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

Published by Jonathan Novak

Mathematician at UC San Diego.

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