_A(z) =\det(zI-A),$ where $I \in \mathrm{End}(A)$ is the identity operator, can also be expressed as

$P_A(z) = \sum\limits_{d=0}^n z^{n-d} (-1)^d \mathrm{Tr}(A^{\wedge d}).$

Yes, the power of $-1$ is correct.

Problem 3. Prove that the following two statements are equivalent: every linear operator on a finite-dimensional Hilbert space has nonempty spectrum; every polynomial function of a complex variable has a zero.

Problem 4. Let $V$ be a finite-dimensional Hilbert space with orthonormal basis $e_1,\dots,e_n.$ For $1 \leq d \leq n$ and $I \subseteq \{1,\dots,n\}$ a subset with $|I|=d,$ let $V_I$ be the subspace of $V$ spanned by $\{e_i \colon i \in I\}$ and let $A_I$ be the restriction of $A$ to $V_I.$ Prove that

$\mathrm{Tr}(A^{\wedge d}) = \sum\limits_{\substack{I \subseteq \{1,\dots,n\} \\ |I|=d}} \det(A_I).$

Problem 5. With notation as in Problem 2, write $P_A(z)=(z-\alpha_1)\dots (z-\alpha_n).$ Determine the eigenvalues of $A^{\otimes d},$ $A^{\wedge d},$ and $A^{\vee d}$ in terms of $\alpha_1,\dots,\alpha_n.$

Math 202C: May 20 Lecture

*** Problems in this lecture due 05/24/2026 at 23:59 ***

Let $G$ be an arbitrary group, and let $(V,\varphi)$ be a finite-dimensional unitary representation of $G$ . Then, there exists an orthogonal direct sum decomposition

$V = U_1 \oplus \dots \oplus U_k$

of $V$ into nonzero $G$ -invariant and $G$ -irrreducible subspaces. This fact requires nothing from the group $G$ , what matters is that the representation space $V$ is a finite-dimensional Hilbert space and the action $\varphi$ represents $G$ as a group of unitary transformations.

A better way to organize this decomposition is to introduce a set $\Lambda$ parameterizing isomorphism classes of irreducible finite-dimensional unitary representations of $G$ . Since we are making no assumptions on $G$ , nothing more specific can be said about the parameter set $\Lambda$ (as we know from Math 202B, if $G$ is a finite group then $\Lambda$ is a finite set whose cardinality is equal to the number of conjugacy classes in $G$ , but we are not assuming that here). For each $\lambda \in \Lambda,$ fix a representative $(U^\lambda,\rho^\lambda)$ of the corresponding isomorphism class of irreducible finite-dimensional unitary representations of $G$ . Then, we can organize the above decomposition of $(V,\varphi)$ as

$V= \bigoplus\limits_{\lambda \in \Lambda} V^\lambda,$

where each constituent of this orthogonal (internal) direct sum further decomposes as

$V^\lambda = U_1^\lambda \oplus \dots \oplus U^\lambda_{m^\lambda(V)},$

with each $U_i^\lambda$ isomorphic to the irrep $U^\lambda.$ We allow the possibility $m^\lambda(V)=0,$ which means precisely that $V^\lambda = \{0_V\}.$ What we are doing here is building the same binary tree of pairs of complementary subspaces that produces the decomposition $V=U_1 \oplus \dots \oplus \dots U_k,$ and then grouping the leaves of the tree corresponding to their isomorphism type $\lambda \in \Lambda.$ This means that the decomposition

$V = \bigoplus\limits_{\lambda \in \Lambda} V^\lambda$

into isotypic components is canonical, but the decomposition of each isotypic component $V^\lambda$ itself is not. What we can say is that the isotypic decomposition of $V$ can instead be written as the external direct sum

$V \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda,$

where this is not an equality but an isomorphism of representations

Lately we have argued that a still better way to think about this is to replace the numerical multiplicity $m^\lambda(V)$ with a multiplicity space: we write the isotypic decomposition of $(V,\varphi)$ as

$V \simeq \bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda$

with $M^\lambda(V) = \mathrm{Hom}_G(U^\lambda,V).$ Thanks to Schur’s Lemma, each nonzero linear transformation $A \in \mathrm{Hom}_G(U^\lambda,V)$ is an injection of $U^\lambda$ into $V$ , so we can think of the multiplicity space $M^\lambda(V)$ as parameterizing all ways in which the irreducible representation $U^\lambda$ can be realized inside the representation $V$ in a $G$ -equivariant way. In particular, there is an evaluation map

$M^\lambda(V) \otimes U^\lambda \longrightarrow V^\lambda$

given by $A \otimes u \mapsto Au.$ To make this spacial (as opposed to numerical) version of the isotypic decomposition legitimate, we still have to say how each of its components $M^\lambda(V) \otimes U^\lambda$ is being viewed as a unitary representation of $G.$ This means that $M^\lambda(V) \otimes U^\lambda$ must be a Hilbert space on which $G$ acts by linear isometries. Concerning the Hilbert space structure on $M^\lambda(V) \otimes U^\lambda$ , we have discussed in lecture how to make a tensor product of Hilbert spaces into a Hilbert space in lecture, and also $U^\lambda$ is by definition a Hilbert space, so all that is left here is to define a scalar product on $M^\lambda(V)=\mathrm{Hom}_G(U^\lambda,V).$ To do this, we simply take the Frobenius-Hilbert-Schmidt scalar product on $\mathrm{Hom}(U^\lambda,V),$ in which $M^\lambda(V)$ sits as a subspace. Now for the action of $G$ on the Hilbert space $M^\lambda(V) \otimes U^\lambda$ we take this simplest possible recipe, namely

$g(A \otimes u) = A \otimes \varphi(g)u.$

Now we want to write down a proof of our proposed spacial reformulation

$V \simeq \bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda$

of the numerical isotypic decomposition

$V \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda.$

This means that we should give an isomormphism of representations

$\bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda.$

Problem 1: Reduce the above to giving an isomorphism of representations $M^\lambda(V) \otimes U^\lambda \simeq m^\lambda(V)U^\lambda$ for each $\lambda \in \Lambda$ , and give the required isomorphism.

