Quantum groups

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13a. Quantum groups

We have seen so far that classical probability has a “twin sister”, which is Voiculescu's free probability theory. The relation between the two comes from an almost perfect symmetry between the main limiting theorems in both theories, which can be axiomatized. At a more concrete level, passed a few technical manipulations, the main limiting laws are as follows, with the vertical correspondence being the Bercovici-Pata bijection:

[[math]] \xymatrix@R=16pt@C=18pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{.}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

All this remains however a bit abstract. Fortunately, beasts like random matrices and quantum groups are there, providing us with explicit models for the above laws, and for what is going on, in general. In what regards quantum groups, we have:

Theorem

The main limiting laws in classical and free probability come from

[[math]] \xymatrix@R=16pt@C=16pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

as asymptotic laws, with [math]N\to\infty[/math], of the corresponding truncated characters.

Show Proof

This is something that we know from chapter 4, for the lower face of the cube. In what regards the upper face, this is something which remains to be clarified.

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Our purpose in this chapter and in the next one will be to discuss the details of this result, and then further build on it, by expanding the theory into a more general correspondence between classical geometry and free geometry. Then later, in chapters 15-16, we will discuss invariance questions, and we will add as well to the picture some further beasts, which are of even more tricky type, namely the Jones subfactors.

As a starting point, we have the following key definition, from :

Definition

A Woronowicz algebra is a [math]C^*[/math]-algebra [math]A[/math], given with a unitary matrix [math]v\in M_N(A)[/math] whose coefficients generate [math]A[/math], such that the formulae

[[math]] \Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj}\quad,\quad \varepsilon(v_{ij})=\delta_{ij}\quad,\quad S(v_{ij})=v_{ji}^* [[/math]]

define morphisms of [math]C^*[/math]-algebras [math]\Delta:A\to A\otimes A[/math], [math]\varepsilon:A\to\mathbb C[/math], [math]S:A\to A^{opp}[/math].

This definition is in fact a modified version of Woronowicz' main definition in , which best fits our purposes here, covering well the objects in Theorem 13.1. More on this later. We say that [math]A[/math] is cocommutative when [math]\Sigma\Delta=\Delta[/math], where [math]\Sigma(a\otimes b)=b\otimes a[/math] is the flip. We have the following result, which justifies the terminology and axioms:

Proposition

The following are Woronowicz algebras:

• [math]C(G)[/math], with [math]G\subset U_N[/math] compact Lie group. Here the structural maps are:
  [[math]] \Delta(\varphi)=[(g,h)\to \varphi(gh)]\quad,\quad \varepsilon(\varphi)=\varphi(1)\quad,\quad S(\varphi)=[g\to\varphi(g^{-1})] [[/math]]
• [math]C^*(\Gamma)[/math], with [math]F_N\to\Gamma[/math] finitely generated group. Here the structural maps are:
  [[math]] \Delta(g)=g\otimes g\quad,\quad \varepsilon(g)=1\quad,\quad S(g)=g^{-1} [[/math]]

Moreover, we obtain in this way all the commutative/cocommutative algebras.

Show Proof

In both cases, we have to indicate a certain matrix [math]v[/math]. For the first assertion, we can use the matrix [math]v=(v_{ij})[/math] formed by matrix coordinates of [math]G[/math], given by:

[[math]] g=\begin{pmatrix} v_{11}(g)&\ldots&v_{1N}(g)\\ \vdots&&\vdots\\ v_{N1}(g)&\ldots&v_{NN}(g) \end{pmatrix} [[/math]]

As for the second assertion, we can use here the diagonal matrix formed by generators:

[[math]] v=\begin{pmatrix} g_1&&0\\ &\ddots&\\ 0&&g_N \end{pmatrix} [[/math]]

Finally, the last assertion follows from the Gelfand theorem, in the commutative case. In the cocommutative case this follows from the Peter-Weyl theory, explained below.

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In view of Proposition 13.3, we can formulate the following definition:

Definition

Given a Woronowicz algebra [math]A[/math], we formally write

[[math]] A=C(G)=C^*(\Gamma) [[/math]]

and call [math]G[/math] compact quantum group, and [math]\Gamma[/math] discrete quantum group.

When [math]A[/math] is both commutative and cocommutative, [math]G[/math] is a compact abelian group, [math]\Gamma[/math] is a discrete abelian group, and these groups are dual to each other:

[[math]] G=\widehat{\Gamma}\quad,\quad\Gamma=\widehat{G} [[/math]]

In general, we still agree to write the formulae [math]G=\widehat{\Gamma},\Gamma=\widehat{G}[/math], but in a formal sense. Finally, let us make as well the following convention:

Definition

We identify two Woronowicz algebras [math](A,v)[/math] and [math](B,w)[/math], as well as the corresponding quantum groups, when we have an isomorphism of [math]*[/math]-algebras

[[math]] \lt v_{ij} \gt \simeq \lt w_{ij} \gt [[/math]]

mapping standard coordinates to standard coordinates.

This convention is here for avoiding amenability issues, as for any compact or discrete quantum group to correspond to a unique Woronowicz algebra. More on this later.

Moving ahead now, let us call corepresentation of [math]A[/math] any unitary matrix [math]u\in M_n(\mathcal A)[/math], where [math]\mathcal A= \lt v_{ij} \gt [/math], satisfying the same conditions as those satisfied by [math]u[/math], namely:

[[math]] \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj}\quad,\quad \varepsilon(u_{ij})=\delta_{ij}\quad,\quad S(u_{ij})=u_{ji}^* [[/math]]

We have the following key result, due to Woronowicz :

Theorem

Any Woronowicz algebra has a unique Haar integration functional,

[[math]] \left(\int_G\otimes id\right)\Delta=\left(id\otimes\int_G\right)\Delta=\int_G(.)1 [[/math]]

which can be constructed by starting with any faithful positive form [math]\varphi\in A^*[/math], and setting

[[math]] \int_G=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k} [[/math]]

where [math]\phi*\psi=(\phi\otimes\psi)\Delta[/math]. Moreover, for any corepresentation [math]u\in M_n(\mathbb C)\otimes A[/math] we have

[[math]] \left(id\otimes\int_G\right)u=P [[/math]]

where [math]P[/math] is the orthogonal projection onto [math]Fix(u)=\{\xi\in\mathbb C^n|u\xi=\xi\}[/math].

