15c. Basic operations

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

All the above is quite encouraging, so time now to take easiness very seriously, and develop some general abstract theory for the easy groups. Let us first discuss some basic composition operations. We will be mainly interested in the following operations:

Definition

The closed subgroups of [math]U_N[/math] are subject to intersection and generation operations, constructed as follows:

• Intersection: [math]H\cap K[/math] is the usual intersection of [math]H,K[/math].
• Generation: [math] \lt H,K \gt [/math] is the closed subgroup generated by [math]H,K[/math].

Alternatively, we can define these operations at the function algebra level, by performing certain operations on the associated ideals, as follows:

Proposition

Assuming that we have presentation results as follows,

[[math]] C(H)=C(U_N)/I\quad,\quad C(K)=C(U_N)/J [[/math]]

the groups [math]H\cap K[/math] and [math] \lt H,K \gt [/math] are given by the following formulae,

[[math]] C(H\cap K)=C(U_N)/ \lt I,J \gt [[/math]]

[[math]] C( \lt H,K \gt )=C(U_N)/(I\cap J) [[/math]]

at the level of the associated algebras of functions.

Show Proof

This is indeed clear from the definition of the operations [math]\cap[/math] and [math] \lt \,,\, \gt [/math], as formulated above, and from the Stone-Weierstrass theorem.

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In what follows we will need Tannakian formulations of the above two operations. The result here, coming from the general Tannakian duality result established in chapter 14, and that we have in fact already used a couple of times in the above, is as follows:

Theorem

The intersection and generation operations [math]\cap[/math] and [math] \lt \,, \gt [/math] can be constructed via the Tannakian correspondence [math]G\to C_G[/math], as follows:

• Intersection: defined via [math]C_{G\cap H}= \lt C_G,C_H \gt [/math].
• Generation: defined via [math]C_{ \lt G,H \gt }=C_G\cap C_H[/math].

Show Proof

This follows from Proposition 15.15, and from Tannakian duality. Indeed, it follows from Tannakian duality that given a closed subgroup [math]G\subset U_N[/math], with fundamental representation [math]v[/math], the algebra of functions [math]C(G)[/math] has the following presentation:

[[math]] C(G)=C(U_N)\Big/\left \lt T\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,\forall l,\forall T\in Hom(v^{\otimes k},v^{\otimes l})\right \gt [[/math]]

In other words, given a closed subgroup [math]G\subset U_N[/math], we have a presentation of the following type, with [math]I_G[/math] being the ideal coming from the Tannakian category of [math]G[/math]:

[[math]] C(G)=C(U_N)/I_G [[/math]]

But this leads to the conclusion in the statement.

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In relation now with our easiness questions, we first have the following result:

Proposition

Assuming that [math]H,K[/math] are easy, then so is [math]H\cap K[/math], and we have

[[math]] D_{H\cap K}= \lt D_H,D_K \gt [[/math]]

at the level of the corresponding categories of partitions.

Show Proof

We have indeed the following computation:

[[math]] \begin{eqnarray*} C_{H\cap K} &=& \lt C_H,C_K \gt \\ &=& \lt span(D_H),span(D_K) \gt \\ &=&span( \lt D_H,D_K \gt ) \end{eqnarray*} [[/math]]

Thus, by Tannakian duality we obtain the result.

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Regarding now the generation operation, the situation here is more complicated, due to a number of technical reasons, and we only have the following statement:

Proposition

Assuming that [math]H,K[/math] are easy, we have an inclusion

[[math]] \lt H,K \gt \subset\{H,K\} [[/math]]

coming from an inclusion of Tannakian categories as follows,

[[math]] C_H\cap C_K\supset span(D_H\cap D_K) [[/math]]

where [math]\{H,K\}[/math] is the easy group having as category of partitions [math]D_H\cap D_K[/math].

Show Proof

This follows from the definition and properties of the generation operation, explained above, and from the following computation:

[[math]] \begin{eqnarray*} C_{ \lt H,K \gt } &=&C_H\cap C_K\\ &=&span(D_H)\cap span(D_K)\\ &\supset&span(D_H\cap D_K) \end{eqnarray*} [[/math]]

Indeed, by Tannakian duality we obtain from this all the assertions.

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It is not clear if the inclusions in Proposition 15.18 are isomorphisms or not, and this even under a supplementary [math]N \gt \gt 0[/math] assumption. Technically speaking, the problem comes from the fact that the operation [math]\pi\to T_\pi[/math] does not produce linearly independent maps, and so all that we are doing is sensitive to the value of [math]N\in\mathbb N[/math]. The subject here is quite technical, to be further developed in chapter 16 below, with probabilistic motivations in mind, without however solving the present algebraic questions.

