15a. Easy groups

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

We have seen in the previous chapter that the Tannakian duals of the groups [math]O_N,U_N[/math] are very simple objects. To be more precise, the Brauer theorem for these two groups states that we have equalities as follows, with [math]D=P_2,\mathcal P_2[/math] respectively:

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

Our goal here will be that of axiomatizing and studying the closed subgroups [math]G\subset U_N[/math] which are of this type, but with [math]D[/math] being allowed to be, more generally, a category of partitions. We will call such groups “easy”, and our results will be as follows:

(1) At the level of the continuous examples, we will see that besides [math]O_N,U_N[/math], we have the bistochastic groups [math]B_N,C_N[/math]. This is something which is interesting, and also instructive, making it clear why we have to upgrade, from pairings to partitions.

(2) At the level of discrete examples, we have none so far, but we will see that the symmetric group [math]S_N[/math], the hyperoctahedral group [math]H_N[/math], and more generally the complex reflection groups [math]H_N^s[/math] with [math]s\in\mathbb N\cup\{\infty\}[/math], are all easy, in the above generalized sense.

(3) Still at the level of the basic examples, some key Lie groups such as [math]SU_2,SO_3[/math], or the symplectic group [math]Sp_N[/math], are not easy, but the point is that these are however covered by a suitable “super-easiness” version of the easiness, as defined above.

(4) At the level of the general theory, we will develop some algebraic theory in this chapter, for the most in relation with various product operations, the idea being that in the easy case, everything eventually reduces to computations with partitions.

(5) Also at the level of the general theory, we will develop as well some analytic theory, in the next chapter, based on the same idea, namely that in the easy case, everything eventually reduces to some elementary computations with partitions.

All this sounds quite exciting, good theory that we will be developing here, hope you agree with me. In order to get started now, let us formulate the following key definition, extending to the case of arbitrary partitions what we already know about pairings:

Definition

Given a partition [math]\pi\in P(k,l)[/math] and an integer [math]N\in\mathbb N[/math], we define

[[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.

To be more precise here, in order to compute the Kronecker type symbols [math]\delta_\pi(^i_j)\in\{0,1\}[/math], we proceed exactly as in the pairing case, namely by putting the multi-indices [math]i=(i_1,\ldots,i_k)[/math] and [math]j=(j_1,\ldots,j_l)[/math] on the legs of [math]\pi[/math], in the obvious way. In case all the blocks of [math]\pi[/math] contain 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].

With the above notion in hand, we can now formulate the following key definition, from , motivated by the Brauer theorems for [math]O_N,U_N[/math], as indicated before:

Definition

A closed subgroup [math]G\subset U_N[/math] is called easy when

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

for any two colored integers [math]k,l=\circ\bullet\circ\bullet\ldots\,[/math], for certain sets of partitions

[[math]] D(k,l)\subset P(k,l) [[/math]]

where [math]\pi\to T_\pi[/math] is the standard implementation of the partitions, as linear maps.

In other words, we call a group [math]G[/math] easy when its Tannakian category appears in the simplest possible way: from the linear maps associated to partitions. The terminology is quite natural, because Tannakian duality is basically our only serious tool.

As basic examples, the orthogonal and unitary groups [math]O_N,U_N[/math] are both easy, coming respectively from the following collections of sets of partitions:

[[math]] P_2=\bigsqcup_{k,l}P_2(k,l)\quad,\quad \mathcal P_2=\bigsqcup_{k,l}\mathcal P_2(k,l) [[/math]]

In the general case now, as an important theoretical remark, in the context of Definition 15.2, consider the following collection of sets of partitions:

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

This collection of sets [math]D[/math] obviously determines [math]G[/math], but the converse is not true. Indeed, at [math]N=1[/math] for instance, both the choices [math]D=P_2,\mathcal P_2[/math] produce the same easy group, namely [math]G=\{1\}[/math]. We will be back to this issue on several occasions, with results about it.

In order to advance, our first goal will be that of establishing a duality between easy groups and certain special classes of collections of sets as above, namely:

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

Let us begin with a general definition, from , as follows:

Definition

Let [math]P(k,l)[/math] be the set of partitions 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(k,l)[/math] is called a category of partitions 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].

As before, this is something that we already met in chapter 14, but for the pairings only. Observe the similarity with the axioms for Tannakian categories, also from chapter 14. We will see in a moment that this similarity can be turned into something very precise, the idea being that such a category produces a family of easy quantum groups [math](G_N)_{N\in\mathbb N}[/math], one for each [math]N\in\mathbb N[/math], via the formula in Definition 15.1, and Tannakian duality.

As basic examples, that we have already met in chapter 14, in connection with the representation theory of [math]O_N,U_N[/math], we have the categories [math]P_2,\mathcal P_2[/math] of pairings, and of matching pairings. Further basic examples include the categories [math]P,P_{even}[/math] of all partitions, and of all partitions whose blocks have even size. We will see in a moment that these latter categories are related to the symmetric and hyperoctahedral groups [math]S_N,H_N[/math].

The relation with the Tannakian categories comes from the following 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] are certain integers, coming from the erased components in the middle.

Show Proof

This is something that we already know for the pairings, from chapter 14, and the proof in general is similar, with the only axiom where some slight changes appear being the composition one. Here the computation is as follows, as before for pairings, with [math]c(\pi,\sigma)\in\mathbb N[/math] counting the middle components, which are not necessarily circles:

[[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]]

Thus, our correspondence is indeed categorical, as claimed.