Show Proof

Following , this can be done in 3 steps, as follows:

(1) Given [math]\varphi\in A^*[/math], our claim is that the following limit converges, for any [math]a\in A[/math]:

[[math]] \int_\varphi a=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k}(a) [[/math]]

Indeed, by linearity we can assume that [math]a\in A[/math] is the coefficient of certain corepresentation, [math]a=(\tau\otimes id)u[/math]. But in this case, an elementary computation gives the following formula, with [math]P_\varphi[/math] being the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi)u[/math]:

[[math]] \left(id\otimes\int_\varphi\right)u=P_\varphi [[/math]]

(2) Since [math]u\xi=\xi[/math] implies [math][(id\otimes\varphi)u]\xi=\xi[/math], we have [math]P_\varphi\geq P[/math], where [math]P[/math] is the orthogonal projection onto the fixed point space in the statement, namely:

[[math]] Fix(u)=\left\{\xi\in\mathbb C^n\Big|u\xi=\xi\right\} [[/math]]

The point now is that when [math]\varphi\in A^*[/math] is faithful, by using a standard positivity trick, we can prove that we have [math]P_\varphi=P[/math], exactly as in the classical case.

(3) With the above formula in hand, the left and right invariance of [math]\int_G=\int_\varphi[/math] is clear on coefficients, and so in general, and this gives all the assertions. See .

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We can now develop, again following , the Peter-Weyl theory for the corepresentations of [math]A[/math]. Consider the dense subalgebra [math]\mathcal A\subset A[/math] generated by the coefficients of the fundamental corepresentation [math]v[/math], and endow it with the following scalar product:

[[math]] \lt a,b \gt =\int_Gab^* [[/math]]

With this convention, we have the following result, from :

Theorem

We have the following Peter-Weyl type results:

• Any corepresentation decomposes as a sum of irreducible corepresentations.
• Each irreducible corepresentation appears inside a certain [math]v^{\otimes k}[/math].
• [math]\mathcal A=\bigoplus_{u\in Irr(A)}M_{\dim(u)}(\mathbb C)[/math], the summands being pairwise orthogonal.
• The characters of irreducible corepresentations form an orthonormal system.

Show Proof

All these results are from , the idea being as follows:

(1) Given [math]u\in M_n(A)[/math], the intertwiner algebra [math]End(u)=\{T\in M_n(\mathbb C)|Tu=uT\}[/math] is a finite dimensional [math]C^*[/math]-algebra, and so decomposes as [math]End(u)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_r}(\mathbb C)[/math]. But this gives a decomposition of type [math]u=u_1+\ldots+u_r[/math], as desired.

(2) Consider the Peter-Weyl corepresentations, [math]v^{\otimes k}[/math] with [math]k[/math] colored integer, defined by [math]v^{\otimes\emptyset}=1[/math], [math]v^{\otimes\circ}=v[/math], [math]v^{\otimes\bullet}=\bar{v}[/math] and multiplicativity. The coefficients of these corepresentations span the dense algebra [math]\mathcal A[/math], and by using (1), this gives the result.

(3) Here the direct sum decomposition, which is a [math]*[/math]-coalgebra isomorphism, follows from (2). As for the second assertion, this follows from the fact that [math](id\otimes\int_G)u[/math] is the orthogonal projection [math]P_u[/math] onto the space [math]Fix(u)[/math], for any corepresentation [math]u[/math].

(4) Let us define indeed the character of [math]u\in M_n(A)[/math] to be the trace, [math]\chi_u=Tr(u)[/math]. Since this character is a coefficient of [math]u[/math], the orthogonality assertion follows from (3). As for the norm 1 claim, this follows once again from [math](id\otimes\int_G)u=P_u[/math].

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We can now solve a problem that we left open before, namely:

Proposition

The cocommutative Woronowicz algebras appear as the quotients

[[math]] C^*(\Gamma)\to A\to C^*_{red}(\Gamma) [[/math]]

given by [math]A=C^*_\pi(\Gamma)[/math] with [math]\pi\otimes\pi\subset\pi[/math], with [math]\Gamma[/math] being a discrete group.

Show Proof

This follows from the Peter-Weyl theory, and clarifies a number of things said before, notably in Proposition 13.3. Indeed, for a cocommutative Woronowicz algebra the irreducible corepresentations are all 1-dimensional, and this gives the results.

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As another consequence of the above results, once again by following Woronowicz , we have the following statement, dealing with functional analysis aspects, and extending what we already knew about the [math]C^*[/math]-algebras of the usual discrete groups:

Theorem

Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A[/math], and [math]A_{red}[/math] be the quotient of [math]A[/math] by the null ideal of the Haar integration. The following are then equivalent:

• The Haar functional of [math]A_{full}[/math] is faithful.
• The projection map [math]A_{full}\to A_{red}[/math] is an isomorphism.
• The counit map [math]\varepsilon:A_{full}\to\mathbb C[/math] factorizes through [math]A_{red}[/math].
• We have [math]N\in\sigma(Re(\chi_v))[/math], the spectrum being taken inside [math]A_{red}[/math].

If this is the case, we say that the underlying discrete quantum group [math]\Gamma[/math] is amenable.

Show Proof

This is well-known in the group dual case, [math]A=C^*(\Gamma)[/math], with [math]\Gamma[/math] being a usual discrete group. In general, the result follows by adapting the group dual case proof:

[math](1)\iff(2)[/math] This simply follows from the fact that the GNS construction for the algebra [math]A_{full}[/math] with respect to the Haar functional produces the algebra [math]A_{red}[/math].