Summarizing, we have some problems here, and we must proceed as follows:

Theorem

The intersection and easy generation operations [math]\cap[/math] and [math]\{\,,\}[/math] can be constructed via the Tannakian correspondence [math]G\to D_G[/math], as follows:

• Intersection: defined via [math]D_{G\cap H}= \lt D_G,D_H \gt [/math].
• Easy generation: defined via [math]D_{\{G,H\}}=D_G\cap D_H[/math].

Show Proof

Here the situation is as follows:

(1) This is a true and honest result, coming from Proposition 15.17.

(2) This is more of an empty statement, coming from Proposition 15.18.

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As already mentioned, there is some interesting mathematics still to be worked out, in relation with all this, and we will be back to this later, with further details. With the above notions in hand, however, even if not fully satisfactory, we can formulate a nice result, which improves our main result so far, namely Theorem 15.13, as follows:

Theorem

The basic unitary and reflection groups, namely

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

are all easy, and they form an intersection and easy generation diagram, in the sense that the above square diagram satisfies [math]U_N=\{K_N,O_N\}[/math], and [math]H_N=K_N\cap O_N[/math].

Show Proof

We know from Theorem 15.13 that the groups in the statement are 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]]

Now observe that this latter diagram is an intersection and generation diagram. By using Theorem 15.19, this reformulates into the fact that the diagram of quantum groups is an intersection and easy generation diagram, as claimed.

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It is possible to further improve the above result, by proving that the diagram there is actually a plain generation diagram. However, this is something more technical, and for a discussion here, you can check for instance my quantum group book .

Moving forward, as a continuation of the above, it is possible to develop some more general theory, along the above lines. Given a closed subgroup [math]G\subset U_N[/math], we can talk about its “easy envelope”, which is the smallest easy group [math]\widetilde{G}[/math] containing [math]G[/math]. This easy envelope appears by definition as an intermediate closed subgroup, as follows:

[[math]] G\subset\widetilde{G}\subset U_N [[/math]]

With this notion in hand, Proposition 15.18 can be refined into a result stating that given two easy groups [math]H,K[/math], we have inclusions as follows:

[[math]] \lt H,K \gt \subset\widetilde{ \lt H,K \gt }\subset\{H,K\} [[/math]]

In order to discuss all this, let us start with the following definition:

Definition

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

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

with [math]S_N\subset U_N[/math] being the standard embedding, via permutation matrices.

We will be interested in such groups, which cover for instance all the easy groups, and many more. At the Tannakian level, we have the following result:

Theorem

The homogeneous groups [math]S_N\subset G\subset U_N[/math] are in one-to-one correspondence with the intermediate tensor categories

[[math]] span\left(T_\pi\Big|\pi\in\mathcal P_2\right)\subset C\subset span\left(T_\pi\Big|\pi\in P\right) [[/math]]

where [math]P[/math] is the category of all partitions, [math]\mathcal P_2[/math] is the category of the matching pairings, and [math]\pi\to T_\pi[/math] is the standard implementation of partitions, as linear maps.

Show Proof

This follows from Tannakian duality, and from the Brauer type results for [math]S_N,U_N[/math]. To be more precise, we know from Tannakian duality that each closed subgroup [math]G\subset U_N[/math] can be reconstructed from its Tannakian category [math]C=(C(k,l))[/math], as follows:

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

Thus we have a one-to-one correspondence [math]G\leftrightarrow C[/math], given by Tannakian duality, and since the endpoints [math]G=S_N,U_N[/math] are both easy, corresponding to the categories [math]C=span(T_\pi|\pi\in D)[/math] with [math]D=P,\mathcal P_2[/math], this gives the result.

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Our purpose now will be that of using the Tannakian result in Theorem 15.22, in order to introduce and study a combinatorial notion of “easiness level”, for the arbitrary intermediate groups [math]S_N\subset G\subset U_N[/math]. Let us begin with the following simple fact:

Proposition

Given a homogeneous group [math]S_N\subset G\subset U_N[/math], with associated Tannakian category [math]C=(C(k,l))[/math], the sets

[[math]] D^1(k,l)=\left\{\pi\in P(k,l)\Big|T_\pi\in C(k,l)\right\} [[/math]]

form a category of partitions, in the sense of Definition 15.3.

