12c. Brauer theorems

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The above results are certainly interesting from a graph theory perspective, but from a group theory perspective, they remain a bit anecdotical. We would like to present now a series of alternative results, going in the other sense, that is, featuring less graphs, or rather featuring some combinatorial objects which are more complicated and abstract than graphs, but which can be extremely useful for the study of finite groups.

The idea will be a bit the same as for the Frucht theorem, namely that of viewing an arbitrary finite group [math]G[/math] as symmetry group of a combinatorial object, [math]G=G(X)[/math]. What will change, however, is the nature of [math]X[/math], the general principle being as follows: \begin{principle} Any finite, or even general compact group [math]G[/math] appears as the symmetry group of its corresponding Tannakian category [math]C_G[/math],

[[math]] G=G(C_G) [[/math]]

and by suitably delinearizing [math]C_G[/math], say via a Brauer theorem of type [math]C_G=span(D_G)[/math], we can view [math]G[/math] as symmetry group of a certain combinatorial object [math]D_G[/math]. \end{principle} Excited about this? Does not look easy, all this material, with both Tannaka and Brauer being quite scary names, in the context of group theory. But, believe me, all this is worth learning, and there is nothing better, when doing graphs or any kind of other algebraic discipline, to have in your bag some cutting-edge technology regarding the groups, such as the results of Tannaka and Brauer. So, we will go for this.

Getting started now, we will develop our theory as a continuation of the Peter-Weyl theory developed in chapter 11. That theory was developed for the finite groups, but with some minimal changes, and we will leave clarifying the details here to you, everything works in fact for a compact group [math]G[/math]. So, our starting point will be:

Theorem

We have the following Peter-Weyl results, valid for any compact group [math]G\subset_u U_N[/math], with the orthogonality being with respect to the Haar integration:

• Any representation decomposes as a sum of irreducible representations.
• Each irreducible representation appears inside a certain tensor power [math]u^{\otimes k}[/math].
• [math]C(G)=\overline{\bigoplus}_{v\in Irr(G)}M_{\dim(v)}(\mathbb C)[/math], the summands being pairwise orthogonal.
• The characters of irreducible representations form an orthonormal system.

Show Proof

As explained above, this is something that we know from chapter 11, in the finite group case, and in the general compact group case the proof is similar, with the only technical point being that of proving, somewhere between (1,2) and (3,4), the existence of the Haar measure. But this can be proved by using the arguments from chapter 11, with that chapter being written precisely with this idea in mind, namely that of having a quite straightforward extension to the compact group case, whenever needed.

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Going now towards Tannakian duality, let us start with:

Definition

A tensor category over [math]H=\mathbb C^N[/math] is a collection [math]C=(C_{kl})[/math] of linear spaces [math]C_{kl}\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_{kk}[/math] contains the identity operator.
• [math]C_''ptyset 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, as usual, 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]]

We have already met such categories, when dealing with the Tannakian categories of the closed subgroups [math]G\subset U_N[/math], and our knowledge can be summarized as follows:

Proposition

Given a closed subgroup [math]G\subset U_N[/math], its Tannakian category

[[math]] C_{kl}=\left\{T\in\mathcal L(H^{\otimes k},H^{\otimes l})\Big|Tg^{\otimes k}=g^{\otimes l}T,\forall g\in G\right\} [[/math]]

is a tensor category over [math]H=\mathbb C^N[/math]. Conversely, given a tensor category [math]C[/math] over [math]\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_{kl}\right\} [[/math]]

is a closed subgroup of [math]U_N[/math].

Show Proof

This is something that we basically know, the idea being as follows:

(1) Regarding the first assertion, we have to check here the axioms (1-6) in Definition 12.21. The axioms (1-4) being all clear from definitions, let us establish (5). But 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 12.21:

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

Regarding now the condition in Definition 12.21 (6), this comes from the fact that the matrix coefficients [math]g\to g_{ij}[/math] and their conjugates [math]g\to\bar{g}_{ij}[/math] commute with each other.

(2) Regarding the second assertion, we have to check that the subset [math]G\subset U_N[/math] constructed in the statement is a closed subgroup. But, assuming [math]g,h\in G[/math], we have [math]gh\in G[/math], due to the following computation, valid for any [math]k,l[/math] and any [math]T\in C_{kl}[/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]]

Also, we have [math]1\in G[/math], trivially. And also, assuming [math]g\in G[/math], we have [math]g^{-1}\in G[/math], due to:

[[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]]

Finally, the fact that our subgroup [math]G\subset U_N[/math] is closed is clear from definitions.