[math](2)\iff(3)[/math] Here [math]\implies[/math] is trivial, and conversely, a counit map [math]\varepsilon:A_{red}\to\mathbb C[/math] produces an isomorphism [math]A_{red}\to A_{full}[/math], via a formula of type [math](\varepsilon\otimes id)\Phi[/math]. See .

[math](3)\iff(4)[/math] Here [math]\implies[/math] is clear, coming from [math]\varepsilon(N-Re(\chi (v)))=0[/math], and the converse can be proved by doing some functional analysis. Once again, we refer here to .

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Let us discuss now some interesting examples. Following Wang , we have:

Proposition

The following universal algebras are Woronowicz algebras,

[[math]] C(O_N^+)=C^*\left((v_{ij})_{i,j=1,\ldots,N}\Big|v=\bar{v},v^t=v^{-1}\right) [[/math]]

[[math]] C(U_N^+)=C^*\left((v_{ij})_{i,j=1,\ldots,N}\Big|v^*=v^{-1},v^t=\bar{v}^{-1}\right) [[/math]]

so the underlying compact quantum spaces [math]O_N^+,U_N^+[/math] are compact quantum groups.

Show Proof

This follows from the elementary fact that if a matrix [math]v=(v_{ij})[/math] is orthogonal or biunitary, then so must be the following matrices:

[[math]] v^\Delta_{ij}=\sum_kv_{ik}\otimes v_{kj}\quad,\quad v^\varepsilon_{ij}=\delta_{ij}\quad,\quad v^S_{ij}=v_{ji}^* [[/math]]

Thus, we can indeed define morphisms [math]\Delta,\varepsilon,S[/math] as in Definition 13.2, by using the universal properties of [math]C(O_N^+)[/math], [math]C(U_N^+)[/math], and this gives the result.

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There is a connection here with group duals, coming from:

Proposition

Given a closed subgroup [math]G\subset U_N^+[/math], consider its “diagonal torus”, which is the closed subgroup [math]T\subset G[/math] constructed as follows:

[[math]] C(T)=C(G)\Big/\left \lt v_{ij}=0\Big|\forall i\neq j\right \gt [[/math]]

This torus is then a group dual, [math]T=\widehat{\Lambda}[/math], where [math]\Lambda= \lt g_1,\ldots,g_N \gt [/math] is the discrete group generated by the elements [math]g_i=v_{ii}[/math], which are unitaries inside [math]C(T)[/math].

Show Proof

Since [math]u[/math] is unitary, its diagonal entries [math]g_i=v_{ii}[/math] are unitaries inside [math]C(T)[/math]. Moreover, from [math]\Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj}[/math] we obtain, when passing inside the quotient:

[[math]] \Delta(g_i)=g_i\otimes g_i [[/math]]

It follows that we have [math]C(T)=C^*(\Lambda)[/math], modulo identifying as usual the [math]C^*[/math]-completions of the various group algebras, and so that we have [math]T=\widehat{\Lambda}[/math], as claimed.

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With this notion in hand, we have the following result:

Theorem

The diagonal tori of the basic rotation groups are as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ U_N\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&O_N^+\ar[u] }\qquad \item[a]ymatrix@R=7mm@C=15mm{\\ :\\} \qquad \item[a]ymatrix@R=14mm@C=15mm{ \mathbb T^N\ar[r]&\widehat{F_N}\\ \mathbb Z_2^N\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u] } [[/math]]

where [math]F_N[/math] is the free group on [math]N[/math] generators, and [math]*[/math] is a group-theoretical free product.

Show Proof

This is clear indeed from [math]U_N^+[/math], and the other results can be obtained by imposing to the generators of [math]F_N[/math] the relations defining the corresponding quantum groups.

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As a conclusion to all this, the above results, coming from , suggest developing a theory of “noncommutative geometry”, covering both the classical and the free geometry, by using compact quantum groups. We will be back to this in chapter 14.

Getting now into more examples, we have the following key result, coming from the work in , , , , , covering the basic rotation and reflection groups:

Theorem

The classical and free, real and complex quantum rotation groups can be complemented with quantum reflection groups, as follows,

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

with [math]H_N=\mathbb Z_2\wr S_N[/math] and [math]K_N=\mathbb T\wr S_N[/math] being the hyperoctahedral group and the full complex reflection group, and [math]H_N^+=\mathbb Z_2\wr_*S_N^+[/math] and [math]K_N^+=\mathbb T\wr_*S_N^+[/math] being their free versions.

Show Proof

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

(1) The first observation is that [math]S_N[/math], regarded as group of permutations of the [math]N[/math] coordinate axes of [math]\mathbb R^N[/math], is a group of orthogonal matrices, [math]S_N\subset O_N[/math]. The corresponding coordinate functions [math]v_{ij}:S_N\to\{0,1\}[/math] form a matrix [math]v=(v_{ij})[/math] which is “magic”, in the sense that its entries are projections, summing up to 1 on each row and each column. In fact, by using the Gelfand theorem, we have the following presentation result:

[[math]] C(S_N)=C^*_{comm}\left((v_{ij})_{i,j=1,\ldots,N}\Big|v={\rm magic}\right) [[/math]]

(2) Based on the above, and following Wang's paper , we can construct the free analogue [math]S_N^+[/math] of the symmetric group [math]S_N[/math] via the following formula:

[[math]] C(S_N^+)=C^*\left((v_{ij})_{i,j=1,\ldots,N}\Big|v={\rm magic}\right) [[/math]]

Here the fact that we have indeed a Woronowicz algebra is standard, exactly as for the free rotation groups in Proposition 13.10, because if a matrix [math]v=(v_{ij})[/math] is magic, then so are the matrices [math]v^\Delta,v^\varepsilon,v^S[/math] constructed there, and this gives the existence of [math]\Delta,u,S[/math].

