14a. Tensor categories

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

We have seen that the representations of a closed subgroup [math]G\subset U_N[/math] are subject to a number of non-trivial results, collectively known as Peter-Weyl theory. To be more precise, the main ideas of Peter-Weyl theory were as follows:

• The representations of [math]G[/math] split as sums of irreducibles, and the irreducibles can be found inside the tensor products [math]u^{\otimes k}[/math] between the fundamental representation [math]u:G\subset U_N[/math] and its adjoint [math]\bar{u}:G\subset U_N[/math], called Peter-Weyl representations.
• The main problem is therefore that of splitting the Peter-Weyl representations [math]u^{\otimes k}[/math] into irreducibles. Technically speaking, this leads to the question of explicitely computing the corresponding fixed point spaces [math]Fix(u^{\otimes k})[/math].
• From a probabilistic perspective, in connection with characters and truncated characters, which require the explicit knowledge of [math]\int_G[/math], we are led into the same fundamental question, namely the computation of the spaces [math]Fix(u^{\otimes k})[/math].

Summarizing, no matter what we want to do with [math]G[/math], we must compute the spaces [math]Fix(u^{\otimes k})[/math]. As a first idea now, it is technically convenient to slightly enlarge the class of spaces to be computed, by talking about Tannakian categories, as follows:

Definition

The Tannakian category associated to a closed subgroup [math]G\subset_uU_N[/math] is the collection [math]C=(C(k,l))[/math] of vector spaces

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

where the representations [math]u^{\otimes k}[/math] with [math]k=\circ\bullet\bullet\circ\ldots[/math] colored integer, defined by

[[math]] u^{\otimes\emptyset}=1\quad,\quad u^{\otimes\circ}=u\quad,\quad u^{\otimes\bullet}=\bar{u} [[/math]]

and multiplicativity, [math]u^{\otimes kl}=u^{\otimes k}\otimes u^{\otimes l}[/math], are the Peter-Weyl representations.

Here are a few examples of such representations, namely those coming from the colored integers of length 2, to be often used in what follows:

[[math]] u^{\otimes\circ\circ}=u\otimes u\quad,\quad u^{\otimes\circ\bullet}=u\otimes\bar{u} [[/math]]

[[math]] u^{\otimes\bullet\circ}=\bar{u}\otimes u\quad,\quad u^{\otimes\bullet\bullet}=\bar{u}\otimes\bar{u} [[/math]]

As a first observation, the knowledge of the Tannakian category is more or less the same thing as the knowledge of the fixed point spaces, which appear as:

[[math]] Fix(u^{\otimes k})=C(0,k) [[/math]]

Indeed, these latter spaces fully determine all the spaces [math]C(k,l)[/math], because of the Frobenius isomorphisms, which for the Peter-Weyl representations read:

[[math]] \begin{eqnarray*} C(k,l) &=&Hom(u^{\otimes k},u^{\otimes l})\\ &\simeq&Hom(1,\bar{u}^{\otimes k}\otimes u^{\otimes l})\\ &=&Hom(1,u^{\otimes\bar{k}l})\\ &=&Fix(u^{\otimes\bar{k}l}) \end{eqnarray*} [[/math]]

In order to get started now, let us make a summary of what we have so far, regarding these spaces [math]C(k,l)[/math], coming from the general theory developed in chapter 13. In order to formulate our result, let us start with an abstract definition, as follows:

Definition

Let [math]H[/math] be a finite dimensional Hilbert space. A tensor category over [math]H[/math] is a collection [math]C=(C(k,l))[/math] of linear spaces

[[math]] C(k,l)\subset\mathcal L(H^{\otimes k},H^{\otimes l}) [[/math]]

satisfying the following conditions:

• [math]S,T\in C[/math] implies [math]S\otimes T\in C[/math].
• If [math]S,T\in C[/math] are composable, then [math]ST\in C[/math].
• [math]T\in C[/math] implies [math]T^*\in C[/math].
• Each [math]C(k,k)[/math] contains the identity operator.
• [math]C(\emptyset,k)[/math] with [math]k=\circ\bullet,\bullet\circ[/math] contain the operator [math]R:1\to\sum_ie_i\otimes e_i[/math].
• [math]C(kl,lk)[/math] with [math]k,l=\circ,\bullet[/math] contain the flip operator [math]\Sigma:a\otimes b\to b\otimes a[/math].

