4d. Brauer theorems

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

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As a second application of Tannakian duality, let us study now the representation theory of [math]O_N^+,U_N^+[/math]. In order to get started, let us get back to the operators [math]R,R^*[/math], from the beginning of this chapter. We know that these two operators must be present in any Tannakian category, and in what concerns [math]U_N^+[/math], which is the biggest [math]N\times N[/math] compact quantum group, a converse of this fact holds, by contravariant functoriality, as follows:

Proposition

The tensor category [math] \lt R,R^* \gt [/math] generated by the operators

[[math]] R:1\to\sum_ie_i\otimes e_i [[/math]]

[[math]] R^*(e_i\otimes e_j)=\delta_{ij} [[/math]]

produces via Tannakian duality the algebra [math]C(U_N^+)[/math].

Show Proof

This follows from the results from the beginning of this chapter, via the Tannakian duality established above. To be more precise, we know from Proposition 4.5 that the intertwining relations coming from the operators [math]R,R^*[/math], and so from any element of the tensor category [math] \lt R,R^* \gt [/math], hold automatically. Thus the quotient operation in Proposition 4.8 is trivial, and we obtain the algebra [math]C(U_N^+)[/math] itself, as stated.

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As a conclusion, in order to compute the Tannakian category of [math]U_N^+[/math], we must simply solve a linear algebra question, namely computing the category [math] \lt R,R^* \gt [/math]. Regarding now [math]O_N^+[/math], the result here is similar, as follows:

Proposition

The tensor category [math] \lt R,R^* \gt [/math] generated by the operators

[[math]] R:1\to\sum_ie_i\otimes e_i [[/math]]

[[math]] R^*(e_i\otimes e_j)=\delta_{ij} [[/math]]

with identifying the colors, [math]\circ=\bullet[/math], produces via Tannakian duality the algebra [math]C(O_N^+)[/math].

Show Proof

By Proposition 4.5 the intertwining relations coming from [math]R,R^*[/math], and so from any element of the tensor category [math] \lt R,R^* \gt [/math], hold automatically, so the quotient operation in Proposition 4.8 is trivial, and we obtain [math]C(O_N^+)[/math] itself, as stated.

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Our goal now will be that of reaching to a better understanding of [math]R,R^*[/math]. In order to do so, we use a diagrammatic formalism, as follows:

Definition

Let [math]k,l[/math] be two colored integers, having lengths [math]|k|,|l|\in\mathbb N[/math].

• [math]P_2(k,l)[/math] is the set of pairings between an upper row of [math]|k|[/math] points, and a lower row of [math]|l|[/math] points, with these two rows of points colored by [math]k,l[/math].
• [math]\mathcal{P}_2(k,l)\subset P_2(k,l)[/math] is the set of matching pairings, whose horizontal strings connect [math]\circ-\circ[/math] or [math]\bullet-\bullet[/math], and whose vertical strings connect [math]\circ-\bullet[/math].
• [math]NC_2(k,l)\subset P_2(k,l)[/math] is the set of pairings which are noncrossing, in the sense that we can draw the pairing as for the strings to be noncrossing.
• [math]\mathcal{NC}_2(k,l)\subset P_2(k,l)[/math] is the subset of noncrossing matching pairings, obtained as an intersection, [math]\mathcal{NC}_2(k,l)=NC_2(k,l)\cap\mathcal P_2(k,l)[/math].

The relation with the Tannakian categories of linear maps comes from the fact that we can associate linear maps to the pairings, as in , as follows:

Definition

Associated to any pairing [math]\pi\in P_2(k,l)[/math] and any integer [math]N\in\mathbb N[/math] is the linear map

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

given 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 Kronecker symbols [math]\delta_\pi\in\{0,1\}[/math] depending on whether the indices fit or not.

To be more precise here, in the definition of the Kronecker symbols, we agree to put the two multi-indices on the two rows of points of the pairing, in the obvious way. The Kronecker symbols are then defined by [math]\delta_\pi=1[/math] when all the strings of [math]\pi[/math] join equal indices, and by [math]\delta_\pi=0[/math] otherwise. Observe that all this is independent of the coloring.

Here are a few basic examples of such linear maps:

Proposition

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

• [math]T_\cap=R[/math].
• [math]T_\cup=R^*[/math].
• [math]T_{||\ldots||}=id[/math].
• [math]T_{\slash\hskip-1.5mm\backslash}=\Sigma[/math].

Show Proof

We can assume if we want that all the upper and lower legs of [math]\pi[/math] are colored [math]\circ[/math]. With this assumption made, the proof goes as follows:

(1) We have [math]\cap\in P_2(\emptyset,\circ\circ)[/math], and so the corresponding operator is a certain linear map [math]T_\cap:\mathbb C\to\mathbb C^N\otimes\mathbb C^N[/math]. The formula of this map is as follows:

[[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]]

We recognize here the formula of [math]R(1)[/math], and so we have [math]T_\cap=R[/math], as claimed.

(2) Here we have [math]\cup\in P_2(\circ\circ,\emptyset)[/math], and so the corresponding operator is a certain linear form [math]T_\cap:\mathbb C^N\otimes\mathbb C^N\to\mathbb C[/math]. The formula of this linear form is as follows:

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

Since this is the same as [math]R^*(e_i\otimes e_j)[/math], we have [math]T_\cup=R^*[/math], as claimed.