(3) Consider now the group [math]H_N^s\subset U_N[/math] consisting of permutation-like matrices having as entries the [math]s[/math]-th roots of unity. This group decomposes as follows:

[[math]] H_N^s=\mathbb Z_s\wr S_N [[/math]]

It is straightforward then to construct a free analogue [math]H_N^{s+}\subset U_N^+[/math] of this group, for instance by formulating a definition as follows, with [math]\wr_*[/math] being a free wreath product:

[[math]] H_N^{s+}=\mathbb Z_s\wr_*S_N^+ [[/math]]

(4) In order to finish, besides the case [math]s=1[/math], of particular interest are the cases [math]s=2,\infty[/math]. Here the corresponding reflection groups are as follows:

[[math]] H_N=\mathbb Z_2\wr S_N\quad,\quad K_N=\mathbb T\wr S_N [[/math]]

As for the corresponding quantum groups, these are denoted as follows:

[[math]] H_N^+=\mathbb Z_2\wr_*S_N^+\quad,\quad K_N^+=\mathbb T\wr_*S_N^+ [[/math]]

Thus, we are led to the conclusions in the statement. See , .

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13b. Diagrams, easiness

Getting now towards easiness, let us start with the following definition:

Definition

The Tannakian category associated to a Woronowicz algebra [math](A,v)[/math] is the collection [math]C_A=(C_A(k,l))[/math] of vector spaces

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

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

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

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

As a key remark, the fact that [math]v\in M_N(A)[/math] is biunitary translates into the following conditions, where [math]R:\mathbb C\to\mathbb C^N\otimes\mathbb C^N[/math] is the linear map given by [math]R(1)=\sum_ie_i\otimes e_i[/math]:

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

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

We are therefore led to the following abstract definition, summarizing the main properties of the categories appearing from Woronowicz algebras:

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 subspaces

[[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,\circ\bullet)[/math] and [math]C(\emptyset,\bullet\circ)[/math] contain the operator [math]R:1\to\sum_ie_i\otimes e_i[/math].

The point now is that conversely, we can associate a Woronowicz algebra to any tensor category in the sense of Definition 13.15, in the following way:

Proposition

Given a tensor category [math]C=(C(k,l))[/math] over [math]\mathbb C^N[/math], as above,

[[math]] A_C=C^*\left((v_{ij})_{i,j=1,\ldots,N}\Big|T\in Hom(v^{\otimes k},v^{\otimes l}),\forall k,l,\forall T\in C(k,l)\right) [[/math]]

is a Woronowicz algebra.

Show Proof

This is something standard, because the relations [math]T\in Hom(v^{\otimes k},v^{\otimes l})[/math] determine a Hopf ideal, so they allow the construction of [math]\Delta,\varepsilon,S[/math] as in Definition 13.2.

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With the above constructions in hand, we have the following result:

Theorem

The Tannakian duality constructions

[[math]] C\to A_C\quad,\quad A\to C_A [[/math]]

are inverse to each other, modulo identifying full and reduced versions.

Show Proof

The idea is that we have [math]C\subset C_{A_C}[/math], for any algebra [math]A[/math], and so we are left with proving that we have [math]C_{A_C}\subset C[/math], for any category [math]C[/math]. But this follows from a long series of algebraic manipulations, and for details we refer to Malacarne , and also to Woronowicz , where this result was first proved, by using other methods.

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In practice now, all this is quite abstract, and we will rather need Brauer type results, for the specific quantum groups that we are interested in. Let us start with:

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].

In other words, what we have here are the same axioms as in chapter 4, but with the condition that [math]P(k,\bar{k})[/math] with [math]|k|=2[/math] must contain the crossing partition [math]\slash\hskip-2.0mm\backslash[/math] removed. At the level of examples, there are many of them, and we will get to this in a moment.

Observe the similarity with Definition 13.15. In fact Definition 13.18 is a delinearized version of Definition 13.15, the relation with the Tannakian categories coming from:

Proposition

Given a partition [math]\pi\in P(k,l)[/math], consider the linear map

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

given by the following formula, where [math]e_1,\ldots,e_N[/math] is 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 then 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

The concatenation property 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]]

As for the other two formulae in the statement, their proofs are similar.

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In relation with quantum groups, we have the following result, from :

Theorem

Each category of partitions [math]D=(D(k,l))[/math] produces a family of compact quantum groups [math]G=(G_N)[/math], one for each [math]N\in\mathbb N[/math], via the following formula:

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

To be more precise, the spaces on the right form a Tannakian category, and so produce a certain closed subgroup [math]G_N\subset U_N^+[/math], via the Tannakian duality correspondence.

Show Proof

This follows indeed from Woronowicz's Tannakian duality, in its “soft” form from Malacarne , as explained in Theorem 13.17. Indeed, let us set:

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

By using the various axioms in Definition 13.18, and the categorical properties of the operation [math]\pi\to T_\pi[/math], from Proposition 13.19, we deduce that [math]C=(C(k,l))[/math] is a Tannakian category. Thus the Tannakian duality applies, and gives the result.

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Philosophically speaking, the quantum groups appearing as in Theorem 13.20 are the simplest, from the perspective of Tannakian duality, so let us formulate:

Definition

A closed subgroup [math]G\subset U_N^+[/math] is called easy when we have

[[math]] Hom(v^{\otimes k},v^{\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].

In other words, we adhere here to the same philosophy as before in chapter 4, in the classical case, namely that easiness means easiness at the Tannakian level.

Getting now to examples, we have the following Brauer type result, coming from the work in , , , , , covering the basic rotation and reflection groups:

Theorem

The basic quantum rotation and reflection groups,

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

are all easy, the corresponding categories of partitions being as follows,

[[math]] \xymatrix@R=18pt@C5pt{ &\mathcal{NC}_{even}\ar[dl]\ar[dd]&&\mathcal {NC}_2\ar[dl]\ar[ll]\ar[dd]\\ NC_{even}\ar[dd]&&NC_2\ar[dd]\ar[ll]\\ &\mathcal P_{even}\ar[dl]&&\mathcal P_2\ar[dl]\ar[ll]\\ P_{even}&&P_2\ar[ll] } [[/math]]

with on top, the symbol [math]NC[/math] standing everywhere for noncrossing partitions.