Here the tensor powers [math]H^{\otimes k}[/math], which are Hilbert spaces depending on a colored integer [math]k=\circ\bullet\bullet\circ\ldots\,[/math], are defined by the following formulae, and multiplicativity:

[[math]] H^{\otimes\emptyset}=\mathbb C\quad,\quad H^{\otimes\circ}=H\quad,\quad H^{\otimes\bullet}=\bar{H}\simeq H [[/math]]

With these conventions, we have the following result, summarizing our knowledge on the subject, coming from the results from the previous chapter:

Theorem

For a closed subgroup [math]G\subset_uU_N[/math], the associated Tannakian category

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

is a tensor category over the Hilbert space [math]H=\mathbb C^N[/math].

Show Proof

We know that the fundamental representation [math]u[/math] acts on the Hilbert space [math]H=\mathbb C^N[/math], and that its conjugate [math]\bar{u}[/math] acts on the Hilbert space [math]\bar{H}=\mathbb C^N[/math]. Now by multiplicativity we conclude that any Peter-Weyl representation [math]u^{\otimes k}[/math] acts on the Hilbert space [math]H^{\otimes k}[/math], so that we have embeddings as in Definition 14.2, as follows:

[[math]] C(k,l)\subset\mathcal L(H^{\otimes k},H^{\otimes l}) [[/math]]

Regarding now the fact that the axioms (1-6) in Definition 14.2 are indeed satisfied, this is something that we basically already know, as follows:

(1,2,3) These results follow from definitions, and were explained in chapter 13.

(4) This is something trivial, coming from definitions.

(5) This follows from the fact that each element [math]g\in G[/math] is a unitary, which can be reformulated as follows, with [math]R:1\to\sum_ie_i\otimes e_i[/math] being the map in Definition 14.2:

[[math]] R\in Hom(1,g\otimes\bar{g})\quad,\quad R\in Hom(1,\bar{g}\otimes g) [[/math]]

Indeed, given an arbitrary matrix [math]g\in M_N(\mathbb C)[/math], we have the following computation:

[[math]] \begin{eqnarray*} (g\otimes\bar{g})(R(1)\otimes1) &=&\left(\sum_{ijkl}e_{ij}\otimes e_{kl}\otimes g_{ij}\bar{g}_{kl}\right)\left(\sum_ae_a\otimes e_a\otimes 1\right)\\ &=&\sum_{ika}e_i\otimes e_k\otimes g_{ia}\bar{g}_{ka}^*\\ &=&\sum_{ik}e_i\otimes e_k\otimes(gg^*)_{ik} \end{eqnarray*} [[/math]]

We conclude from this that we have the following equivalence:

[[math]] \begin{eqnarray*} R\in Hom(1,g\otimes\bar{g}) &\iff&gg^*=1 \end{eqnarray*} [[/math]]

By replacing [math]g[/math] with its conjugate matrix [math]\bar{g}[/math], we have as well:

[[math]] R\in Hom(1,\bar{g}\otimes g)\iff\bar{g}g^t=1 [[/math]]

Thus, the two intertwining conditions in Definition 14.2 (5) are both equivalent to the fact that [math]g[/math] is unitary, and so these conditions are indeed satisfied, as desired.

(6) This is again something elementary, coming from the fact that the various matrix coefficients [math]g\to g_{ij}[/math] and their complex conjugates [math]g\to\bar{g}_{ij}[/math] commute with each other. To be more precise, with [math]\Sigma:a\otimes b\to b\otimes a[/math] being the flip operator, we have:

[[math]] \begin{eqnarray*} (g\otimes h)(\Sigma\otimes id)(e_a\otimes e_b\otimes 1) &=&\left(\sum_{ijkl}e_{ij}\otimes e_{kl}\otimes g_{ij}h_{kl}\right)(e_b\otimes e_a\otimes1)\\ &=&\sum_{ik}e_i\otimes e_k\otimes g_{ib}h_{ka} \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} (\Sigma\otimes id)(h\otimes g)(e_a\otimes e_b\otimes 1) &=&(\Sigma\otimes id)\left(\sum_{ijkl}e_{ij}\otimes e_{kl}\otimes h_{ij}g_{kl}\right)(e_a\otimes e_b\otimes1)\\ &=&(\Sigma\otimes id)\left(\sum_{ik}e_i\otimes e_k\otimes h_{ia}g_{kb}\right)\\ &=&\sum_{ik}e_k\otimes e_i\otimes h_{ia}g_{kb}\\ &=&\sum_{ik}e_i\otimes e_k\otimes h_{ka}g_{ib} \end{eqnarray*} [[/math]]

Now since functions commute, [math]g_{ib}h_{ka}=h_{ka}g_{ib}[/math], this gives the result.