Once Problem 1 is complete you can live out the rest of your days with the true isotypic decomposition in your pocket, ready to be deployed anytime.

Math 202C: May 18 Lecture

*** Problems in this lecture due May 24 at 23:59 ***

Problem 18.1. A tensor $t \in V \times V$ is said to be decomposable (or pure) if it can be written as $t=v \otimes w$ for some $v,w \in V$ , and indecomposable otherwise. Show that indecomposable tensors exist.

Problem 18.2. Given a Hilbert space $V$ , establish the orthogonal direct sum decomposition $V \otimes V = V \wedge V \oplus V \vee V.$ For $V$ finite-dimensional, calculate the dimensions of all spaces involved in this decomposition.

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.

Math 202: Endomorphism Algebras

Math 202A was an expression of the belief that the quantization of a finite set $X$ is the Hilbert space $V=\mathcal{F}(X)$ containing $X$ as an orthonormal basis. Math 202B extends the gospel from Hilbert spaces to algebras, taking $\mathcal{F}(X)$ as the prototypical commutative algebra and $\mathrm{End}(V)=\mathrm{End}\mathcal{F}(X)$ as the prototypical noncommutative algebra.

The endomorphism algebra $\mathrm{End}(V)=\mathrm{End}\mathcal{F}(X)$ contains a subalgebra isomorphic to $\mathcal{F}(X),$ namely the diagonal subalgebra $\mathcal{D}(X)$ consisting of operators on $V$ such that

$Ax=\widehat{A}(x)x, \quad x \in X,$

where $\widehat{A}(x) \in \mathbb{C}$ is a scalar. Indeed, the isomorphism

$\Phi \colon \mathcal{D}(X) \longrightarrow \mathcal{F}(X)$

defined by $\Phi(A)=\widehat{A}$ is the Fourier transform on $\mathcal{D}(X)$ , and it can equivalently be described in terms of a basis of $\mathrm{End}(V)$ constructed from the orthonormal basis $X \subset V$ , namely the basis of elementary operators

$E_{xy}v=x\langle y,v\rangle, \quad x,y \in X,\ v \in V.$

These operators multiply as

$E_{xy}E_{wz}=E_{xz}\langle y,w\rangle, \quad x,y,z,w \in X,$

and conjugate as

$E_{xy}^*=E_{yx}, \quad x,y \in X,$

making $\{E_{xx} \colon x \in X\}$ a set of pairwise orthogonal projections that spans the diagonal subalgebra $\mathcal{D}(X).$ The Fourier transform $\Phi \colon \mathcal{D}(X) \to \mathcal{F}(X)$ is $\Phi(E_{xx})=E_x.$

The algebra $\mathrm{End}(V)=\mathrm{End}\mathcal{F}(X)$ is bigger and harder to understand than $\mathcal{F}(X),$ which is why the algebra of endomorphisms on a finite-dimensional Hilbert space is the subject of entire courses and the algebra of functions on a finite set is not. The first problem treated is generally: find necessary and sufficient conditions on an operator $A \in \mathrm{End}(V)$ such that $A \in \mathcal{D}(Y)$ for some orthonormal basis $Y \subset V$ . Typically, this is posed as a problem about matrices: you are given the matrix of $A$ relative to the computational basis $X \subset V$ , and almost certainly the entity who gave it to you this matrix ensured it was not diagonal, so $A \not\in \mathcal{D}(X).$

Theorem E1. There exists and orthonormal basis $Y \subset V$ such that $A \in \mathcal{D}(Y)$ if and only if $A$ is normal.

Consequently, if $A$ is normal then it lives in $\mathcal{D}(Y)$ and its Fourier transform $\widehat{A} \in \mathcal{F}(Y)$ outputs eigenvalues, $Ay=\widehat{A}(y)y.$ Our general taxonomy of special elements in an abstract algebra gives you a number of conditions which imply that an operator is normal.

The diagonalization problem puts a particular focus on diagonal subalgebras of $\mathrm{End}(V)$ , and hence it is useful to characterize these objects from a second point of view.

Theorem E2. A commutative subalgebra $\mathcal{A} \leq \mathrm{End}(V)$ is a maximal commutative subalgebra if and only if $\mathcal{A}=\mathcal{D}(Y)$ for some orthonormal basis $Y \subset V.$

So the diagonal subalgebras of $\mathrm{End}(V)$ are exactly the maximal elements of the poset of commutative subalgebras of $\mathrm{End}(V)$ . The natural question is then whether one has a Fourier transform defined on every commutative subalgebra of $\mathrm{End}(V)$ which converts it into a function algebra.