Show Proof

We already know, from chapter 4, the results for the lower face of the cube. In what regards the results for the upper face, the idea is as follows:

(1) Let us first discuss the easiness property of [math]O_N^+,U_N^+[/math]. The quantum group [math]U_N^+[/math] is by definition constructed via the following relations:

[[math]] v^*=v^{-1}\quad,\quad v^t=\bar{v}^{-1} [[/math]]

Thus, the following operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\ ,\ \pi={\ }^{\,\cap}_{\circ\bullet}\quad,\quad T_\pi\ ,\ \pi={\ }^{\,\cap}_{\bullet\circ} [[/math]]

It follows that 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 NC}_2 [[/math]]

Now by imposing the extra relation [math]v=\bar{v}[/math], we obtain the easiness of [math]O_N^+[/math] as well.

(2) In what regards now the easiness property of [math]H_N^+,K_N^+[/math], this follows again like in the classical case. Indeed, the first observation is that the magic condition satisfied by a matrix [math]v[/math] can be reformulated as follows, with [math]\mu\in P(2,1)[/math] being the fork partition:

[[math]] T_\mu\in Hom(v^{\otimes 2},v) [[/math]]

Now by proceeding as in the proof for [math]U_N^+[/math] discussed above, we conclude that the quantum group [math]S_N^+[/math] is indeed easy, the associated category of partitions being:

[[math]] D= \lt NC_2,\mu \gt =NC [[/math]]

With this in hand, we can pass to the quantum groups [math]H_N^+,K_N^+[/math] in a standard way, and we are led to easiness, and the categories in the statement. See , .

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There are many other examples of easy quantum groups, as for instance the real and complex, classical and free bishochastic quantum groups [math]B_N,C_N,B_N^+,C_N^+[/math], or various intermediate liberations [math]G_N\subset G_N^\times\subset G_N^+[/math] of the easy groups that we know. However, those in Theorem 13.22 remain the most important ones. In order to discuss this, classification results for the easy quantum groups, let us start with a basic result from :

Theorem

The classical and free uniform orthogonal easy quantum groups, [math]S_N\subset G\subset O_N^+[/math], with inclusions between them, are as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ &H_N^+\ar[rr]&&O_N^+\\ S_N^+\ar[rr]\ar[ur]&&B_N^+\ar[ur]\\ &H_N\ar[rr]\ar[uu]&&O_N\ar[uu]\\ S_N\ar[uu]\ar[ur]\ar[rr]&&B_N\ar[uu]\ar[ur] } [[/math]]

Moreover, this is an intersection/easy generation diagram, in the sense that for any of its square subdiagrams [math]P\subset Q,R\subset S[/math] we have [math]P=Q\cap R[/math] and [math] \lt Q,R \gt =S[/math].

Show Proof

There are several things to be proved, the idea being as follows:

(1) To start with, regarding the terminology and notations, the notion of uniformity in the statement is a straightforward compact quantum group extension of the notion of uniformity that we met in chapter 4, for the compact Lie groups.

(2) Also regarding the statement, [math]B_N\subset O_N[/math] is the real bistochastic group, consisting of matrices whose entries sum up to 1, on each row and column, and [math]B_N^+\subset O_N^+[/math] is its straightforward liberation, obtained by imposing the condition [math]v\xi=\xi[/math], with [math]\xi\in\mathbb C^N[/math] being the all-one vector. It is routine to check that [math]B_N,B_N^+[/math] are indeed easy, coming respectively from the categories [math]P_{12},NC_{12}[/math], with [math]12[/math] standing for “singletons and pairings”.

(3) Finally, the easy generation operation [math] \lt \,, \gt [/math] is defined by saying that if [math]G,H\subset U_N^+[/math] are easy, coming from categories of partitions [math]D_G,D_H[/math], then [math] \lt G,H \gt \subset U_N^+[/math] is the easy quantum group coming from the category of partitions [math]D=D_G\cap D_H[/math].

(4) Regarding now the proof, we know that the quantum groups in the statement are indeed easy and uniform, the corresponding categories of partitions being as follows:

[[math]] \xymatrix@R=20pt@C6pt{ &NC_{even}\ar[dl]\ar[dd]&&NC_2\ar[dl]\ar[ll]\ar[dd]\\ NC\ar[dd]&&NC_{12}\ar[dd]\ar[ll]\\ &P_{even}\ar[dl]&&P_2\ar[dl]\ar[ll]\\ P&&P_{12}\ar[ll] } [[/math]]

Since this latter diagram is an intersection and generation diagram, we conclude that we have an intersection and easy generation diagram of quantum groups, as stated.

(5) Regarding now the classification, consider first an easy group [math]S_N\subset G_N\subset O_N[/math]. This must come from a certain category [math]P_2\subset D\subset P[/math], and if we assume [math]G=(G_N)[/math] to be uniform, then [math]D[/math] is uniquely determined by the subset [math]L\subset\mathbb N[/math] consisting of the sizes of the blocks of the partitions in [math]D[/math]. Our claim is that the admissible sets are as follows:

-- [math]L=\{2\}[/math], producing [math]O_N[/math].

-- [math]L=\{1,2\}[/math], producing [math]B_N[/math].

-- [math]L=\{2,4,6,\ldots\}[/math], producing [math]H_N[/math].

-- [math]L=\{1,2,3,\ldots\}[/math], producing [math]S_N[/math].