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With the above in hand, our purpose now will be that of showing that any closed subgroup [math]G\subset U_N[/math] is uniquely determined by its Tannakian category [math]C=(C(k,l))[/math]:

[[math]] G\leftrightarrow C [[/math]]

This result, known as Tannakian duality, is something quite deep, and very useful. Indeed, the idea is that what we would have here is a “linearization” of [math]G[/math], allowing us to do combinatorics, and ultimately reach to very concrete and powerful results, regarding [math]G[/math] itself. And as a consequence, solve our probability questions left.

Speaking linearization, there is also a comment to be made here, in relation with Lie algebras. Remember a discussion that we had some time ago, at the beginning of chapter 10, with me arguing that yes, a good idea for the study of the closed subgroups [math]G\subset U_N[/math] would be the consideration of the tangent space at the origin [math]\mathfrak g=T_1(G)[/math], called Lie algebra of [math]G[/math], which is a very good “linearization” of [math]G[/math], but no need to get head-first into that, because there are perhaps some other methods for linearizing [math]G[/math]? Well, time now to justify this claim, with some general theory of the correspondence [math]G\leftrightarrow C[/math].

Getting started now, we want to construct a correspondence [math]G\leftrightarrow C[/math], and we already know from Theorem 14.3 how the correspondence [math]G\to C[/math] appears, namely via:

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

Regarding now the construction in the other sense, [math]C\to G[/math], this is something very simple as well, coming from the following elementary result:

Theorem

Given a tensor category [math]C=(C(k,l))[/math] over the space [math]H\simeq\mathbb C^N[/math],

[[math]] G=\left\{g\in U_N\Big|Tg^{\otimes k}=g^{\otimes l}T\ ,\ \forall k,l,\forall T\in C(k,l)\right\} [[/math]]

is a closed subgroup [math]G\subset U_N[/math].

Show Proof

Consider indeed the closed subset [math]G\subset U_N[/math] constructed in the statement. We want to prove that [math]G[/math] is indeed a group, and the verifications here go as follows:

(1) Given two matrices [math]g,h\in G[/math], their product satisfies [math]gh\in G[/math], due to the following computation, valid for any [math]k,l[/math] and any [math]T\in C(k,l)[/math]:

[[math]] \begin{eqnarray*} T(gh)^{\otimes k} &=&Tg^{\otimes k}h^{\otimes k}\\ &=&g^{\otimes l}Th^{\otimes k}\\ &=&g^{\otimes l}h^{\otimes l}T\\ &=&(gh)^{\otimes l}T \end{eqnarray*} [[/math]]

(2) Also, we have [math]1\in G[/math], trivially. Finally, for [math]g\in G[/math] and [math]T\in C(k,l)[/math], we have:

[[math]] \begin{eqnarray*} T(g^{-1})^{\otimes k} &=&(g^{-1})^{\otimes l}[g^{\otimes l}T](g^{-1})^{\otimes k}\\ &=&(g^{-1})^{\otimes l}[Tg^{\otimes k}](g^{-1})^{\otimes k}\\ &=&(g^{-1})^{\otimes l}T \end{eqnarray*} [[/math]]

Thus we have [math]g^{-1}\in G[/math], and so [math]G[/math] is a group, as claimed.

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Summarizing, we have so far precise axioms for the tensor categories [math]C=(C(k,l))[/math], given in Definition 14.2, as well as correspondences as follows:

[[math]] G\to C\quad,\quad C\to G [[/math]]

We will show in what follows that these correspondences are inverse to each other. In order to get started, we first have the following technical result:

Theorem

If we denote the correspondences in Theorem 14.3 and 14.4, between closed subgroups [math]G\subset U_N[/math] and tensor categories [math]C=(C(k,l))[/math] over [math]H=\mathbb C^N[/math], as

[[math]] G\to C_G\quad,\quad C\to G_C [[/math]]

then we have embeddings as follows, for any [math]G[/math] and [math]C[/math] respectively,

[[math]] G\subset G_{C_G}\quad,\quad C\subset C_{G_C} [[/math]]

and proving that these correspondences are inverse to each other amounts in proving

[[math]] C_{G_C}\subset C [[/math]]

for any tensor category [math]C=(C(k,l))[/math] over the space [math]H=\mathbb C^N[/math].