Theorem E3. A subalgebra $\mathcal{A} \leq \mathrm{End}(V)$ is commutative if and only if it is isomorphic to a subalgebra of $\mathcal{F}(Y)$ for some orthonormal basis $Y \subset V.$

The philistines call this result “simultaneous diagonalization,” because the non-obvious direction is proved as follows: take a basis $A_1,\dots,A_m$ of $\mathcal{A}$ , use the general fact that an algebra is commutative if and only if all its elements are normal, and using Theorem E1 show that there exists an orthonormal basis $Y \subset V$ such that $A_1,\dots,A_m \in \mathcal{D}(Y).$ Since all subalgebras of function algebras are function algebras, Theorem E3 proves that all commutative subalgebras of $\mathrm{End}(V)$ are function algebras.

The classification of noncommutative subalgebras of $\mathrm{End}(V)$ is more complicated, and this is precisely the point where linear algebra starts to look like representation theory: in order to understand noncommutative subalgebras $\mathcal{A} \leq \mathrm{End}(V)$ it is no longer sufficient to look for one-dimensional $\mathcal{A}$ -invariant subspaces of $V.$ A fundamental result due to Burnside (sometimes called the fundamental theorem of noncommutative algebra) asserts that every subalgebra $\mathcal{A}$ of $\mathrm{End}(V)$ except $\mathrm{End}(V)$ itself admits a proper nontrivial invariant subspace.

Theorem E4. We have $\mathcal{A}=\mathrm{End}(V)$ if and only if $\mathcal{A}$ has exactly two invariant subspaces. In fact, if $\mathcal{A}$ is a proper subalgebra of $\mathrm{End}(V)$ , it has at least four invariant subspaces.

The first part of Theorem E4 is Burnside’s theorem, the second is Maschke’s theorem: $W$ is $\mathcal{A}$ -invariant if and only if $W^\perp$ is.

Given a subalgebra $\mathcal{A}$ of $\mathrm{End}(V)$ , it is helpful to order the set of all $\mathcal{A}$ -invariant subspaces by inclusion. This forms an induced sublattice of the lattice of all subspaces of $V$ with maximal element $V$ . Nonzero elements $W$ of the lattice of $\mathcal{A}$ -invariant subspaces are called representations of $\mathcal{A}$ because restricting every $A \in \mathcal{A}$ to $W$ gives an algebra homomomorphism $\rho \colon \mathcal{A} \to \mathrm{End}(W).$ A representation of $\mathcal{A}$ is called irreducible if the only element below it in the lattice of $\mathcal{A}$ -invariant subspaces is $\{0_V\}.$

Theorem E5. For every $\mathcal{A} \leq \mathrm{End}(V)$ , there exists an orthogonal decomposition of $V$ into irreducible representations of $\mathcal{A}$ .

Note that Theorem E5 is a noncommutative generalization of Theorem E3, which gives an orthogonal decomposition of $V$ into one-dimensional invariant subspaces of a commutative subalgebra $\mathcal{A} \leq \mathrm{End}(V)$ . However, while Theorem E3 is enough to classify commutative subalgebras of $\mathrm{End}(V)$ , Theorem E5 is not quite enough to classify all subalgebras of $\mathrm{End}(V),$ and you are not expected to know the classification of subalgebras of $\mathrm{End}(V).$

We did however classify states and traces on $\mathrm{End}(V)=\mathrm{End}\mathcal{F}(X)$ . First, we proved that the function defined by

$\mathrm{Tr}_X(A) = \sum\limits_{x \in X} \langle x,Ax\rangle, \quad A \in \mathrm{End}(V)$ ,$

is a trace on $\mathrm{End}(V)$ . Second, we proved that if $Y \subset V$ is any orthonormal basis, the linear functional

$\mathrm{Tr}_Y(A)\sum\limits_{y \in Y} \langle y,Ay\rangle$

coincides with $\mathrm{Tr}_X.$ Therefore, we simply denote by $\mathrm{Tr}$ the function defined by $\mathrm{Tr}_X.$

Theorem E6. Up to scaling, $\mathrm{Tr}$ is the unique trace on $\mathrm{End}(V).$

The classification of states on $\mathrm{End}(V)$ is more flexible but still very structured. Recall from Part $\mathcal{A}$ that an element $P$ of an abstract algebra $\mathcal{A}$ is said to be nonnegative if it can be factored in the form $P=A^*A.$ In the particular case of the endomorphism algebra $\mathrm{End}(V)$ , a nonngegative operator $P \in \mathrm{End}(V)$ is called a density operator if it satisfies $\mathrm{Tr}(P)=1.$ Note that this condition is equivalent to

$\mathrm{Tr} P =\sum\limits_{x \in X} \langle x,Px\rangle= \sum\limits_{x \in X} \langle Ax,Ax\rangle = 1,$

which is equivalent to saying that the function in $\mathcal{F}(X)$ defined by $x \mapsto \langle x,Px\rangle$ is a probability function in $\mathcal{F}(X),$ which makes density operators a natural noncommutative analogue of probability functions.