(6) Indeed, in one sense, this follows from our easiness results for [math]O_N,B_N,H_N,S_N[/math]. In the other sense now, assume that [math]L\subset\mathbb N[/math] is such that the set [math]P_L[/math] consisting of partitions whose sizes of the blocks belong to [math]L[/math] is a category of partitions. We know from the axioms of the categories of partitions that the semicircle [math]\cap[/math] must be in the category, so we have [math]2\in L[/math]. We claim that the following conditions must be satisfied as well:

[[math]] k,l\in L,\,k \gt l\implies k-l\in L [[/math]]

[[math]] k\in L,\,k\geq 2\implies 2k-2\in L [[/math]]

(7) Indeed, we will prove that both conditions follow from the axioms of the categories of partitions. Let us denote by [math]b_k\in P(0,k)[/math] the one-block partition:

[[math]] b_k=\left\{\begin{matrix}\sqcap\hskip-0.7mm \sqcap&\ldots&\sqcap\\ 1\hskip2mm 2&\ldots&k\end{matrix} \right\} [[/math]]

For [math]k \gt l[/math], we can write [math]b_{k-l}[/math] in the following way:

[[math]] b_{k-l}=\left\{\begin{matrix}\sqcap\hskip-0.7mm \sqcap&\ldots&\ldots&\ldots&\ldots&\sqcap\\ 1\hskip2mm 2&\ldots&l&l+1&\ldots&k\\ \sqcup\hskip-0.7mm \sqcup&\ldots&\sqcup&|&\ldots&|\\ &&&1&\ldots&k-l\end{matrix}\right\} [[/math]]

In other words, we have the following formula:

[[math]] b_{k-l}=(b_l^*\otimes |^{\otimes k-l})b_k [[/math]]

Since all the terms of this composition are in [math]P_L[/math], we have [math]b_{k-l}\in P_L[/math], and this proves our first claim. As for the second claim, this can be proved in a similar way, by capping two adjacent [math]k[/math]-blocks with a [math]2[/math]-block, in the middle.

(8) With these conditions in hand, we can conclude in the following way:

\underline{Case 1}. Assume [math]1\in L[/math]. By using the first condition with [math]l=1[/math] we get:

[[math]] k\in L\implies k-1\in L [[/math]]

This condition shows that we must have [math]L=\{1,2,\ldots,m\}[/math], for a certain number [math]m\in\{1,2,\ldots,\infty\}[/math]. On the other hand, by using the second condition we get:

[[math]] \begin{eqnarray*} m\in L &\implies&2m-2\in L\\ &\implies&2m-2\leq m\\ &\implies&m\in\{1,2,\infty\} \end{eqnarray*} [[/math]]

The case [math]m=1[/math] being excluded by the condition [math]2\in L[/math], we reach to one of the two sets producing the groups [math]S_N,B_N[/math].

\underline{Case 2}. Assume [math]1\notin L[/math]. By using the first condition with [math]l=2[/math] we get:

[[math]] k\in L\implies k-2\in L [[/math]]

This condition shows that we must have [math]L=\{2,4,\ldots,2p\}[/math], for a certain number [math]p\in\{1,2,\ldots,\infty\}[/math]. On the other hand, by using the second condition we get:

[[math]] \begin{eqnarray*} 2p\in L &\implies&4p-2\in L\\ &\implies&4p-2\leq 2p\\ &\implies&p\in\{1,\infty\} \end{eqnarray*} [[/math]]

Thus [math]L[/math] must be one of the two sets producing [math]O_N,H_N[/math], and we are done.

(9) In the free case, [math]S_N^+\subset G_N\subset O_N^+[/math], the situation is quite similar, the admissible sets being once again the above ones, producing this time [math]O_N^+,B_N^+,H_N^+,S_N^+[/math]. See .

■

The above classification is something quite simple, but when when lifting the uniformity assumption, or when looking at the unitary case, or, more generally, when looking at the unitary case without the uniformity assumption, things become quite complicated. However, a classification is still possible, and we refer here to Tarrago-Weber .

This was for the story of the classification of easy quantum groups, in the classical and free cases. When looking at intermediate liberations [math]G_N\subset G_N^\times\subset G_N^+[/math] things become quite complicated, and we refer here to Raum-Weber and subsequent papers.

Quite remarkably, however, by tricking a bit, we have the following result:

Theorem (Ground Zero)

Under a collection of suitable extra assumptions

[[math]] \xymatrix@R=16pt@C=16pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

are the unique easy quantum groups.

Show Proof

This is something quite technical, and it is beyond our purposes here to get into the details of the proof, or even into the full details of the statement. Let us mention, however, that in what regards the exact assumptions, these are as follows:

(1) Easiness. This is the key assumption, bringing into the picture partitions and combinatorics, and classification techniques in the spirit of those used above.

(2) Uniformity. With this being, as before, the straightforward quantum group extension of the uniformity notion that we met in chapter 4, for the classical groups.

(3) Twistability. With this meaning that we have an inclusion [math]H_N\subset G[/math], which is something which is normally needed, in order to twist [math]G[/math].

(4) Orientability. With this meaning that [math]H_N\subset G\subset U_N^+[/math], which can be thought of as living inside the cube, can be recovered out of its projections on the edges.

So, this was for the general idea. As for the precise statement, and then of course for the proof, and for the whole story in general, with all this, we refer here to .

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13c. Weingarten formula

With the above understood, let us discuss now the probabilistic consequences of our general easiness theory, in the spirit of the work done in chapter 4, in the classical case. In what regards the asymptotic laws of the main characters, we have here:

Theorem

For an easy quantum group [math]G=(G_N)[/math], coming from a category of partitions [math]D=(D(k,l))[/math], the asymptotic moments of the character [math]\chi=\sum_iv_{ii}[/math] are

[[math]] \lim_{N\to\infty}\int_{G_N}\chi^k=|D(k)| [[/math]]

where [math]D(k)=D(\emptyset,k)[/math], with the limiting sequence on the left consisting of certain integers, and being stationary at least starting from the [math]k[/math]-th term.

Show Proof

This is something elementary, which follows straight from Peter-Weyl theory, by using the linear independence result for the vectors [math]\xi_\pi[/math] from chapter 4, as follows:

[[math]] \begin{eqnarray*} \lim_{N\to\infty}\int_{G_N}\chi^k &=&\lim_{N\to\infty}\dim\left(Fix(v^{\otimes k})\right)\\ &=&\lim_{N\to\infty}\dim\left(span\left(\xi_\pi\big|\pi\in D(k)\right)\right)\\ &=&|D(k)| \end{eqnarray*} [[/math]]

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

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In practice now, for the basic rotation and reflection groups, we obtain:

Theorem

The character laws for basic rotation and reflection groups are

[[math]] \xymatrix@R=20pt@C=20pt{ &\mathfrak B_1\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_1\ar@{-}[dd]\\ \beta_1\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_1\ar@{-}[dd]\ar@{-}[ur]\\ &B_1\ar@{-}[rr]\ar@{-}[uu]&&G_1\ar@{-}[uu]\\ b_1\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_1\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

in the [math]N\to\infty[/math] limit, corresponding to the basic probabilistic limiting theorems, at [math]t=1[/math].

