14c. Brauer theorems

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

As a basic illustration for the Tannakian correspondence, we will work out Brauer theorems for [math]O_N,U_N[/math]. These are very classical results, and there are many possible proofs for them. We will follow here the modern approach from . Let us start with:

Definition

Given a pairing [math]\pi\in P_2(k,l)[/math] and an integer [math]N\in\mathbb N[/math], we can construct a linear map between tensor powers of [math]\mathbb C^N[/math],

[[math]] T_\pi:(\mathbb C^N)^{\otimes k}\to(\mathbb C^N)^{\otimes l} [[/math]]

by the following formula, with [math]e_1,\ldots,e_N[/math] being the standard basis of [math]\mathbb C^N[/math],

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l} [[/math]]

and with the coefficients on the right being Kronecker type symbols,

[[math]] \delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}\in\{0,1\} [[/math]]

whose values depend on whether the indices fit or not.

To be more precise here, we put the multi-indices [math]i=(i_1,\ldots,i_k)[/math] and [math]j=(j_1,\ldots,j_l)[/math] on the legs of our pairing [math]\pi[/math], in the obvious way. In the case where all strings of [math]\pi[/math] join pairs of equal indices of [math]i,j[/math], we set [math]\delta_\pi(^i_j)=1[/math]. Otherwise, we set [math]\delta_\pi(^i_j)=0[/math].

The point with the above definition comes from the fact that most of the “familiar” maps, in the Tannakian context, are of the above form. Here are some examples:

Proposition

The correspondence [math]\pi\to T_\pi[/math] has the following properties:

• [math]T_\cap=(1\to\sum_ie_i\otimes e_i)[/math].
• [math]T_\cup=(e_i\otimes e_j\to\delta_{ij})[/math].
• [math]T_{||\ldots||}=id[/math].
• [math]T_{\slash\hskip-1.5mm\backslash}=(e_a\otimes e_b\to e_b\otimes e_a)[/math].

Show Proof

We can assume that all legs of [math]\pi[/math] are colored [math]\circ[/math], and then:

(1) We have [math]\cap\in P_2(\emptyset,\circ\circ)[/math], so the corresponding linear map is as follows:

[[math]] T_\cap:\mathbb C\to\mathbb C^N\otimes\mathbb C^N [[/math]]

The formula of this linear map is then, as claimed:

[[math]] \begin{eqnarray*} T_\cap(1) &=&\sum_{ij}\delta_\cap(i\ j)e_i\otimes e_j\\ &=&\sum_{ij}\delta_{ij}e_i\otimes e_j\\ &=&\sum_ie_i\otimes e_i \end{eqnarray*} [[/math]]

(2) Here we have [math]\cup\in P_2(\circ\circ,\emptyset)[/math], so the corresponding linear map is as follows:

[[math]] T_\cap:\mathbb C^N\otimes\mathbb C^N\to\mathbb C [[/math]]

The formula of this linear form is then as follows:

[[math]] T_\cap(e_i\otimes e_j) =\delta_\cap(i\ j) =\delta_{ij} [[/math]]

(3) Consider indeed the “identity” pairing [math]||\ldots||\in P_2(k,k)[/math], with [math]k=\circ\circ\ldots\circ\circ[/math]. The corresponding linear map is then the identity, because we have:

[[math]] \begin{eqnarray*} T_{||\ldots||}(e_{i_1}\otimes\ldots\otimes e_{i_k}) &=&\sum_{j_1\ldots j_k}\delta_{||\ldots||}\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_k\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_k}\\ &=&\sum_{j_1\ldots j_k}\delta_{i_1j_1}\ldots\delta_{i_kj_k}e_{j_1}\otimes\ldots\otimes e_{j_k}\\ &=&e_{i_1}\otimes\ldots\otimes e_{i_k} \end{eqnarray*} [[/math]]