Show Proof

This is something trivial, with the embeddings [math]G\subset G_{C_G}[/math] and [math]C\subset C_{G_C}[/math] being both clear from definitions, and with the last assertion coming from this.

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In order to establish Tannakian duality, we will need some abstract constructions. Following Malacarne , let us start with the following elementary fact:

Proposition

Given a tensor category [math]C=C((k,l))[/math] over a Hilbert space [math]H[/math],

[[math]] E_C =\bigoplus_{k,l}C(k,l) \subset\bigoplus_{k,l}B(H^{\otimes k},H^{\otimes l}) \subset B\left(\bigoplus_kH^{\otimes k}\right) [[/math]]

is a closed [math]*[/math]-subalgebra. Also, inside this algebra,

[[math]] E_C^{(s)} =\bigoplus_{|k|,|l|\leq s}C(k,l) \subset\bigoplus_{|k|,|l|\leq s}B(H^{\otimes k},H^{\otimes l}) =B\left(\bigoplus_{|k|\leq s}H^{\otimes k}\right) [[/math]]

is a finite dimensional [math]*[/math]-subalgebra.

Show Proof

This is clear indeed from the categorical axioms from Definition 14.2, which, since satisfied, prove that the various linear spaces in the statement are stable under both the multiplication operation, and under taking the adjoints.

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Now back to our reconstruction question, we want to prove [math]C=C_{G_C}[/math], which is the same as proving [math]E_C=E_{C_{G_C}}[/math]. We will use a standard commutant trick, as follows:

Theorem

For any [math]*[/math]-algebra [math]A\subset M_N(\mathbb C)[/math] we have the equality

[[math]] A=A'' [[/math]]

where prime denotes the commutant, [math]X'=\left\{T\in M_N(\mathbb C)\big|Tx=xT,\forall x\in X\right\}[/math].

Show Proof

This is a particular case of von Neumann's bicommutant theorem, which follows from the explicit description of [math]A[/math] worked out in chapter 13, namely:

[[math]] A=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

Indeed, the center of each matrix algebra being reduced to the scalars, the commutant of this algebra is as follows, with each copy of [math]\mathbb C[/math] corresponding to a matrix block:

[[math]] A'=\mathbb C\oplus\ldots\oplus\mathbb C [[/math]]

Now when taking once again the commutant, the computation is trivial, and we obtain in this way [math]A[/math] itself, and this leads to the conclusion in the statement.

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By using now the bicommutant theorem, we have:

Proposition

Given a Tannakian category [math]C[/math], the following are equivalent:

• [math]C=C_{G_C}[/math].
• [math]E_C=E_{C_{G_C}}[/math].
• [math]E_C^{(s)}=E_{C_{G_C}}^{(s)}[/math], for any [math]s\in\mathbb N[/math].
• [math]E_C^{(s)'}=E_{C_{G_C}}^{(s)'}[/math], for any [math]s\in\mathbb N[/math].

In addition, the inclusions [math]\subset[/math], [math]\subset[/math], [math]\subset[/math], [math]\supset[/math] are automatically satisfied.

Show Proof

This follows from the above results, as follows:

[math](1)\iff(2)[/math] This is clear from definitions.

[math](2)\iff(3)[/math] This is clear from definitions as well.

[math](3)\iff(4)[/math] This comes from the bicommutant theorem. As for the last assertion, we have indeed [math]C\subset C_{G_C}[/math] from Theorem 14.5, and this shows that we have as well:

[[math]] E_C\subset E_{C_{G_C}} [[/math]]

We therefore obtain by truncating [math]E_C^{(s)}\subset E_{C_{G_C}}^{(s)}[/math], and by taking the commutants, this gives [math]E_C^{(s)}\supset E_{C_{G_C}}^{(s)}[/math]. Thus, we are led to the conclusion in the statement.

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Summarizing, we would like to prove that we have [math]E_C^{(s)'}\subset E_{C_{G_C}}^{(s)'}[/math]. Let us first study the commutant on the right. As a first observation, we have:

Proposition

We have the following equality,

[[math]] E_{C_G}^{(s)}=End\left(\bigoplus_{|k|\leq s}u^{\otimes k}\right) [[/math]]

between subalgebras of [math]B\left(\bigoplus_{|k|\leq s}H^{\otimes k}\right)[/math].