Show Proof

This follows indeed from Theorem 13.22 and Theorem 13.25, by using the known moment formulae for the laws in the statement, at [math]t=1[/math].

■

In the free case, the convergence can be shown to be stationary starting from [math]N=4[/math]. The “fix” comes by looking at truncated characters, constructed as follows:

[[math]] \chi_t=\sum_{i=1}^{[tN]}v_{ii} [[/math]]

In order to investigate these truncated characters, we can use the Weingarten formula, which is very similar to the one from the classical case, as follows:

Theorem

For an easy quantum group [math]G\subset_vU_N^+[/math], coming from a category of partitions [math]D=(D(k,l))[/math], we have the Weingarten formula

[[math]] \int_Gv_{i_1j_1}^{e_1}\ldots v_{i_kj_k}^{e_k}=\sum_{\pi,\nu\in D(k)}\delta_\pi(i)\delta_\nu(j)W_{kN}(\pi,\nu) [[/math]]

for any [math]k=e_1\ldots e_k[/math] and any [math]i,j[/math], where [math]D(k)=D(\emptyset,k)[/math], [math]\delta[/math] are usual Kronecker type symbols, checking whether the indices match, and [math]W_{kN}=G_{kN}^{-1}[/math], with

[[math]] G_{kN}(\pi,\nu)=N^{|\pi\vee\nu|} [[/math]]

where [math]|.|[/math] is the number of blocks.

Show Proof

This is something very standard, coming from the fact that the above integrals form altogether the orthogonal projection [math]P^k[/math] onto the following space:

[[math]] Fix(v^{\otimes k})=span(D(k)) [[/math]]

Consider indeed the following linear map, with [math]D(k)[/math] being as in the statement:

[[math]] E(x)=\sum_{\pi\in D(k)} \lt x,\xi_\pi \gt \xi_\pi [[/math]]

By a standard linear algebra computation, it follows that we have [math]P=WE[/math], where [math]W[/math] is the inverse of the restriction of [math]E[/math] to the following space:

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

But this restriction is the linear map given by the matrix [math]G_{kN}[/math], and so [math]W[/math] is the linear map given by the inverse matrix [math]W_{kN}=G_{kN}^{-1}[/math], and this gives the result.

■

Now back to characters, we have the following final result on the subject, with the convergence being non-stationary at [math]t \lt 1[/math], in both the classical and free cases:

Theorem

The truncated characters for the basic quantum groups

[[math]] \xymatrix@R=17pt@C=17pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

are in the [math]N\to\infty[/math] limit the following laws,

[[math]] \xymatrix@R=19pt@C=21pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{.}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

which are the main laws in classical and free probability.

Show Proof

As before with other results, this is something that we know from chapter 4 for the lower face of the cube, and the proof for the upper face is similar. To be more precise, the point is that we have in the present quantum group setting we have:

[[math]] \begin{eqnarray*} \int_{G_N}\chi_t^k &\simeq&\sum_{\pi\in D(k)}W_{kN}(\pi,\pi)G_{k[tN]}(\pi,\pi)\\ &\simeq&\sum_{\pi\in D(k)}N^{-|\pi|}(tN)^{|\pi|}\\ &=&\sum_{\pi\in D(k)}t^{|\pi|} \end{eqnarray*} [[/math]]

But this leads to the laws in the statement, via results that we already know.

■

We refer to and related papers for full details on all the above. Also, we refer to , , , for more general theory for the easy quantum groups.

Finally, as a consequence of this, and of the Ground Zero theorem, we have: \begin{conclusion} Under suitable combinatorial assumptions,

[[math]] \xymatrix@R=19pt@C=21pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{.}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

are the unique main laws in noncommutative probability. \end{conclusion} To be more precise, this conclusion, while being obviously something a bit informal and philosophical, is in fact, technically speaking, more of a mathematical theorem, coming by putting together Theorem 13.24 and Theorem 13.28. So, very nice all this, we eventually managed to understand how general noncommutative probability works.

13d. Gram determinants

As a last topic for this chapter, let us discuss, following and related papers, the computation of Gram determinants for the easy quantum groups. We already know from chapter 4 that for the group [math]S_N[/math] the formula of the Gram determinant is as follows:

Theorem

The determinant of the Gram matrix of [math]S_N[/math] is given by

[[math]] \det(G_{kN})=\prod_{\pi\in P(k)}\frac{N!}{(N-|\pi|)!} [[/math]]

with the convention that in the case [math]N \lt k[/math] we obtain [math]0[/math].

Show Proof

This is something that we know from chapter 4, the idea being that [math]G_{kN}[/math] decomposes as a product of an upper triangular and lower triangular matrix.

■

For the orthogonal group [math]O_N[/math], the combinatorics is that of the Young diagrams. We denote by [math]|.|[/math] the number of boxes, and we use quantity [math]f^\lambda[/math], which gives the number of standard Young tableaux of shape [math]\lambda[/math]. The result is then as follows:

Theorem

The determinant of the Gram matrix of [math]O_N[/math] is given by

[[math]] \det(G_{kN})=\prod_{|\lambda|=k/2}f_N(\lambda)^{f^{2\lambda}} [[/math]]

where the quantities on the right are [math]f_N(\lambda)=\prod_{(i,j)\in\lambda}(N+2j-i-1)[/math].