■

Time now to put everyting together. All the above was pure combinatorics, and in relation with the compact groups, we have the following result:

Theorem

Each category of partitions [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 15.1, 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 the formulae in Proposition 15.4, and to our axioms for the categories of partitions, from Definition 15.3, this collection of spaces [math]C=(C_{kl})[/math] satisfies the axioms for the Tannakian categories, from chapter 14. 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.

■

In relation with the easiness property, we can now formulate a key result, which can serve as an alternative definition for the easy groups, as follows:

Theorem

A closed subgroup [math]G\subset U_N[/math] is easy precisely when

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

for any colored integers [math]k,l[/math], for a certain category of partitions [math]D\subset P[/math].

Show Proof

This basically follows from Theorem 15.5, as follows:

(1) In one sense, we know from Theorem 15.5 that any category of partitions [math]D\subset P[/math] produces a family of closed groups [math]G\subset U_N[/math], one for each [math]N\in\mathbb N[/math], according to Tannakian duality and to the Hom space formula there, namely:

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

But these groups [math]G\subset U_N[/math] are indeed easy, in the sense of Definition 15.2.

(2) In the other sense now, assume that [math]G\subset U_N[/math] is easy, in the sense of Definition 15.2, coming via the above Hom space formula, from a collection of sets as follows:

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

Consider now the category of partitions [math]\widetilde{D}= \lt D \gt [/math] generated by this family. This is by definition the smallest category of partitions containing [math]D[/math], whose existence follows by starting with [math]D[/math], and performing the various categorical operations, namely horizontal and vertical concatenation, and upside-down turning. It follows then, via another application of Tannakian duality, that we have the following formula, for any [math]k,l[/math]:

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

Thus, our group [math]G\subset U_N[/math] can be viewed as well as coming from [math]\widetilde{D}[/math], and so appearing as particular case of the construction in Theorem 15.5, and this gives the result.

■

As already mentioned above, Theorem 15.6 can be regarded as an alternative definition for easiness, with the assumption that [math]D\subset P[/math] must be a category of partitions being added. In what follows we will rather use this new definition, which is more precise.

Generally speaking, the same comments as before apply. First, [math]G[/math] is easy when its Tannakian category appears in the simplest possible way: from a category of partitions. The terminology is quite natural, because Tannakian duality is our only serious tool.

Also, the category of partitions [math]D[/math] is not unique, for instance because at [math]N=1[/math] all the categories of partitions produce the same easy group, namely [math]G=\{1\}[/math]. We will be back to this issue on several occasions, with various results about it.

We will see in what follows that many interesting examples of compact quantum groups are easy. Moreover, most of the known series of “basic” compact quantum groups, [math]G=(G_N)[/math] with [math]N\in\mathbb N[/math], can be in principle made fit into some suitable extensions of the easy quantum group formalism. We will discuss this too, in what follows.

The notion of easiness goes back to the results of Brauer in regarding the orthogonal group [math]O_N[/math], and the unitary group [math]U_N[/math], which reformulate as follows:

Theorem

We have the following results:

• The unitary group [math]U_N[/math] is easy, coming from the category [math]\mathcal P_2[/math].
• The orthogonal group [math]O_N[/math] is easy as well, coming from the category [math]P_2[/math].

Show Proof

This is something that we already know, from chapter 14, based on Tannakian duality, the idea of the proof 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 conclusion in the statement.

■

As already mentioned in the beginning of this chapter, there are many other examples of easy groups, and we will gradually explore this. To start with, we have the following result, dealing with the orthogonal and unitary bistochastic groups [math]B_N,C_N[/math]:

Theorem

We have the following results:

• The unitary bistochastic group [math]C_N[/math] is easy, coming from the category [math]\mathcal P_{12}[/math] of matching singletons and pairings.
• The orthogonal bistochastic group [math]B_N[/math] is easy, coming from the category [math]P_{12}[/math] of singletons and pairings.

Show Proof

The proof here is similar to the proof of Theorem 15.7. To be more precise, we can use the results there, and the proof goes as follows:

(1) The group [math]C_N\subset U_N[/math] is defined by imposing the following relations, with [math]\xi[/math] being the all-one vector, which correspond to the bistochasticity condition:

[[math]] u\xi=\xi\quad,\quad \bar{u}\xi=\xi [[/math]]

But these relations tell us precisely that the following two operators, with the partitions on the right being singletons, must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad:\quad\pi=|_{\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} \gt =\mathcal P_{12} [[/math]]

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

(2) In order to deal now with the real bistochastic group [math]B_N[/math], we can either use a similar argument, or simply use the following intersection formula:

[[math]] B_N=C_N\cap O_N [[/math]]

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

[[math]] D = \lt \mathcal P_{12},P_2 \gt =P_{12} [[/math]]

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

■

As a comment here, we have used in the above the fact, which is something quite trivial, that the category of partitions associated to an intersection of easy quantum groups is generated by the corresponding categories of partitions. We will be back to this, and to some other product operations as well, with similar results, later on.

We can put now the results that we have together, as follows:

Theorem

The basic unitary and bistochastic groups,

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

are all easy, coming from the various categories of singletons and pairings.

Show Proof

We know from the above that the groups in the statement are indeed easy, the corresponding diagram of categories of partitions being as follows:

[[math]] \xymatrix@R=16mm@C=18mm{ \mathcal P_{12}\ar[d]&\mathcal P_2\ar[l]\ar[d]\\ P_{12}&P_2\ar[l]} [[/math]]

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

■

Summarizing, what we have so far is a general notion of “easiness”, coming from the Brauer theorems for [math]O_N,U_N[/math], and their straightforward extensions to [math]B_N,C_N[/math].

General references

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