16b. Spectral measures

[math] \newcommand{\mathds}{\mathbb}[/math]

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Before getting into the case where the index is big, [math]N \gt \gt 0[/math], let us comment on one of the key ingredients for the above classification results, at [math]N \lt 6[/math]. This is the Jones annular theory of subfactors, which is something very beautiful and useful, regarding the case where the index is arbitrary, [math]N\in[1,\infty)[/math]. The main result is as follows:

Theorem

The theta series of a subfactor of index [math]N \gt 4[/math], which is given by

[[math]] \Theta (q)=q+\frac{1-q}{1+q}\, f\left( \frac{q}{(1+q)^2}\right) [[/math]]

with [math]f=\sum_k\dim(P_k)z^k[/math] being the Poincaré series, has positive coefficients.

Show Proof

This is something quite advanced, the idea being that [math]\Theta[/math] is the generating series of a certain series of multiplicities associated to the subfactor, and more specifically associated to the canonical inclusion [math]TL_N\subset P[/math]. We refer here to Jones' paper .

■

In relation to this, and to some questions from physics as well, coming from conformal field theory, an interesting question is that of computing the “blowup” of the spectral measure of the subfactor, via the Jones change of variables, namely:

[[math]] z\to\frac{q}{1+q^2} [[/math]]

This question makes sense in any index, meaning both [math]N\in[1,4][/math], where Theorem 16.6 does not apply, and [math]N\in(4,\infty)[/math], where Theorem 16.6 does apply. We will discuss in what follows both these questions, by starting with the small index one, [math]N\in[1,4][/math].

Following and related papers, it is convenient to stay, at least for the beginning, at a purely elementary level, and associate such series to any rooted bipartite graph. Let us start with the following definition, which is something straightforward, inspired by the definition of the Poincaré series of a subfactor, and by Theorem 16.2:

Definition

The Poincaré series of a rooted bipartite graph [math]X[/math] is

[[math]] f(z)=\sum_{k=0}^\infty{\rm loop}_X(2k)z^k [[/math]]

where [math]{\rm loop}_X(2k)[/math] is the number of [math]2k[/math]-loops based at the root.

In the case where [math]X[/math] is the principal graph of a subfactor [math]A_0\subset A_1[/math], this series [math]f[/math] is the Poincaré series of the subfactor, in the usual sense:

[[math]] f(z)=\sum_{k=0}^\infty\dim(A_0'\cap A_k)z^k [[/math]]

In general, the Poincaré series should be thought of as being a basic representation theory invariant of the underlying group-like object. For instance for the Wassermann type subfactor associated to a compact Lie group [math]G\subset U_N[/math], the Poincaré series is:

[[math]] f(z)=\int_G\frac{1}{1-Tr(g)z}\,dg [[/math]]

Regarding now the theta series, this can introduced as a version of the Poincaré series, via the change of variables [math]z^{-1/2}=q^{1/2}+q^{-1/2}[/math], as follows:

Definition

The theta series of a rooted bipartite graph [math]X[/math] is

[[math]] \Theta(q)=q+\frac{1-q}{1+q}f\left(\frac{q}{(1+q)^2}\right) [[/math]]

where [math]f[/math] is the Poincaré series.

The theta series can be written as [math]\Theta(q)=\sum a_rq^r[/math], and it follows from the above formula, via some simple manipulations, that its coefficients are integers:

[[math]] a_r\in\mathbb Z [[/math]]

In fact, we have the following explicit formula from Jones' paper , relating the coefficients of [math]\Theta(q)=\sum a_rq^r[/math] to those of the Poincaré series [math]f(z)=\sum c_kz^k[/math]:

[[math]] a_r=\sum_{k=0}^r(-1)^{r-k}\frac{2r}{r+k}\begin{pmatrix}r+k\cr r-k\end{pmatrix}c_k [[/math]]

In the case where [math]X[/math] is the principal graph of a subfactor [math]A_0\subset A_1[/math] of index [math]N \gt 4[/math], it is known from that the numbers [math]a_r[/math] are certain multiplicities associated to the planar algebra inclusion [math]TL_N\subset P[/math], as explained in Theorem 16.6 and its proof. In particular, the coefficients of the theta series are in this case positive integers:

[[math]] a_r\in\mathbb N [[/math]]

Before getting into computations, let us discuss as well the measure-theoretic versions of the above invariants. Once again, we start with an arbitrary rooted bipartite graph [math]X[/math]. We can first introduce a measure [math]\mu[/math], whose Stieltjes transform is [math]f[/math], as follows:

Definition

The real measure [math]\mu[/math] of a rooted bipartite graph [math]X[/math] is given by

[[math]] f(z)=\int_0^\infty\frac{1}{1-xz}\,d\mu(x) [[/math]]

where [math]f[/math] is the Poincaré series.