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Summarizing, we have so far precise axioms for the tensor categories [math]C=(C_{kl})[/math], given in Definition 12.21, as well as correspondences as follows:

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

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

Proposition

Consider the following conditions:

• [math]C=C_{G_C}[/math], for any tensor category [math]C[/math].
• [math]G=G_{C_G}[/math], for any closed subgroup [math]G\subset U_N[/math].

We have then [math](1)\implies(2)[/math]. Also, [math]C\subset C_{G_C}[/math] is automatic.

Show Proof

Given [math]G\subset U_N[/math], we have [math]G\subset G_{C_G}[/math]. On the other hand, by using (1) we have [math]C_G=C_{G_{C_G}}[/math]. Thus, we have an inclusion of closed subgroups of [math]U_N[/math], which becomes an isomorphism at the level of the associated Tannakian categories, so [math]G=G_{C_G}[/math]. Finally, the fact that we have an inclusion [math]C\subset C_{G_C}[/math] is clear from definitions.

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The point now is that it is possible to prove that we have [math]C_{G_C}\subset C[/math], by doing some abstract algebra, and we are led in this way to the following conclusion:

Theorem

The Tannakian duality constructions

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

are inverse to each other.

Show Proof

This is something quite tricky, the idea being as follows:

(1) According to Proposition 12.23, we must prove [math]C_{G_C}\subset C[/math]. For this purpose, given a tensor category [math]C=(C_{kl})[/math] over a Hilbert space [math]H[/math], consider the following [math]*[/math]-algebra:

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

Consider also, inside this [math]*[/math]-algebra, the following [math]*[/math]-subalgebra:

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

(2) It is then routine to check that we have equivalences as follows:

[[math]] \begin{eqnarray*} C_{G_C}\subset C &\iff&E_{C_{G_C}}\subset E_C\\ &\iff&E_{C_{G_C}}^{(s)}\subset E_C^{(s)},\forall s\\ &\iff&E_{C_{G_C}}^{(s)'}\supset E_C^{(s)'},\forall s \end{eqnarray*} [[/math]]

(3) Summarizing, we would like to prove that we have [math]E_C^{(s)'}\subset E_{C_{G_C}}^{(s)'}[/math]. But this can be done by doing some abstract algebra, and we refer here to Malacarne , or to the paper of Woronowicz . For more on all this, you have as well my book .

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With this piece of general theory in hand, let us go back to Principle 12.19, and develop the second idea there, namely delinearization and Brauer theorems. We have:

Definition

A category of crossing partitions is a collection [math]D=\bigsqcup_{k,l}D(k,l)[/math] of subsets [math]D(k,l)\subset P(k,l)[/math], having 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 Definition 12.21, and more on this in a moment. In order now to construct a Tannakian category out of such a category, we will need:

Proposition

Each partition [math]\pi\in P(k,l)[/math] produces a 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 type symbols [math]\delta_\pi\in\{0,1\}[/math] depending on whether the indices fit or not. 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]}\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] are certain integers, coming from the erased components in the middle.

Show Proof

This is something elementary, the computations being 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 key theoretical result, as follows:

Theorem

Any category of crossing partitions [math]D\subset P[/math] produces a series of compact groups [math]G=(G_N)[/math], with [math]G_N\subset U_N[/math] for any [math]N\in\mathbb N[/math], via the formula

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

for any [math]k,l[/math], and Tannakian duality. We call such groups easy.

Show Proof

Indeed, once we fix an integer [math]N\in\mathbb N[/math], the various axioms in Definition 12.25 show, via Proposition 12.26, that the following spaces form a Tannakian category:

[[math]] span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

Thus, Tannakian duality applies, and provides us with a closed subgroup [math]G_N\subset U_N[/math] such that the following equalities are satisfied, for any colored integers [math]k,l[/math]:

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

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

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At the level of basic examples of easy groups, these are the real and complex rotation groups, coming from the following key theorem of Brauer:

Theorem

We have the following results:

• [math]U_N[/math] is easy, coming from the category of all matching pairings [math]\mathcal P_2[/math].
• [math]O_N[/math] is easy too, coming from the category of all pairings [math]P_2[/math].