Theorem E7. For every density operator $P$ , the linear functional $\sigma(A)=\mathrm{Tr}(PA)$ is a state on $\mathrm{End}(V),$ and every state on $\mathrm{End}(V)$ is of this form. Moreover, $\sigma$ is faithful if and only if $\mathrm{rank}(P) = \dim V.$

The significance of faithful states is that they allow us to define a noncommutative analogue of the Fourier transform that maps abstract algebras to subalgebras of endomorphism algebras. Namely, let $\mathcal{A}$ be an abstract algebra, and suppose that $\sigma$ is a faithful state on $\mathcal{A}$ — like the situation with a Fourier basis, it is not necessarily the case that a faithful state exists, but suppose one does. Then, we have a left-Frobenius scalar product on $\mathcal{A}$ defined by $\langle A,B \rangle=\sigma(A^*B),$ so that we can view $\mathcal{A}$ as a Hilbert space and consider the corresponding endomorphism algebra $\mathrm{End}(\mathcal{A})$ .

Theorem E8. The function $\Psi \colon \mathcal{A} \longrightarrow \mathrm{End}(\mathcal{A})$ defined by $\Psi(A)B=AB$ , $A,B \in \mathcal{A}$ , is an injective algebra homomorphism.

The map $\Psi$ in Theorem E8 is called the Wedderburn transform of $\mathcal{A}$ relative to $\sigma$ . Theorem 8 says that any algebra which supports a faithful state can be transformed into a subalgebra of an endomorphism algebra using the corresponding Wedderburn transform. This should be compared with the statement that any algebra which supports a Fourier basis can be transformed into a function algebra using the Fourier transform. One consequence of the Wedderburn transform is the following.

Theorem E9. An algebra $\mathcal{A}$ supports a faithful tracial state if and only if it is isomorphic to a subalgebra of an endomorphism algebra.

There is a fundamental relationship between the Fourier transform and the Wedderburn transform which is extremely important in situations where both can be implemented. Let $\mathcal{A}$ be an algebra which admits a Fourier basis $F=\{F^\lambda \colon \lambda \in \Lambda\}.$ Then, the scalar product on $\mathcal{A}$ in which the Fourier basis is orthonormal is a Frobenius scalar product on $\mathcal{A}.$ This means that the Wedderburn transform $\Psi \colon \mathcal{A} \to \mathrm{End}(\mathcal{A})$ with respect to this Frobenius scalar product is defined.

Theorem E10. For every $A \in \mathcal{A}$ , we have $\Psi(A) \in \mathcal{D}(F),$ and moreover $\Psi(A)F^\lambda = \widehat{A}(\lambda)F^\lambda$ for all $\lambda \in \Lambda.$

Math 202C: Lecture 15

Enumeration of topics:

Part $\mathcal{A}$ : Abstract algebras.
Part $\mathcal{F}$ : Function algebras.
Part $\mathcal{E}$ : Endomorphism algebras.
Part $\mathcal{C}$ : Convolution algebras.
Part $\mathcal{Z}$ : Class algebras.
Part $\mathcal{P}$ : Polynomial algebras.
Part $\mathcal{G}$ : Gelfand-Tsetlin algebras.

Math 202: Function algebras

The function algebra of a finite set $X$ is the set $\mathcal{F}(X)$ of functions $A \colon X \to \mathbb{C}$ with operations defined pointwise. A basic family of functions are the elementary functions,

$E_x(y)=\delta_{xy}, \quad x,y \in X.$

Theorem F1. The elementary functions are nonzero pairwise orthogonal projections.

Combining Theorem F1 with Theorem A3, we get that the set $\{E_x \colon x \in X\}$ is a basis of $\mathcal{F}(X)$ .

The algebra $\mathcal{F}(X)$ can be understood completely. In particular, subalgebras can be explicitly classified. A partition of $X$ is a set $\mathfrak{p}=\{P_1,\dots,P_m\}$ of nonempty disjoint subsets of $X$ whose union is $X$ . The elements $P_i$ of $\mathfrak{p}$ are called its blocks. The set $\mathfrak{P}(X)$ of partitions of $X$ is partially ordered set under the refinement order, where $\mathfrak{p} \leq \mathfrak{q}$ if and only if $\mathfrak{q}$ can be obtained from $\mathfrak{p}$ by breaking blocks. Associated to every $\mathfrak{p} \in \mathfrak{P}(X)$ is the subalgebra $\mathcal{A}(\mathfrak{p})$ consisting of functions constant on the blocks of $\mathfrak{p}.$ Note that $\mathcal{A}(\mathfrak{p})$ is isomorphic to $\mathcal{F}(\mathfrak{p}).$

Theorem F2. The set $\{\mathcal{A}(\mathfrak{p}) \colon \mathfrak{p} \in \mathfrak{P}(X)\}$ is the complete collection of subalgebras of $\mathcal{F}(X)$ . Moreover, $\mathcal{A}(\mathfrak{p}) \leq \mathcal{A}(\mathfrak{q})$ if and only if $\mathfrak{p}\leq \mathfrak{q}.$

This classification theorem can be proved in several ways. A nice argument uses the finite-dimensional Stone-Weierstrass theorem. A subalgebra $\mathcal{A} \leq \mathcal{F}(X)$ is said to separate points if, for every $\{x,y\} \subseteq X,$ there exists a function $A \in \mathcal{A}$ such that $A(x) \neq A(y).$

Theorem F3. If $\mathcal{A} \leq \mathcal{F}(X)$ separates points, then $\mathcal{A}=\mathcal{F}(X).$

There is a parallel classification of ideals in $\mathcal{F}(X)$ . Let us order the set $\{S \subseteq X\}$ of all subsets of $X$ by inclusion. Associated to each $S \subseteq X$ is the ideal

$\mathcal{I}(S) = \{A \in \mathcal{F}(X) \colon A(x)=0 \text{ for all }x \in S\}.$

Theorem F4. The set $\{\mathcal{I}(S) \colon S \subseteq X\}$ is the complete collection of ideals in $\mathcal{F}(X)$ . Moreover, $\mathcal{I}(S) \subseteq \mathcal{I}(T)$ if and only if $T \subseteq S.$

Just like Theorem F2 shows that we cannot discover any new algebras by looking at subalgebras of functional algebras, Theorem F4 reveals that we cannot discover any new algebras by taking quotients of function algebras.