In the case where [math]X[/math] is the principal graph of a subfactor [math]A_0\subset A_1[/math], we recover in this way the spectral measure of the subfactor, as introduced in Definition 16.1, with the remark however that the existence of such a measure [math]\mu[/math] was not discussed there. In general, and so also in the particular subfactor case, clarifying the things here, the fact that [math]\mu[/math] as above exists indeed comes from the following simple fact:

Proposition

The real measure [math]\mu[/math] of a rooted bipartite graph [math]X[/math] is given by the following formula, where [math]L=MM^t[/math], with [math]M[/math] being the adjacency matrix of the graph,

[[math]] \mu=law(L) [[/math]]

and with the probabilistic computation being with respect to the expectation

[[math]] A\to \lt A \gt [[/math]]

with [math] \lt A \gt [/math] being the [math](*,*)[/math]-entry of a matrix [math]A[/math], where [math]*[/math] is the root.

Show Proof

With the conventions in the statement, namely [math]L=MM^t[/math], with [math]M[/math] being the adjacency matrix, and with [math] \lt A \gt [/math] being the [math](*,*)[/math]-entry of a matrix [math]A[/math], we have:

[[math]] \begin{eqnarray*} f(z) &=&\sum_{k=0}^\infty{\rm loop}_X(2k)z^k\\ &=&\sum_{k=0}^\infty\left \lt L^k\right \gt z^k\\ &=&\left \lt \frac{1}{1-Lz}\right \gt \end{eqnarray*} [[/math]]

But this shows that we have the formula [math]\mu=law(L)[/math], as desired.

■

In the subfactor case some further interpretations are available as well. For instance in the case of the fixed point subfactors coming from of a compact group [math]G\subset U_N[/math], discussed after Definition 16.7 above, [math]\mu[/math] is the spectral measure of the main character:

[[math]] \mu=law(\chi) [[/math]]

In relation now with the theta series, things are more tricky, in order to introduce its measure-theoretic version. Following , let us introduce the following notion:

Definition

The circular measure [math]\varepsilon[/math] of a rooted bipartite graph [math]X[/math] is given by

[[math]] d\varepsilon(q)=d\mu((q+q^{-1})^2) [[/math]]

where [math]\mu[/math] is the associated real measure.

In other words, the circular measure [math]\varepsilon[/math] is by definition the pullback of the usual real measure [math]\mu[/math] via the following map, coming from the theory of the theta series in :

[[math]] \mathbb R\cup\mathbb T\to\mathbb R_+ [[/math]]

[[math]] q\to (q+q^{-1})^2 [[/math]]

As we will see, all this best works in index [math]N\in[1,4][/math], with the circular measure [math]\varepsilon[/math] being here the best-looking invariant, among all subfactor invariants. In index [math]N \gt 4[/math] things will turn to be quite complicated, but more on this later.

As a basic example for all this, assume that [math]\mu[/math] is a discrete measure, supported by [math]n[/math] positive numbers [math]x_1 \lt \ldots \lt x_n[/math], with corresponding densities [math]p_1,\ldots,p_n[/math]:

[[math]] \mu=\sum_{i=1}^n p_i\delta_{x_i} [[/math]]

For each [math]i\in\{1,\ldots,n\}[/math] the equation [math](q+q^{-1})^2=x_i[/math] has four solutions, that we can denote [math]q_i,q_i^{-1},-q_i,-q_i^{-1}[/math]. With this notation, we have:

[[math]] \varepsilon=\frac{1}{4}\sum_{i=1}^np_i\left(\delta_{q_i}+\delta_{q_i^{-1}}+\delta_{-q_i}+\delta_{-q_i^{-1}}\right) [[/math]]

In general, the basic properties of [math]\varepsilon[/math] can be summarized as follows:

Proposition

The circular measure has the following properties:

• [math]\varepsilon[/math] has equal density at [math]q,q^{-1},-q,-q^{-1}[/math].
• The odd moments of [math]\varepsilon[/math] are [math]0[/math].
• The even moments of [math]\varepsilon[/math] are half-integers.
• When [math]X[/math] has norm [math]\leq 2[/math], [math]\varepsilon[/math] is supported by the unit circle.
• When [math]X[/math] is finite, [math]\varepsilon[/math] is discrete.
• If [math]K[/math] is a solution of [math]L=(K+K^{-1})^2[/math], then [math]\varepsilon={\rm law}(K)[/math].

Show Proof

These results can be deduced from definitions, the idea being that (1-5) are trivial, and that (6) follows from the formula of [math]\mu[/math] from Proposition 16.10.