Show Proof

This can be deduced from Tannakian duality, the idea being as follows:

(1) The group [math]U_N[/math] being defined via the relations [math]u^*=u^{-1}[/math], [math]u^t=\bar{u}^{-1}[/math], 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]]

(2) The group [math]O_N\subset U_N[/math] being defined by imposing the relations [math]u_{ij}=\bar{u}_{ij}[/math], 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 conclusions in the statement.

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Moving now towards finite groups, we first have the following result:

Theorem

The symmetric group [math]S_N[/math], regarded as group of unitary matrices,

[[math]] S_N\subset O_N\subset U_N [[/math]]

via the permutation matrices, is easy, coming from the category of all partitions [math]P[/math].

Show Proof

Consider indeed the group [math]S_N[/math], regarded as a group of unitary matrices, with each permutation [math]\sigma\in S_N[/math] corresponding to the associated permutation matrix:

[[math]] \sigma(e_i)=e_{\sigma(i)} [[/math]]

Consider as well the easy group [math]G\subset O_N[/math] coming from the category of all partitions [math]P[/math]. Since [math]P[/math] is generated by the one-block “fork” partition [math]Y\in P(2,1)[/math], we have:

[[math]] C(G)=C(O_N)\Big/\Big \lt T_Y\in Hom(u^{\otimes 2},u)\Big \gt [[/math]]

The linear map associated to [math]Y[/math] is given by the following formula:

[[math]] T_Y(e_i\otimes e_j)=\delta_{ij}e_i [[/math]]

In order to do the computations, we use the following formulae:

[[math]] u=(u_{ij})_{ij}\quad,\quad u^{\otimes 2}=(u_{ij}u_{kl})_{ik,jl}\quad,\quad T_Y=(\delta_{ijk})_{i,jk} [[/math]]

We therefore obtain the following formula:

[[math]] (T_Yu^{\otimes 2})_{i,jk} =\sum_{lm}(T_Y)_{i,lm}(u^{\otimes 2})_{lm,jk} =u_{ij}u_{ik} [[/math]]

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

[[math]] (uT_Y)_{i,jk} =\sum_lu_{il}(T_Y)_{l,jk} =\delta_{jk}u_{ij} [[/math]]

Thus, the relation defining [math]G\subset O_N[/math] reformulates as follows:

[[math]] T_Y\in Hom(u^{\otimes 2},u)\iff u_{ij}u_{ik}=\delta_{jk}u_{ij},\forall i,j,k [[/math]]

In other words, the elements [math]u_{ij}[/math] must be projections, which must be pairwise orthogonal on the rows of [math]u=(u_{ij})[/math]. We conclude that [math]G\subset O_N[/math] is the subgroup of matrices [math]g\in O_N[/math] having the property [math]g_{ij}\in\{0,1\}[/math]. Thus we have [math]G=S_N[/math], as desired.

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The hyperoctahedral group [math]H_N[/math] is easy as well, the result here being as follows:

Theorem

The hyperoctahedral group [math]H_N[/math], regarded as group of matrices,

[[math]] S_N\subset H_N\subset O_N [[/math]]

is easy, coming from the category of partitions with even blocks [math]P_{even}[/math].

Show Proof

This follows as usual from Tannakian duality. To be more precise, consider the following one-block partition, which, as the name indicates, looks like a [math]H[/math] letter:

[[math]] H\in P(2,2) [[/math]]

The linear map associated to this partition is then given by:

[[math]] T_H(e_i\otimes e_j)=\delta_{ij}e_i\otimes e_i [[/math]]

By using this formula, we have the following computation:

[[math]] \begin{eqnarray*} (T_H\otimes id)u^{\otimes 2}(e_a\otimes e_b) &=&(T_H\otimes id)\left(\sum_{ijkl}e_{ij}\otimes e_{kl}\otimes u_{ij}u_{kl}\right)(e_a\otimes e_b)\\ &=&(T_H\otimes id)\left(\sum_{ik}e_i\otimes e_k\otimes u_{ia}u_{kb}\right)\\ &=&\sum_ie_i\otimes e_i\otimes u_{ia}u_{ib} \end{eqnarray*} [[/math]]

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

[[math]] \begin{eqnarray*} u^{\otimes 2}(T_H\otimes id)(e_a\otimes e_b) &=&\delta_{ab}\left(\sum_{ijkl}e_{ij}\otimes e_{kl}\otimes u_{ij}u_{kl}\right)(e_a\otimes e_a)\\ &=&\delta_{ab}\sum_{ij}e_i\otimes e_k\otimes u_{ia}u_{ka} \end{eqnarray*} [[/math]]