Show Proof

We use the basic categorical properties of the correspondence [math]\pi\to T_\pi[/math] between partitions and linear maps, that we established in the above, namely:

[[math]] T_{[\pi\sigma]}=T_\pi\otimes T_\sigma\quad,\quad T_{[^\sigma_\pi]}\sim T_\pi T_\sigma\quad,\quad T_{\pi^*}=T_\pi^* [[/math]]

Together with the fact that [math]C[/math] is a tensor category, we deduce from these formulae that we have the following implication:

[[math]] \begin{eqnarray*} \pi,\sigma\in D^1 &\implies&T_\pi,T_\sigma\in C\\ &\implies&T_\pi\otimes T_\sigma\in C\\ &\implies&T_{[\pi\sigma]}\in C\\ &\implies&[\pi\sigma]\in D^1 \end{eqnarray*} [[/math]]

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

[[math]] \begin{eqnarray*} \pi,\sigma\in D^1 &\implies&T_\pi,T_\sigma\in C\\ &\implies&T_\pi T_\sigma\in C\\ &\implies&T_{[^\sigma_\pi]}\in C\\ &\implies&[^\sigma_\pi]\in D^1 \end{eqnarray*} [[/math]]

Finally, we have as well the following implication:

[[math]] \begin{eqnarray*} \pi\in D^1 &\implies&T_\pi\in C\\ &\implies&T_\pi^*\in C\\ &\implies&T_{\pi^*}\in C\\ &\implies&\pi^*\in D^1 \end{eqnarray*} [[/math]]

Thus [math]D^1[/math] is indeed a category of partitions, as claimed.

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We can further refine the above observation, in the following way:

Proposition

Given a compact group [math]S_N\subset G\subset U_N[/math], construct [math]D^1\subset P[/math] as above, and let [math]S_N\subset G^1\subset U_N[/math] be the easy group associated to [math]D^1[/math]. Then:

• We have [math]G\subset G^1[/math], as subgroups of [math]U_N[/math].
• [math]G^1[/math] is the smallest easy group containing [math]G[/math].
• [math]G[/math] is easy precisely when [math]G\subset G^1[/math] is an isomorphism.

Show Proof

All this is elementary, the proofs being as follows:

(1) We know that the Tannakian category of [math]G^1[/math] is given by:

[[math]] C_{kl}^1=span\left(T_\pi\Big|\pi\in D^1(k,l)\right) [[/math]]

Thus we have [math]C^1\subset C[/math], and so [math]G\subset G^1[/math], as subgroups of [math]U_N[/math].

(2) Assuming that we have [math]G\subset G'[/math], with [math]G'[/math] easy, coming from a Tannakian category [math]C'=span(D')[/math], we must have [math]C'\subset C[/math], and so [math]D'\subset D^1[/math]. Thus, [math]G^1\subset G'[/math], as desired.

(3) This is a trivial consequence of (2).

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Summarizing, we have now a notion of “easy envelope”, as follows:

Definition

The easy envelope of a homogeneous group [math]S_N\subset G\subset U_N[/math] is the easy group [math]S_N\subset G^1\subset U_N[/math] associated to the category of partitions

[[math]] D^1(k,l)=\left\{\pi\in P(k,l)\Big|T_\pi\in C(k,l)\right\} [[/math]]

where [math]C=(C(k,l))[/math] is the Tannakian category of [math]G[/math].

At the level of examples, most of the known homogeneous groups [math]S_N\subset G\subset U_N[/math] are in fact easy. However, there are non-easy interesting examples as well, such as the generic reflection groups [math]H_N^{sd}[/math] from chapter 10, and we will certainly have an exercise at the end of this chapter, regarding the computation of the corresponding easy envelopes.

As a technical observation now, we can in fact generalize the above construction to any closed subgroup [math]G\subset U_N[/math], and we have the following result:

Proposition

Given a closed subgroup [math]G\subset U_N[/math], construct [math]D^1\subset P[/math] as above, and let [math]S_N\subset G^1\subset U_N[/math] be the easy group associated to [math]D^1[/math]. We have then

[[math]] G^1=( \lt G,S_N \gt )^1 [[/math]]

where [math] \lt G,S_N \gt \subset U_N[/math] is the smallest closed subgroup containing [math]G,S_N[/math].

Show Proof

According to our Tannakian results, the subgroup [math] \lt G,S_N \gt \subset U_N[/math] in the statement exists indeed, and can be obtained by intersecting categories, as follows:

[[math]] C_{ \lt G,S_N \gt }=C_G\cap C_{S_N} [[/math]]

We conclude from this that for any [math]\pi\in P(k,l)[/math] we have:

[[math]] T_\pi\in C_{ \lt G,S_N \gt }(k,l)\iff T_\pi\in C_G(k,l) [[/math]]

It follows that the [math]D^1[/math] categories for the groups [math] \lt G,S_N \gt [/math] and [math]G[/math] coincide, and so the easy envelopes [math]( \lt G,S_N \gt )^1[/math] and [math]G^1[/math] coincide as well, as stated.