■

In addition to the above, we have the following key formula, which gives the even moments of [math]\varepsilon[/math], and makes the connection with the Jones theta series:

Theorem

We have the Stieltjes transform type formula

[[math]] 2\int\frac{1}{1-qu^2}\,d\varepsilon(u)=1+T(q)(1-q) [[/math]]

where the [math]T[/math] series of a rooted bipartite graph [math]X[/math] is by definition given by

[[math]] T(q)=\frac{\Theta(q)-q}{1-q} [[/math]]

with [math]\Theta[/math] being the associated theta series.

Show Proof

This follows by applying the change of variables [math]q\to (q+q^{-1})^2[/math] to the fact that [math]f[/math] is the Stieltjes transform of [math]\mu[/math]. Indeed, we obtain in this way:

[[math]] \begin{eqnarray*} 2\int\frac{1}{1-qu^2}\,d\varepsilon(u) &=&1+\frac{1-q}{1+q}f\left(\frac{q}{(1+q)^2}\right)\\ &=&1+\Theta(q)-q\\ &=&1+T(q)(1-q) \end{eqnarray*} [[/math]]

Thus, we are led to the conclusion in the statement.

■

As a final theoretical result about all these invariants, which is this time something non-trivial, in the subfactor case, we have the following result, due to Jones :

Theorem

In the case where [math]X[/math] is the principal graph of an irreducible subfactor of index [math] \gt 4[/math], the moments of [math]\varepsilon[/math] are positive numbers.

Show Proof

This follows indeed from the result in that the coefficients of [math]\Theta[/math] are positive numbers, as explained in Theorem 16.6, via the formula in Theorem 16.13.

■

Summarizing, we have a whole menagery of subfactor, planar algebra and bipartite graph invariants, which come in several flavors, namely series and measures, and which can be linear or circular, and which all appear as versions of the Poincaré series.

Our claim now is that the circular measure [math]\varepsilon[/math] is the “best” invariant. As a first justification for this claim, let us compute [math]\varepsilon[/math] for the simplest possible graph in the index range [math]N\in[1,4][/math], namely the graph [math]\tilde{A}_{2n}[/math]. We obtain here something nice, as follows:

Theorem

The circular measure of the basic index [math]4[/math] graph, namely

[[math]] \begin{matrix} &\circ&\!\!\!\!-\circ-\circ\cdots\circ-\circ-&\!\!\!\!\circ\cr \tilde{A}_{2n}=&|&&\!\!\!\!|\cr &\bullet&\!\!\!\!-\circ-\circ-\circ-\circ-&\!\!\!\!\circ\cr\cr\cr\end{matrix} [[/math]]

\vskip-7mm is the uniform measure on the [math]2n[/math]-roots of unity.

Show Proof

Let us identify the vertices of [math]X=\tilde{A}_{2n}[/math] with the group [math]\{w^k\}[/math] formed by the [math]2n[/math]-th roots of unity in the complex plane, where [math]w=e^{\pi i/n}[/math]. The adjacency matrix of [math]X[/math] acts then on the functions [math]f\in C(X)[/math] in the following way:

[[math]] Mf(w^s)=f(w^{s-1})+f(w^{s+1}) [[/math]]

But this shows that we have [math]M=K+K^{-1}[/math], where [math]K[/math] is given by:

[[math]] Kf(w^s)=f(w^{s+1}) [[/math]]

Thus we can use the last assertion in Proposition 16.12, and we get [math]\varepsilon={\rm law}(K)[/math], which is the uniform measure on the [math]2n[/math]-roots of unity. See for details.

■

In order to discuss all this more systematically, and for all the ADE graphs, the idea will be that of looking at the combinatorics of the roots of unity. Let us introduce:

Definition

The series of the form

[[math]] \xi(n_1,\ldots,n_s:m_1,\ldots,m_t)=\frac{(1-q^{n_1})\ldots(1-q^{n_s})}{(1-q^{m_1})\ldots(1-q^{m_t})} [[/math]]

with [math]n_i,m_i\in\mathbb N[/math] are called cyclotomic.