(4) For the basic crossing [math]\slash\hskip-2.0mm\backslash\in P_2(\circ\circ,\circ\circ)[/math], the corresponding linear map is as follows:

[[math]] T_{\slash\hskip-1.5mm\backslash}:\mathbb C^N\otimes\mathbb C^N\to\mathbb C^N\otimes\mathbb C^N [[/math]]

This linear map can be computed as follows:

[[math]] \begin{eqnarray*} T_{\slash\hskip-1.5mm\backslash}(e_i\otimes e_j) &=&\sum_{kl}\delta_{\slash\hskip-1.5mm\backslash}\begin{pmatrix}i&j\\ k&l\end{pmatrix}e_k\otimes e_l\\ &=&\sum_{kl}\delta_{il}\delta_{jk}e_k\otimes e_l\\ &=&e_j\otimes e_i \end{eqnarray*} [[/math]]

Thus we obtain the flip operator [math]\Sigma(a\otimes b)=b\otimes a[/math], as claimed.

■

The relation with the Tannakian categories comes from the following key result:

Proposition

The assignement [math]\pi\to T_\pi[/math] is categorical, in the sense that

[[math]] T_\pi\otimes T_\sigma=T_{[\pi\sigma]}\quad,\quad T_\pi T_\sigma=N^{c(\pi,\sigma)}T_{[^\sigma_\pi]}\quad,\quad T_\pi^*=T_{\pi^*} [[/math]]

where [math]c(\pi,\sigma)[/math] is the number of circles appearing in the middle, when concatenating.

Show Proof

The concatenation axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&(T_\pi\otimes T_\sigma)(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r})\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_\pi\begin{pmatrix}i_1&\ldots&i_p\\j_1&\ldots&j_q\end{pmatrix}\delta_\sigma\begin{pmatrix}k_1&\ldots&k_r\\l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_{[\pi\sigma]}\begin{pmatrix}i_1&\ldots&i_p&k_1&\ldots&k_r\\j_1&\ldots&j_q&l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&T_{[\pi\sigma]}(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r}) \end{eqnarray*} [[/math]]

The composition axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&T_\pi T_\sigma(e_{i_1}\otimes\ldots\otimes e_{i_p})\\ &=&\sum_{j_1\ldots j_q}\delta_\sigma\begin{pmatrix}i_1&\ldots&i_p\\j_1&\ldots&j_q\end{pmatrix} \sum_{k_1\ldots k_r}\delta_\pi\begin{pmatrix}j_1&\ldots&j_q\\k_1&\ldots&k_r\end{pmatrix}e_{k_1}\otimes\ldots\otimes e_{k_r}\\ &=&\sum_{k_1\ldots k_r}N^{c(\pi,\sigma)}\delta_{[^\sigma_\pi]}\begin{pmatrix}i_1&\ldots&i_p\\k_1&\ldots&k_r\end{pmatrix}e_{k_1}\otimes\ldots\otimes e_{k_r}\\ &=&N^{c(\pi,\sigma)}T_{[^\sigma_\pi]}(e_{i_1}\otimes\ldots\otimes e_{i_p}) \end{eqnarray*} [[/math]]

Finally, the involution axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&T_\pi^*(e_{j_1}\otimes\ldots\otimes e_{j_q})\\ &=&\sum_{i_1\ldots i_p} \lt T_\pi^*(e_{j_1}\otimes\ldots\otimes e_{j_q}),e_{i_1}\otimes\ldots\otimes e_{i_p} \gt e_{i_1}\otimes\ldots\otimes e_{i_p}\\ &=&\sum_{i_1\ldots i_p}\delta_\pi\begin{pmatrix}i_1&\ldots&i_p\\ j_1&\ldots& j_q\end{pmatrix}e_{i_1}\otimes\ldots\otimes e_{i_p}\\ &=&T_{\pi^*}(e_{j_1}\otimes\ldots\otimes e_{j_q}) \end{eqnarray*} [[/math]]

Summarizing, our correspondence is indeed categorical.