Theorem F5. For any $S \subseteq S,$ the quotient algebra $\mathfrak{F}(X)/\mathcal{I}(S)$ is isomorphic to the function algebra $\mathcal{F}(S).$

Now let us classify von Neumann algebra structures on $\mathcal{F}(X)$ . This means that we will find all Frobenius scalar products on $\mathcal{F}(X)$ ; since $\mathcal{F}(X)$ is commutative there is no distinction between left-Frobenius and right-Frobenius. Equivalently, we classified faithful states on $\mathcal{F}(X)$ , all of which are tracial. A function $P \in \mathcal{F}(X)$ is called a probability distribution if $P(x) \geq 0$ for all $x \in X$ , and $\sum_{x \in X} P(x)=1.$

Theorem F6. States $\sigma$ on $\mathcal{F}(X)$ are in bijection with probability distributions on $X$ : every state is of the form $\sigma(A) = \sum_{x \in X} A(x)P(x)$ for a unique probability distribution $P$ . Moreover, $\sigma$ is faithful if and only if $P$ is nonvanishing.

Theorem F6 is equivalent to the statement that all Frobenius scalar products are of the form

$\langle A,B \rangle = \sum\limits_{x \in X} \overline{A(x)}B(x)P(x)$

for a unique probability distribution $P$ having full support in $X.$

Going beyond understanding the inner workings of a given function algebra $\mathcal{F}(X),$ we can classify all algebras $\mathcal{A}$ which are isomorphic to a function algebra.$ Clearly, any such $\mathcal{A}$ must be commutative. A basis of $\mathcal{A}$ consisting of pairwise orthogonal projections is called a Fourier basis.

Theorem F7. An algebra $\mathcal{A}$ is isomorphic to a function algebra if and only if it admits a Fourier basis.

Theorem F7 is very important, and so is its proof. The fact that $\mathcal{F}(X)$ admits a Fourier basis is clear (the elementary functions). Conversely, suppose that $\mathcal{A}$ is an abstract algebra in which we are able to find a Fourier basis $\{F^\lambda\colon \lambda \in \Lambda\}$ indexed by some finite set $\Lambda$ . Then, every element $A \in \mathcal{A}$ can be expanded in this basis,

$A = \sum\limits_{\lambda \in \Lambda}\widehat{A}(\lambda)F^\lambda$

and the coefficients in this expansion define a function $\widehat{A} \in \mathcal{F}(\Lambda)$ . The mapping

$\Phi \colon \mathcal{A} \longrightarrow \mathcal{F}(\Lambda)$

defined by $\Phi(A) = \widehat{A}$ is an algebra isomorphism called the Fourier transform. The “the” here is justified by the following.

Theorem F8. The only Fourier basis is $\mathcal{F}(X)$ is the basis $\{E_x \colon x \in X\}$ of elementary functions.

Theorem F8 means that if a Fourier transform $\Phi \colon \mathcal{A} \to \mathcal{F}(\Lambda)$ exists, then it is unique up to relabeling the label set $\Lambda$ .

Math 202C: Lecture 14

Let us begin by reviewing some basic material from Math 202AB. Let $X$ be an orthonormal basis of a Hilbert space $V$ . The corresponding diagonal subalgebra $\mathcal{D}(X) \subset \mathrm{End}(V)$ is the set of all operators which act diagonally on $X.$ That is, $A \in \mathcal{D}(X)$ if and only if $Ax = \widehat{A}(x)x$ for each $x \in X,$ where $\widehat{A}(x) \in \mathbb{C}$ is a scalar. Equivalently,

$A = \sum\limits_{x\in X}\widehat{A}(x)P_x,$

where $P_x \in \mathcal{E}(V)$ is the orthogonal projection of $V$ onto the line spanned by $x.$ A physicist would write this projection operator as $P_x=|x\rangle \langle x|,$ and so that in Dirac notation the above decomposition becomes

$A = \sum\limits_{x \in X} \widehat{A}(x)|x\rangle\langle x|.$

We have that $\mathcal{D}(X)$ is isomorphic to $\mathcal{F}(X)$ via $A \mapsto \widehat{A}$ . This is a particularly transparent instance of the Fourier transform.