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In order now to fine-tune all this, by using an arbitrary parameter [math]p\in\mathbb N[/math], which can be thought of as being an “easiness level”, we can proceed as follows:

Definition

Given a compact group [math]S_N\subset G\subset U_N[/math], and an integer [math]p\in\mathbb N[/math], we construct the family of linear spaces

[[math]] E^p(k,l)=\left\{\alpha_1T_{\pi_1}+\ldots+\alpha_pT_{\pi_p}\in C(k,l)\Big|\alpha_i\in\mathbb C,\pi_i\in P(k,l)\right\} [[/math]]

and we denote by [math]C^p[/math] the smallest tensor category containing [math]E^p=(E^p(k,l))[/math], and by [math]S_N\subset G^p\subset U_N[/math] the compact group corresponding to this category [math]C^p[/math].

As a first observation, at [math]p=1[/math] we have [math]C^1=E^1=span(D^1)[/math], where [math]D^1[/math] is the category of partitions constructed in Proposition 15.24. Thus the group [math]G^1[/math] constructed above coincides with the “easy envelope” of [math]G[/math], from Definition 15.25.

In the general case, [math]p\in\mathbb N[/math], the family [math]E^p=(E^p(k,l))[/math] constructed above is not necessarily a tensor category, but we can of course consider the tensor category [math]C^p[/math] generated by it, as indicated. Finally, in the above definition we have used of course the Tannakian duality results, in order to perform the operation [math]C^p\to G^p[/math].

In practice, the construction in Definition 15.27 is often something quite complicated, and it is convenient to use the following observation:

Proposition

The category [math]C^p[/math] constructed above is generated by the spaces

[[math]] E^p(l)=\left\{\alpha_1T_{\pi_1}+\ldots+\alpha_pT_{\pi_p}\in C(l)\Big|\alpha_i\in\mathbb C,\pi_i\in P(l)\right\} [[/math]]

where [math]C(l)=C(0,l),P(l)=P(0,l)[/math], with [math]l[/math] ranging over the colored integers.

Show Proof

We use the well-known fact, that we know from chapter 13, that given a closed subgroup [math]G\subset U_N[/math], we have a Frobenius type isomorphism, as follows:

[[math]] Hom(u^{\otimes k},u^{\otimes l})\simeq Fix(u^{\otimes\bar{k}l}) [[/math]]

If we apply this to the group [math]G^p[/math], we obtain an isomorphism as follows:

[[math]] C(k,l)\simeq C(\bar{k}l) [[/math]]

On the other hand, we have as well an isomorphism [math]P(k,l)\simeq P(\bar{k}l)[/math], obtained by performing a counterclockwise rotation to the partitions [math]\pi\in P(k,l)[/math]. According to the above definition of the spaces [math]E^p(k,l)[/math], this induces an isomorphism as follows:

[[math]] E^p(k,l)\simeq E^p(\bar{k}l) [[/math]]

We deduce from this that for any partitions [math]\pi_1,\ldots,\pi_p\in C(k,l)[/math], having rotated versions [math]\rho_1,\ldots,\rho_p\in C(\bar{k}l)[/math], and for any scalars [math]\alpha_1,\ldots,\alpha_p\in\mathbb C[/math], we have:

[[math]] \alpha_1T_{\pi_1}+\ldots+\alpha_pT_{\pi_p}\in C(k,l)\iff\alpha_1T_{\rho_1}+\ldots+\alpha_pT_{\rho_p}\in C(\bar{k}l) [[/math]]

But this gives the conclusion in the statement, and we are done.

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The main properties of the construction [math]G\to G^p[/math] can be summarized as follows:

Theorem

Given a compact group [math]S_N\subset G\subset U_N[/math], the compact groups [math]G^p[/math] constructed above form a decreasing family, whose intersection is [math]G[/math]:

[[math]] G=\bigcap_{p\in\mathbb N}G^p [[/math]]

Moreover, [math]G[/math] is easy when this decreasing limit is stationary, [math]G=G^1[/math].

Show Proof

By definition of [math]E^p(k,l)[/math], and by using Proposition 15.28, these linear spaces form an increasing filtration of [math]C(k,l)[/math]. The same remains true when completing into tensor categories, and so we have an increasing filtration, as follows:

[[math]] C=\bigcup_{p\in\mathbb N}C^p [[/math]]

At the compact group level now, we obtain the decreasing intersection in the statement. Finally, the last assertion is clear from Proposition 15.28.

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As a main consequence of the above results, we can now formulate:

Definition

We say that a homogeneous compact group

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

is easy at order [math]p[/math] when [math]G=G^p[/math], with [math]p[/math] being chosen minimal with this property.

Observe that the order 1 notion corresponds to the usual easiness. In general, all this is quite abstract, but there are several explicit examples, that can be worked out. For more on all this, you can check my quantum group book .

General references

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