■

The above result suggests the following general definition, from :

Definition

Let [math]P_2(k,l)[/math] be the set of pairings between an upper colored integer [math]k[/math], and a lower colored integer [math]l[/math]. A collection of subsets

[[math]] D=\bigsqcup_{k,l}D(k,l) [[/math]]

with [math]D(k,l)\subset P_2(k,l)[/math] is called a category of pairings when it has the following properties:

• Stability under the horizontal concatenation, [math](\pi,\sigma)\to[\pi\sigma][/math].
• Stability under vertical concatenation [math](\pi,\sigma)\to[^\sigma_\pi][/math], with matching middle symbols.
• Stability under the upside-down turning [math]*[/math], with switching of colors, [math]\circ\leftrightarrow\bullet[/math].
• Each set [math]P(k,k)[/math] contains the identity partition [math]||\ldots||[/math].
• The sets [math]P(\emptyset,\circ\bullet)[/math] and [math]P(\emptyset,\bullet\circ)[/math] both contain the semicircle [math]\cap[/math].
• The sets [math]P(k,\bar{k})[/math] with [math]|k|=2[/math] contain the crossing partition [math]\slash\hskip-2.0mm\backslash[/math].

Observe the similarity with the axioms for Tannakian categories, from the beginning of this chapter. We will see in a moment that this similarity can be turned into something very precise, with the categories of pairings producing Tannakian categories.

As basic examples of such categories, that we have already met in the above, we have the categories [math]P_2,\mathcal P_2[/math] of pairings, and of matching pairings, with the convention that a matching pairing must pair [math]\circ-\bullet[/math] on the horizontal, and [math]\circ-\circ[/math] or [math]\bullet-\bullet[/math] on the vertical. There are many other examples, and we will discuss this gradually, in what follows.

In relation with the compact groups, we have the following result:

Theorem

Each category of pairings, in the above sense,

[[math]] D=(D(k,l)) [[/math]]

produces a family of compact groups [math]G=(G_N)[/math], one for each [math]N\in\mathbb N[/math], via the formula

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

and the Tannakian duality correspondence.

Show Proof

Given an integer [math]N\in\mathbb N[/math], consider the correspondence [math]\pi\to T_\pi[/math] constructed in Definition 14.17, and then the collection of linear spaces in the statement, namely:

[[math]] C_{kl}=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

According to Proposition 14.19, and to our axioms for the categories of partitions, from Definition 14.20, this collection of spaces [math]C=(C_{kl})[/math] satisfies the axioms for the Tannakian categories, from the beginning of this chapter. Thus the Tannakian duality result there applies, and provides us with a closed subgroup [math]G_N\subset U_N[/math] such that:

[[math]] C_{kl}=Hom(u^{\otimes k},u^{\otimes l}) [[/math]]

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

■

The above result is something fundamental, and suggests formulating:

Definition

Assuming that a closed subgroup [math]G\subset_uU_N[/math] has the property

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

for a certain category of pairings [math]D=(D(k,l))[/math], we say that [math]G[/math] is easy.

This definition, from , is motivated by the fact that, from the point of view of Tannakian duality, the above groups are indeed the “easiest” possible ones. Of course, this might sound a bit strange, after all the quite complicated things that we did in this chapter. But hey, there is a beginning for everything. We will get to know better Tannakian duality and easiness, and their applications, in what follows, and please believe me, you will reach too to the conclusion that Definition 14.22 is justified.

As another comment, it is possible to talk about more general easy groups, by using general categories of partitions, instead of just categories of pairings. We will be back to all this, with a systematic study of easiness, in chapter 15 below.

As a technical remark now, to be always kept in mind, when dealing with easiness, the category of pairings producing an easy group is not unique, for instance because at [math]N=1[/math] all the possible categories of pairings produce the same easy group, namely the trivial group [math]G=\{1\}[/math]. Thus, some subtleties are going on here. More on this later.