We know from Math 202B that $\mathcal{D}(X)$ is a maximal abelian subalgebra of $\mathrm{End}(V)$ , and moreover that every MASA in $\mathrm{End}(V)$ is $\mathcal{D}(X)$ for some orthonormal basis $X$ . You should know why this is true, and moreover you should be able to prove that it is true to someone who does not believe you. Subsequently, you can and should taunt them with your understanding of the algebra $\mathcal{D}(X) \cap \mathcal{D}(Y).$

A basic problem in linear algebra is: given $A \in \mathrm{End}(V),$ does there exist an orthonormal basis $X \subset V$ such that $A \in \mathcal{D}(X)$ ? According to the Spectral Theorem, the answer is “yes” if and only if $A$ is normal. People who live on the wrong side of the tracks refer to this as “unitary diagonalizability,” but you should not mock their intellectual poverty: even if you enjoy thinking in austere algebraic terms an occasional trip to the wrong side of the tracks can be rewarding.

With that polemic out of our system, let us return to our current object of study: the tower of symmetric groups

$S_1 \subset S_2 \subset S_3 \subset \dots.$

For each $n \in \mathbb{N},$ choose an arbitrary parameterization of the isomorphism classes of irreducible unitary representations of $S_n$ by the set $\mathrm{Par}_n$ of partitions of $n,$ and for each $\lambda \in \mathrm{Par}_n$ choose an arbitrary representative $V^\lambda$ of the corresponding isomorphism class. These choices being made, we define a graph $\mathfrak{B}$ called the branching graph of the tower of symmetric groups. The vertex set of $\mathfrak{B}$ is the set

$\mathrm{Par} = \bigsqcup\limits_{n=1}^\infty \mathrm{Par}_n$

of all partitions, so $\mathfrak{B}$ is an infinite graph. Moreover, this graph is graded: each $\mathrm{Par}_n$ is an independent set, and every edge of the graph joins a vertex of $\mathrm{Par}_n$ to a vertex of $\mathrm{Par}_{n+1}$ for some $n \in \mathbb{N}.$ The precise adjacency relation is the following: if $\mu \in \mathrm{Par}_n$ and $\lambda \in \mathrm{Par}_{n+1},$ then $\{\lambda,\mu\}$ is an edge of $\mathfrak{B}$ if and only if when $V^\lambda$ is restricted to a representation of $S_{n-1}$ it contains an irreducible subspace $V^\lambda[\mu]$ isomorphic as a representation of $S_{n-1}$ to $V^\mu.$

The work that we have done over the last two lectures tells us that the graph $\mathfrak{B}$ we have now constructed is simple: it has no multiple edges. Thus, by construction, for any positive integers $1 \leq k <n$ and any $\lambda \in \mathrm{Par}_n$ we have an orthogonal decomposition

$V^\lambda = \bigoplus\limits_{\gamma \in \mathrm{Geo}(\mathrm{Par}_k,\lambda)} V^\lambda[\gamma],$

where $\mathrm{Geo}(\mathrm{Par}_k,\lambda)$ is the set of geodesics in $\mathfrak{B}$ joining a point of $\mathrm{Par}_k$ to the specified point $\lambda \in \mathrm{Par}_n$ , and for each such geodesic $V^\lambda[\gamma]$ is a nonzero subspace of $V^\lambda$ in which $S_k$ acts irreducibly. Note that while all of these subspaces $V^\lambda[\gamma]$ are distinct (indeed, pairwise orthogonal), sum of them may be isomorphic to one another as representations of $S_k$ ; indeed, for any particular $\mu \in \mathrm{Par}_k$ the number of spaces in the above decomposition isomorphic to $V^\mu$ is $|\mathrm{Geo}(\mu,\lambda)|.$

The most extreme case of this is the case $k=1,$ where

$V^\lambda=\bigoplus\limits_{\gamma \in \mathrm{Geo}(1,\lambda)} V^\lambda[\gamma]$

is an orthogonal decomposition of $V^\lambda$ into one-dimensional subspaces known as the Gelfand-Tsetlin lines of $V^\lambda.$ If we choose a unit vector $x^\lambda_\gamma$ from each Gelfand-Tsetlin line $V^\lambda[\gamma],$ we obtain an orthonormal basis of $V^\lambda$ called a Gelfand-Tsetlin basis. Typically one ignores the phase ambiguity implicit in such a choice and simply calls any such choice $X_n^\lambda=\{x^\lambda_\gamma \colon \gamma \in \mathrm{Geo}(1,\lambda)\}$ “the” Gelfand-Tsetlin basis of $V^\lambda,$ which really means that the basis is uniquely defined up to the phase ambiguity. In any case, this gives our first handle on the dimensions of the irreducible representations of the symmetric groups: we now know that

$\dim V^\lambda = |\mathrm{Geo}(1,\lambda)|.$

Now, for each $n \in \mathbb{N}$ and every $\lambda \in \mathrm{Par}_n,$ let $\mathcal{G}_n^\lambda$ be the subalgebra of $\mathcal{C}(S_n)$ consisting of all elements $|A\rangle$ whose image $A^\lambda \in \mathrm{End}(V^\lambda)$ belongs to the diagonal subalgebra $\mathcal{D}(X_n^\lambda)$ of the GT-basis $X_n^\lambda \subset V^\lambda.$ In other words, the image

$A = \bigoplus\limits_{\lambda \in \mathrm{Par}_n} A^\lambda$

of $|A\rangle \in \mathcal{G}_n^\lambda$ under the Wedderburn isomorphism

$\mathcal{C}(S_n) \simeq \bigoplus\limits_{\lambda \in \mathrm{Par}_n} \mathrm{End}(V^\lambda),$

has the feature that, if we look at it as a block matrix relative to the $GT$ -basis in every irreducible representations, the $\lambda$ -block of $A$ is a diagonal matrix. Thus, if we define the Gelfand-Tsetlin subalgebra of $\mathcal{C}(S_n)$ to be