We conclude from this that we have the following equivalence:

[[math]] T_H\in End(u^{\otimes 2})\iff \delta_{ik}u_{ia}u_{ib}=\delta_{ab}u_{ia}u_{ka},\forall i,k,a,b [[/math]]

But the relations on the right tell us that the entries of [math]u=(u_{ij})[/math] must satisfy [math]\alpha\beta=0[/math] on each row and column of [math]u[/math], and so that the corresponding closed subgroup [math]G\subset O_N[/math] consists of the matrices [math]g\in O_N[/math] which are permutation-like, with [math]\pm1[/math] nonzero entries. Thus, the corresponding group is [math]G=H_N[/math], and as a conclusion to this, we have:

[[math]] C(H_N)=C(O_N)\Big/\Big \lt T_H\in End(u^{\otimes 2})\Big \gt [[/math]]

But this means that the hyperoctahedral group [math]H_N[/math] is easy, coming from the category of partitions [math]D= \lt H \gt =P_{even}[/math]. Thus, we are led to the conclusion in the statement.

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More generally now, we have in fact the following result, regarding the series of complex reflection groups [math]H_N^s[/math], which covers both the groups [math]S_N,H_N[/math]:

Theorem

The complex reflection group [math]H_N^s=\mathbb Z_s\wr S_N[/math] is easy, the corresponding category [math]P^s[/math] consisting of the partitions satisfying the condition

[[math]] \#\circ=\#\bullet(s) [[/math]]

as a weighted sum, in each block. In particular, we have the following results:

• [math]S_N[/math] is easy, coming from the category [math]P[/math].
• [math]H_N=\mathbb Z_2\wr S_N[/math] is easy, coming from the category [math]P_{even}[/math].
• [math]K_N=\mathbb T\wr S_N[/math] is easy, coming from the category [math]\mathcal P_{even}[/math].

Show Proof

This is something that we already know at [math]s=1,2[/math], from Theorems 12.29 and 12.30. In general, the proof is similar, based on Tannakian duality. To be more precise, in what regards the main assertion, the idea here is that the one-block partition [math]\pi\in P(s)[/math], which generates the category of partitions [math]P^s[/math] in the statement, implements the relations producing the subgroup [math]H_N^s\subset S_N[/math]. As for the last assertions, these are all elementary:

(1) At [math]s=1[/math] we know that we have [math]H_N^1=S_N[/math]. Regarding now the corresponding category, here the condition [math]\#\circ=\#\bullet(1)[/math] is automatic, and so [math]P^1=P[/math].

(2) At [math]s=2[/math] we know that we have [math]H_N^2=H_N[/math]. Regarding now the corresponding category, here the condition [math]\#\circ=\#\bullet(2)[/math] reformulates as follows:

[[math]] \#\circ+\,\#\bullet=0(2) [[/math]]

Thus each block must have even size, and we obtain, as claimed, [math]P^2=P_{even}[/math].

(3) At [math]s=\infty[/math] we know that we have [math]H_N^\infty=K_N[/math]. Regarding now the corresponding category, here the condition [math]\#\circ=\#\bullet(\infty)[/math] reads:

[[math]] \#\circ=\#\bullet [[/math]]

But this is the condition defining [math]\mathcal P_{even}[/math], and so [math]P^\infty=\mathcal P_{even}[/math], as claimed.

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Summarizing, we have many examples. In fact, our list of easy groups has currently become quite big, and here is a selection of the main results that we have so far:

Theorem

We have a diagram of compact groups as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ K_N\ar[r]&U_N\\ H_N\ar[u]\ar[r]&O_N\ar[u]} [[/math]]

where [math]H_N=\mathbb Z_2\wr S_N[/math] and [math]K_N=\mathbb T\wr S_N[/math], and all these groups are easy.

Show Proof

This follows from the above results. To be more precise, we know that the above groups are all easy, the corresponding categories of partitions being as follows:

[[math]] \xymatrix@R=16mm@C=18mm{ \mathcal P_{even}\ar[d]&\mathcal P_2\ar[l]\ar[d]\\ P_{even}&P_2\ar[l]} [[/math]]

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

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Summarizing, we have reached to the conclusions formulated in Principle 12.19. All this remains of course a bit away from graph theory, but we will make a good use of what we learned here, later on in this book, especially when talking quantum groups.

General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].