Getting back now to concrete things, the point now is that with the above ingredients in hand, and as a first application of Tannakian duality, we can establish a useful result, namely the Brauer theorem for the unitary group [math]U_N[/math]. The statement is a follows:

Theorem

For the unitary group [math]U_N[/math] we have

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in\mathcal P_2(k,l)\right) [[/math]]

where [math]\mathcal P_2[/math] denotes as usual the category of all matching pairings.

Show Proof

This is something very old and classical, due to Brauer , and in what follows we will present a simplified proof for it, based on the easiness technology developed above. Consider the spaces on the right in the statement, namely:

[[math]] C_{kl}=span\left(T_\pi\Big|\pi\in\mathcal P_2(k,l)\right) [[/math]]

According to Proposition 14.19 these spaces form a tensor category. Thus, by Tannakian duality, these spaces must come from a certain closed subgroup [math]G\subset U_N[/math]. To be more precise, if we denote by [math]v[/math] the fundamental representation of [math]G[/math], then:

[[math]] C_{kl}=Hom(v^{\otimes k},v^{\otimes l}) [[/math]]

We must prove that we have [math]G=U_N[/math]. For this purpose, let us recall that the unitary group [math]U_N[/math] is defined via the following relations:

[[math]] u^*=u^{-1}\quad,\quad u^t=\bar{u}^{-1} [[/math]]

But these relations tell us precisely that the following two operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad:\quad \pi={\ }^{\,\cap}_{\circ\bullet}\ ,\ {\ }^{\,\cap}_{\bullet\circ} [[/math]]

Thus the associated Tannakian category is [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt {\ }^{\,\cap}_{\circ\bullet}\,\,,{\ }^{\,\cap}_{\bullet\circ} \gt =\mathcal P_2 [[/math]]

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

■

Regarding the orthogonal group [math]O_N[/math], we have here a similar result, as follows:

Theorem

For the orthogonal group [math]O_N[/math] we have

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in P_2(k,l)\right) [[/math]]

where [math]P_2[/math] denotes as usual the category of all pairings.

Show Proof

As before with Theorem 14.23, regarding [math]U_N[/math], this is something very old and classical, due to Brauer , that we can now prove by using the easiness technology developed above. Consider the spaces on the right in the statement, namely:

[[math]] C_{kl}=span\left(T_\pi\Big|\pi\in P_2(k,l)\right) [[/math]]

According to Proposition 14.19 these spaces form a tensor category. Thus, by Tannakian duality, these spaces must come from a certain closed subgroup [math]G\subset U_N[/math]. To be more precise, if we denote by [math]v[/math] the fundamental representation of [math]G[/math], then:

[[math]] C_{kl}=Hom(v^{\otimes k},v^{\otimes l}) [[/math]]

We must prove that we have [math]G=O_N[/math]. For this purpose, let us recall that the orthogonal group [math]O_N\subset U_N[/math] is defined by imposing the following relations:

[[math]] u_{ij}=\bar{u}_{ij} [[/math]]

But these relations tell us precisely that the following two operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad:\quad\pi=|^{\hskip-1.32mm\circ}_{\hskip-1.32mm\bullet}\ ,\ |_{\hskip-1.32mm\circ}^{\hskip-1.32mm\bullet} [[/math]]

Thus the associated Tannakian category is [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt \mathcal P_2,|^{\hskip-1.32mm\circ}_{\hskip-1.32mm\bullet},|_{\hskip-1.32mm\circ}^{\hskip-1.32mm\bullet} \gt =P_2 [[/math]]

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

■

We will see later, in chapter 16 below, applications of the above results, to integration problems over [math]O_N,U_N[/math], by using the Peter-Weyl methods from chapter 13.

General references

Banica, Teo (2024). "Linear algebra and group theory". arXiv:2206.09283 [math.CO].