It is technically convenient to allow as well [math]1+q^n[/math] factors, to be designated by [math]n^+[/math] symbols in the above writing. For instance we have, by definition:

[[math]] \xi(2^+:3)=\xi(4:2,3) [[/math]]

Also, it is convenient in what follows to use the following notations:

[[math]] \xi'=\frac{\xi}{1-q}\quad,\quad \xi''=\frac{\xi}{1-q^2} [[/math]]

The Poincaré series of the ADE graphs are given by quite complicated formulae. However, the corresponding [math]T[/math] series are all cyclotomic, as follows:

Theorem

The [math]T[/math] series of the ADE graphs are as follows:

• For [math]A_{n-1}[/math] we have [math]T=\xi(n-1:n)[/math].
• For [math]D_{n+1}[/math] we have [math]T=\xi(n-1^+:n^+)[/math].
• For [math]\tilde{A}_{2n}[/math] we have [math]T=\xi'(n^+:n)[/math].
• For [math]\tilde{D}_{n+2}[/math] we have [math]T=\xi''(n+1^+:n)[/math].
• For [math]E_6[/math] we have [math]T=\xi(8:3,6^+)[/math].
• For [math]E_7[/math] we have [math]T=\xi(12:4,9^+)[/math].
• For [math]E_8[/math] we have [math]T=\xi(5^+,9^+:15^+)[/math].
• For [math]\tilde{E}_6[/math] we have [math]T=\xi(6^+:3,4)[/math].
• For [math]\tilde{E}_7[/math] we have [math]T=\xi(9^+:4,6)[/math].
• For [math]\tilde{E}_8[/math] we have [math]T=\xi(15^+:6,10)[/math].

Show Proof

These formulae were obtained in , by counting loops, then by making the change of variables [math]z^{-1/2}=q^{1/2}+q^{-1/2}[/math], and factorizing the resulting series. An alternative proof for these formulae can be obtained by using planar algebra methods.

■

Our purpose now will be that of converting the above technical results, regarding the [math]T[/math] series, into some final results, regarding the corresponding circular measures [math]\varepsilon[/math]. For this purpose, we will use the conversion formula in Theorem 16.13.

In order to formulate our results, we will need some more theory. First, we have:

Definition

A cyclotomic measure is a probability measure [math]\varepsilon[/math] on the unit circle, having the following properties:

• [math]\varepsilon[/math] is supported by the [math]2n[/math]-roots of unity, for some [math]n\in\mathbb N[/math].
• [math]\varepsilon[/math] has equal density at [math]q,q^{-1},-q,-q^{-1}[/math].

It follows from Theorem 16.17 that the circular measures of the finite ADE graphs are supported by certain roots of unity, hence are cyclotomic. We will be back to this.

At the general level now, let us introduce as well the following notion:

Definition

The [math]T[/math] series of a cyclotomic measure [math]\varepsilon[/math] is given by:

[[math]] 1+T(q)(1-q)=2\int\frac{1}{1-qu^2}\,d\varepsilon(u) [[/math]]

Observe that this formula is nothing but the one in Theorem 16.13, written now in the other sense. In other words, if the cyclotomic measure [math]\varepsilon[/math] happens to be the circular measure of a rooted bipartite graph, then the [math]T[/math] series as defined above coincides with the [math]T[/math] series as defined before. This is useful for explicit computations.

We are now ready to discuss the circular measures of the various ADE graphs. The idea is that these measures are all cyclotomic, of level [math]\leq 3[/math], and can be expressed in terms of the basic polynomial densities of degree [math]\leq 6[/math], namely:

[[math]] \alpha=Re(1-q^2) [[/math]]

[[math]] \beta=Re(1-q^4) [[/math]]

[[math]] \gamma=Re(1-q^6) [[/math]]

To be more precise, we have the following result, with [math]\alpha,\beta,\gamma[/math] being as above, with [math]d_n[/math] being the uniform measure on the [math]2n[/math]-th roots of unity, and with [math]d_n'=2d_{2n}-d_n[/math] being the uniform measure on the odd [math]4n[/math]-roots of unity:

Theorem

The circular measures of the ADE graphs are given by:

• [math]A_{n-1}\to\alpha_n[/math].
• [math]\tilde{A}_{2n}\to d_n[/math].
• [math]D_{n+1}\to\alpha_n'[/math].
• [math]\tilde{D}_{n+2}\to (d_n+d_1')/2[/math].
• [math]E_6\to\alpha_{12}+(d_{12}-d_6-d_4+d_3)/2[/math].
• [math]E_7\to\beta_9'+(d_1'-d_3')/2[/math].
• [math]E_8\to\alpha_{15}'+\gamma_{15}'-(d_5'+d_3')/2[/math].
• [math]\tilde{E}_{n+3}\to (d_n+d_3+d_2-d_1)/2[/math].

Show Proof

This follows from the [math]T[/math] series formulae in Theorem 16.17, via some routine manipulations, based on the general conversion formulae given above.

■

It is possible to further build along the above lines, with a combinatorial refinement of the formulae in Theorem 16.20, making appear a certain connection with the Deligne work on the exceptional series of Lie groups, which is not understood yet.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].