Show Proof

We know that the category [math]C_G[/math] is by definition given by:

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

Thus, the corresponding algebra [math]E_{C_G}^{(s)}[/math] appears as follows:

[[math]] E_{C_G}^{(s)} =\bigoplus_{|k|,|l|\leq s}Hom(u^{\otimes k},u^{\otimes l}) \subset\bigoplus_{|k|,|l|\leq s}B(H^{\otimes k},H^{\otimes l}) =B\left(\bigoplus_{|k|\leq s}H^{\otimes k}\right) [[/math]]

On the other hand, the algebra of intertwiners of [math]\bigoplus_{|k|\leq s}u^{\otimes k}[/math] is given by:

[[math]] End\left(\bigoplus_{|k|\leq s}u^{\otimes k}\right) =\bigoplus_{|k|,|l|\leq s}Hom(u^{\otimes k},u^{\otimes l}) \subset\bigoplus_{|k|,|l|\leq s}B(H^{\otimes k},H^{\otimes l}) =B\left(\bigoplus_{|k|\leq s}H^{\otimes k}\right) [[/math]]

Thus we have indeed the same algebra, and we are done.

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We have to compute the commutant of the above algebra. For this purpose, we can use the following general result, valid for any representation of a compact group:

Proposition

Given a unitary group representation [math]v:G\to U_n[/math] we have an algebra representation as follows,

[[math]] \pi_v:C(G)^*\to M_n(\mathbb C)\quad,\quad \varphi\to (\varphi(v_{ij}))_{ij} [[/math]]

whose image is given by [math]Im(\pi_v)=End(v)'[/math].

Show Proof

The first assertion is clear, with the multiplicativity claim for [math]\pi_v[/math] coming from the following computation, where [math]\Delta:C(G)\to C(G)\otimes C(G)[/math] is the comultiplication:

[[math]] \begin{eqnarray*} (\pi_v(\varphi*\psi))_{ij} &=&(\varphi\otimes\psi)\Delta(v_{ij})\\ &=&\sum_k\varphi(v_{ik})\psi(v_{kj})\\ &=&\sum_k(\pi_v(\varphi))_{ik}(\pi_v(\psi))_{kj}\\ &=&(\pi_v(\varphi)\pi_v(\psi))_{ij} \end{eqnarray*} [[/math]]

Let us establish now the equality in the statement, namely:

[[math]] Im(\pi_v)=End(v)' [[/math]]

Let us first prove the inclusion [math]\subset[/math]. Given [math]\varphi\in C(G)^*[/math] and [math]T\in End(v)[/math], we have:

[[math]] \begin{eqnarray*} [\pi_v(\varphi),T]=0 &\iff&\sum_k\varphi(v_{ik})T_{kj}=\sum_kT_{ik}\varphi(v_{kj}),\forall i,j\\ &\iff&\varphi\left(\sum_kv_{ik}T_{kj}\right)=\varphi\left(\sum_kT_{ik}v_{kj}\right),\forall i,j\\ &\iff&\varphi((vT)_{ij})=\varphi((Tv)_{ij}),\forall i,j \end{eqnarray*} [[/math]]

But this latter formula is true, because [math]T\in End(v)[/math] means that we have:

[[math]] vT=Tv [[/math]]

As for the converse inclusion [math]\supset[/math], the proof is quite similar. Indeed, by using the bicommutant theorem, this is the same as proving that we have:

[[math]] Im(\pi_v)'\subset End(v) [[/math]]

But, by using the above equivalences, we have the following computation:

[[math]] \begin{eqnarray*} T\in Im(\pi_v)' &\iff&[\pi_v(\varphi),T]=0,\forall\varphi\\ &\iff&\varphi((vT)_{ij})=\varphi((Tv)_{ij}),\forall\varphi,i,j\\ &\iff&vT=Tv \end{eqnarray*} [[/math]]

Thus, we have obtained the desired inclusion, and we are done.

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By combining the above results, we obtain the following technical statement:

Theorem

We have the following equality,

[[math]] E_{C_G}^{(s)'}=Im(\pi_v) [[/math]]

where the representation [math]v[/math] is the following direct sum,

[[math]] v=\bigoplus_{|k|\leq s}u^{\otimes k} [[/math]]

and where the algebra representation [math]\pi_v:C(G)^*\to M_n(\mathbb C)[/math] is given by [math]\varphi\to(\varphi(v_{ij}))_{ij}[/math].

Show Proof

This follows indeed by combining the above results, and more precisely by combining Proposition 14.9 and Proposition 14.10.

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General references

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