(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) In the case of the basic crossing [math]\slash\hskip-2.0mm\backslash\in P_2(\circ\circ,\circ\circ)[/math], the corresponding linear map [math]T_{\slash\hskip-1.5mm\backslash}:\mathbb C^N\otimes\mathbb C^N\to\mathbb C^N\otimes\mathbb C^N[/math] 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.

■

Summarizing, the correspondence [math]\pi\to T_\pi[/math] provides us with some simple formulae for the operators [math]R,R^*[/math] that we are interested in, and for other important operators, such as the flip [math]\Sigma(a\otimes b)=b\otimes a[/math], and has as well some interesting categorical properties.

Let us further explore these properties, and make the link with the Tannakian categories. We have the following key result, from :

Proposition

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

[[math]] T_\pi\otimes T_\sigma=T_{[\pi\sigma]} [[/math]]

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

[[math]] T_\pi^*=T_{\pi^*} [[/math]]

where [math]c(\pi,\sigma)[/math] are certain integers, coming from the erased components in the middle.

Show Proof

The formulae in the statement are all elementary, as follows:

(1) 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]]

(2) 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]]

(3) 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.

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We can now formulate a first non-trivial result regarding [math]O_N^+,U_N^+[/math], which is a Brauer type theorem for these quantum groups, as follows:

Theorem

For the quantum groups [math]O_N^+,U_N^+[/math] we have

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

with the sets on the right being respectively as follows,

[[math]] D=NC_2,\mathcal{NC}_2 [[/math]]

and with the correspondence [math]\pi\to T_\pi[/math] being constructed as above.

Show Proof

We know from Proposition 4.22 above that the quantum group [math]U_N^+[/math] corresponds via Tannakian duality to the following category:

[[math]] C= \lt R,R^* \gt [[/math]]

On the other hand, it follows from the above categorical considerations that this latter category is given by the following formula:

[[math]] C=span\left(T_\pi\Big|\pi\in\mathcal{NC}_2\right) [[/math]]

To be more precise, consider the following collection of vector spaces:

[[math]] C'=span\left(T_\pi\Big|\pi\in\mathcal{NC}_2\right) [[/math]]

According to the various formulae in Proposition 4.27, these vector spaces form a tensor category. But since the two matching semicircles generate the whole collection of matching pairings, via the operations in Proposition 4.27, we obtain from this [math]C=C'[/math].

As for the result from [math]O_N^+[/math], this follows by adding to the picture the self-adjointness condition [math]u=\bar{u}[/math], which corresponds, at the level of pairings, to removing the colors.

■

The above result is very useful, and virtually solves any question about [math]O_N^+,U_N^+[/math]. We will be back to it in the next chapter, and afterwards, with applications, both of algebraic and analytic nature. As an example here, just by counting the dimensions of the spaces in Theorem 4.28, we will be able to compute the laws of the main characters.

By using the same methods, namely the general Tannakian duality result established above, we can recover as well the classical Brauer theorem , as follows:

Theorem

For the groups [math]O_N,U_N[/math] we have

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

with [math]D=P_2,\mathcal P_2[/math] respectively, and with [math]\pi\to T_\pi[/math] being constructed as above.

Show Proof

As already mentioned, this result is due to Brauer , and is closely related to the Schur-Weyl duality . There are several proofs of this result, one classical proof being via classical Tannakian duality, for the usual closed subgroups [math]G\subset U_N[/math].

In the present context, we can deduce this result from the one that we already have, for [math]O_N^+,U_N^+[/math]. The idea is very simple, namely that of “adding crossings”, as follows:

(1) The group [math]U_N\subset U_N^+[/math] is defined via the following relations:

[[math]] [u_{ij},u_{kl}]=0 [[/math]]

[[math]] [u_{ij},\bar{u}_{kl}]=0 [[/math]]

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

[[math]] T_\pi\quad,\quad \pi={\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\circ}_{\hskip-2.5mm\circ\circ} [[/math]]

[[math]] T_\pi\quad,\quad \pi={\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\bullet}_{\hskip-2.5mm\bullet\circ} [[/math]]

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

[[math]] D = \lt \mathcal{NC}_2,{\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\circ}_{\hskip-2.5mm\circ\circ},{\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\bullet}_{\hskip-2.5mm\bullet\circ} \gt =\mathcal P_2 [[/math]]

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

(2) In order to deal now with [math]O_N[/math], we can simply use the following formula:

[[math]] O_N=O_N^+\cap U_N [[/math]]

At the categorical level, this tells us that the associated Tannakian category is given by [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt NC_2,\mathcal P_2 \gt =P_2 [[/math]]

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

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Summarizing, the orthogonal and unitary groups [math]O_N,U_N[/math] and their free analogues [math]O_N^+,U_N^+[/math] appear to be “easy”, in the sense that their associated Tannakian categories appear in the simplest possible way, namely from certain categories of pairings.

We will be exploit this phenomenon in chapters 5-6 below, with a detailed algebraic and analytic study of these quantum groups, based on their “easiness” property. Then, we will be back to this in chapter 7 below, with an axiomatization of the notion of category of pairings, or more generally of a category of partitions, a definition for easiness, some theory, and an exploration of the main examples of easy quantum groups.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].