$\mathcal{G}_n = \bigsqcup\limits_{\lambda \in \mathrm{Par}_n} \mathcal{G}_n^\lambda,$

then we have obtained a maximal commutative subalgebra of $\mathcal{C}(S_n)$ consisting of those elements $|A\rangle$ which act diagonally on the GT-basis $X_n^\lambda$ of every irreducible representation $V^\lambda$ of $S_n.$

In fact, in “Phase I” of our current project (representation theory of the symmetric groups), we met the Gelfand-Tsetlin algebra in a different guise. Recall that the Jucys-Murphy subalgebra $\mathcal{J}_n$ of $\mathcal{C}(S_n)$ was defined to be the algebra generated by

$\mathcal{Z}_1 \cup \dots \cup \mathcal{Z}_n,$

where $\mathcal{Z}_k$ is the center of $\mathcal{C}(S_k).$ We had to put in some significant work to show that $\mathcal{J}_n = \mathbb{C}[|J_1\rangle,\dots,|J_n\rangle]$ , which is shorthand for saying that the Jucys-Murphy specialization of the polynomial algebra $\mathcal{P}_n=\mathbb{C}[x_1,\dots,x_n]$ is a surjection onto $\mathcal{J}_n.$ This effort front loading is now starting to pay off.

Theorem 14.1. We have $\mathcal{G}_n = \mathcal{J}_n.$

Proof: First we prove that $\mathcal{J}_n \subseteq \mathcal{G}_n.$ For this we use the fact, since we have already shown $\mathcal{J}_n=\mathbb{C}[|J_1\rangle,\dots,|J_n\rangle],$ it is sufficient to show that each JM-element $|J_k\rangle$ belongs to the $GT$ -algebra $\mathcal{J}_n.$ Since $|J_k\rangle=|T_k\rangle-|T_{k-1}\rangle,$ this reduces to showing that the sum $|T_k\rangle$ of all transpositions in $S_k$ belongs to $\mathcal{G}_n.$ To demonstrate this, fix $\lambda \vdash n$ and let

$V^\lambda = \bigoplus\limits_{\gamma \in \mathrm{Geo}(\mathrm{Par}_k,\lambda)} V^\lambda[\gamma]$

be its orthogonal decomposition into irreducible $S_k$ -invariant subspaces, as constructed above. On one hand, each vector in the GT-basis $X^\lambda \subset V^\lambda$ belongs to one of the spaces $V^\lambda[\gamma],$ and on the other $|T_k\rangle \in \mathcal{Z}_k$ acts as a scalar operator in each of these spaces by Schur’s Lemma.

Now we prove $\mathcal{J}_n=\mathcal{G}_n$ using finite-dimensional Stone-Weierstrass and the fact that $\mathcal{G}_n$ is isomorphic to the function algebra $\mathcal{F}(\mathrm{Geo}_n)$ , where $\mathrm{Geo}_n$ is the set of all geodesics from $\mathrm{Par}_1$ to $\mathrm{Par}_n$ in the branching graph $\mathfrak{B}$ (make sure you understand why). Under this isomorphism, $\mathcal{J}_n$ becomes a subalgebra $\widehat{\mathcal{J}}_n$ of $\mathcal{F}(\mathrm{Geo}_n),$ and in order to invoke Stone-Weirstrass to prove that $\widehat{\mathcal{J}}_n=\mathcal{F}(\mathrm{Geo}_n)$ we just have to show that $\widehat{\mathcal{J}}_n$ separates points. If $\gamma,\gamma' \in \mathrm{Geo}_n$ are distinct geodesics $\mathrm{Par}_1 \to \mathrm{Par}_n$ in $\mathfrak{B},$ then there necessarily exists $2 \leq k \leq n$ such that $\gamma$ passes through $\mu \in \mathrm{Par}_k$ and $\gamma'$ passes through $\mu' \in \mathrm{Par}_k$ and $\mu \neq \mu'.$ In particular, $V^\mu$ and $V^{\mu'}$ are non-isomorphic irreducible representations of $S_k$ . Now, every element $|C_\nu\rangle \in \mathcal{Z}_k$ acts as a scalar operator in each of these irreducible representations: the eigenvalue of $|C_\nu\rangle$ acting in $V^\mu$ is the central character

$\omega^\mu_\nu=\frac{|C_\nu|}{\dim V^\mu}\chi^{\mu}_\nu$

and the eigenvalue of $|C_\nu\rangle$ acting in $V^{\mu'}$ is the central character

$\omega^{\mu'}_\nu=\frac{|C_\nu|}{\dim V^{\mu'}}\chi^{\mu'}_\nu.$

If it were the case that $\omega^\mu_\nu=\omega^{\mu'}_\nu$ for all $\nu \in \mathrm{Par}_k,$ this would force

$\chi^\mu_\nu = \frac{\dim V^\mu}{\dim V^{\mu'}}\chi^{\mu'}_\nu$

for all $\nu \in \mathrm{Par}_k,$ which would contradict the linear independence of characters of non-isomorphic irreducible representations. $\square$

Math 202C: Lecture 13

*** Problems due May 3 at 23:59 ***

Let $H \leq G$ be finite groups. In Lecture 12, we proved that restriction of irreducible representations of $G$ to $H$ is multiplicity-free if and only if the centralizer $Z(\mathcal{C}(H),\mathcal{C}(G))$ is a commutative subalgebra of $\mathcal{C}(G).$ Our objective now is to use this criterion to establish that restriction of irreducible representations of $S_n$ to $S_{n-1}$ is multiplicity free.

For the moment we stay in the general context where the group $G$ and its subgroup $H$ are arbitrary; eventually we will set $G=S_n$ and $H=S_{n-1}.$ By definition, the centralizer $Z(\mathcal{C}(H),\mathcal{C}(G))$ is the set of all $|A\rangle \in \mathcal{C}(G)$ which commute with every $|B\rangle \in \mathcal{C}(H),$ and it is clear that this condition is equivalent to

$|A\rangle|h\rangle = |h\rangle|A\rangle, \quad h \in H,$

which is in turn equivalent to

$|A\rangle = |h\rangle|A\rangle|h\rangle^*, \quad h \in H,$

with $|h\rangle^*=|h^{-1}\rangle.$ To understand those elements $|A\rangle \in \mathcal{C}(H)$ which have the above property, consider an equivalence relation on $G$ defined by

$g_1 \sim g_2 \iff g_2=hg_1h^{-1} \text{ for some} h \in H.$

That is, equivalence classes under this relation are orbits of $H$ acting on $G$ by conjugation. Let $\Lambda$ be a set parameterizing the distinct $H$ -conjugacy classes in $G$ ,

$\{H-\text{conjugacy classes in }G\} = \{C_\alpha \colon \alpha \in \Lambda\}.$

In the case where $H=G,$ this is just the set of conjugacy classes in $G$ as per usual, making the following a generalization of something we already know.

Problem 13.1. Prove that the set $\{|C_\alpha\rangle \colon \alpha \in \Gamma\}$ is an orthogonal basis of $Z(\mathcal{C}(H),\mathcal{C}(G)).$

In the case $H=G,$ the centralizer $Z(\mathcal{C}(H),\mathcal{C}(G))$ is the center $Z(\mathcal{C}(G)),$ which is commutative by definition, and the above problem recovers the 202B fact that the center of the convolution algebra of a group has a basis consisting of indicator functions of conjugacy classes. At the other extreme, if $H=\{\iota\}$ is the subgroup consisting solely of the identity element, then $Z(\mathcal{C}(H),\mathcal{C}(G))=\mathcal{C}(G),$ which is commutative precisely when $G$ is abelian. In between these two extremes, it may be more difficult to determine whether or not $Z(\mathcal{C}(H),\mathcal{C}(G))$ is commutative, but there is a particular circumstance which will force this to be the case.

Problem 13.2. Prove that if the class basis $\{|C_\alpha\rangle \colon \alpha \in \Lambda\}$ consists of selfadjoint elements, then $Z(\mathcal{C}(H),\mathcal{C}(G))$ is a commutative algebra.

We now specialize to the case $G=S_n$ and $H=S_{n-1}.$ We would like to find a concrete indexing of the $S_{n-1}$ -conjugacy classes in $S_n$ . For permutations, the operation of conjugation has a simple combinatorial meaning: conjugating $g$ by $h$ relabels the elements in the cycles of $g$ according to $h$ . If $h$ is confined to $S_{n-1},$ then it cannot relabel $n$ , hence in order for two permutations $g_1$ and $g_2$ in $S_n$ to be $S_{n-1}$ -conjugates it is necessary that the cycle containing $n$ in each have the same length.

Problem 13.3. Prove that the above condition is also sufficient for $g_1$ and $g_2$ to be $S_{n-1}$ -conjugates.

We can now conclude that $S_{n-1}$ -conjugacy classes in $S_n$ are indexed by pairs $(r,\alpha)$ consisting of an integer $1 \leq r \leq n$ corresponding to the length of the cycle containing $n$ and a partition $\alpha \vdash n-r$ consisting to the cycle-type of the remainder of the permutation. In particular, we get the dimension formula

$\dim Z(\mathcal{C}(S_{n-1},\mathcal{C}(S_n))=\sum\limits_{r=1}^n p(n-r),$

where $p(k)$ is the number of partitions of $k$ . Furthermore, it is easy to see that if an $S_{n-1}$ -conjugacy class $C_{(r,\alpha)}$ contains a permutation $\pi$ , then it also contains $\pi^{-1}$ , since the inverse of a permutation not only has the same cycle type, but actually the same elements within each cycle. Thus, the centralizer $Z(\mathcal{C}(S_{n-1}),\mathcal{C}(S_n))$ is commutative by Problem 13.2.

The significance of this commutativity is that, by Lecture 12, it is equivalent to multiplicity-free branching for irreducible representations of the symmetric groups: the isotypic decomposition of any irreducible representation $V^\lambda$ of $S_n$ when viewed as a representation of $S_{n-1}$ has the form

$V^\lambda = \bigoplus_{\mu \vdash n-1} \mathrm{Mult}(V^\mu,V^\lambda)V^\mu, \quad \mathrm{Mult}(V^\mu,V^\lambda) \in \{0,1\}.$

In Lecture 14, we will use this fact to construct a special orthonormal basis in $V^\lambda$ which we will then exploit mercilessly.

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

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