Show Proof

This follows from the results of Zinn-Justin in . Indeed, it is known from there that the Gram matrix is diagonalizable, as follows:

[[math]] G_{kN}=\sum_{|\lambda|=k/2}f_N(\lambda)P_{2\lambda} [[/math]]

To be more precise, here [math]1=\sum P_{2\lambda}[/math] is the standard partition of unity associated to the Young diagrams having [math]k/2[/math] boxes, and the coefficients [math]f_N(\lambda)[/math] are those in the statement. Now since we have [math]Tr(P_{2\lambda})=f^{2\lambda}[/math], this gives the result. See , .

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For the free orthogonal and symmetric groups, the results, by Di Francesco , are substantially more complicated. But, we can use the following trick:

Proposition

The Gram matrices of [math]NC_2(2k)\simeq NC(k)[/math] are related by

[[math]] G_{2k,n}(\pi,\sigma)=n^k(\Delta_{kn}^{-1}G_{k,n^2}\Delta_{kn}^{-1})(\pi',\sigma') [[/math]]

where [math]\pi\to\pi'[/math] is the shrinking operation, and [math]\Delta_{kn}[/math] is the diagonal of [math]G_{kn}[/math].

Show Proof

In the context of the standard bijection [math]NC_2(2k)\simeq NC(k)[/math], we have:

[[math]] |\pi\vee\sigma|=k+2|\pi'\vee\sigma'|-|\pi'|-|\sigma'| [[/math]]

We therefore have the following formula, valid for any [math]n\in\mathbb N[/math]:

[[math]] n^{|\pi\vee\sigma|}=n^{k+2|\pi'\vee\sigma'|-|\pi'|-|\sigma'|} [[/math]]

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

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Now back to determinants, let us begin with some examples. We first have:

Proposition

The first Gram matrices and determinants for [math]O_N^+[/math] are

[[math]] \det\begin{pmatrix}N^2&N\\N&N^2\end{pmatrix}=N^2(N^2-1) [[/math]]

[[math]] \det\begin{pmatrix} N^3&N^2&N^2&N^2&N\\ N^2&N^3&N&N&N^2\\ N^2&N&N^3&N&N^2\\ N^2&N&N&N^3&N^2\\ N&N^2&N^2&N^2&N^3 \end{pmatrix}=N^5(N^2-1)^4(N^2-2) [[/math]]

with the matrices being written by using the lexicographic order on [math]NC_2(2k)[/math].

Show Proof

The formula at [math]k=2[/math], where [math]NC_2(4)=\{\sqcap\sqcap,\bigcap\hskip-4.9mm{\ }_\cap\,\}[/math], is clear. At [math]k=3[/math] however, things are tricky. We have [math]NC(3)=\{|||,\sqcap|,\sqcap\hskip-3.2mm{\ }_|\,,|\sqcap,\sqcap\hskip-0.7mm\sqcap\}[/math], and the corresponding Gram matrix and its determinant are, according to Theorem 13.30:

[[math]] \det\begin{pmatrix} N^3&N^2&N^2&N^2&N\\ N^2&N^2&N&N&N\\ N^2&N&N^2&N&N\\ N^2&N&N&N^2&N\\ N&N&N&N&N \end{pmatrix}=N^5(N-1)^4(N-2) [[/math]]

By using Proposition 13.32, the Gram determinant of [math]NC_2(6)[/math] is given by:

[[math]] \begin{eqnarray*} \det(G_{6N}) &=&\frac{1}{N^2\sqrt{N}}\times N^{10}(N^2-1)^4(N^2-2)\times\frac{1}{N^2\sqrt{N}}\\ &=&N^5(N^2-1)^4(N^2-2) \end{eqnarray*} [[/math]]

Thus, we have obtained the formula in the statement.

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In general, such tricks won't work, because [math]NC(k)[/math] is strictly smaller than [math]P(k)[/math] at [math]k\geq4[/math]. However, following Di Francesco , we have the following result:

Theorem

The determinant of the Gram matrix for [math]O_N^+[/math] is given by

[[math]] \det(G_{kN})=\prod_{r=1}^{[k/2]}P_r(N)^{d_{k/2,r}} [[/math]]

where [math]P_r[/math] are the Chebycheff polynomials, given by

[[math]] P_0=1\quad,\quad P_1=X\quad,\quad P_{r+1}=XP_r-P_{r-1} [[/math]]

and [math]d_{kr}=f_{kr}-f_{k,r+1}[/math], with [math]f_{kr}[/math] being the following numbers, depending on [math]k,r\in\mathbb Z[/math],

[[math]] f_{kr}=\binom{2k}{k-r}-\binom{2k}{k-r-1} [[/math]]

with the convention [math]f_{kr}=0[/math] for [math]k\notin\mathbb Z[/math].

Show Proof

This is something quite technical, obtained by using a decomposition as follows of the Gram matrix [math]G_{kN}[/math], with the matrix [math]T_{kN}[/math] being lower triangular:

[[math]] G_{kN}=T_{kN}T_{kN}^t [[/math]]

Thus, a bit as in the proof of Theorem 13.30, we obtain the result, but the problem lies however in the construction of [math]T_{kN}[/math], which is non-trivial. See .

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We refer to for further details regarding the above result, including a short proof, based on the bipartite planar algebra combinatorics developed by Jones in . Let us also mention that the Chebycheff polynomials have something to do with all this due to the fact that these are the orthogonal polynomials for the Wigner law. See .

Moving ahead now, regarding [math]S_N^+[/math], we have here the following formula, which is quite similar, obtained via shrinking, also from Di Francesco :

Theorem

The determinant of the Gram matrix for [math]S_N^+[/math] is given by

[[math]] \det(G_{kN})=(\sqrt{N})^{a_k}\prod_{r=1}^kP_r(\sqrt{N})^{d_{kr}} [[/math]]

where [math]P_r[/math] are the Chebycheff polynomials, given by

[[math]] P_0=1\quad,\quad P_1=X\quad,\quad P_{r+1}=XP_r-P_{r-1} [[/math]]

and [math]d_{kr}=f_{kr}-f_{k,r+1}[/math], with [math]f_{kr}[/math] being the following numbers, depending on [math]k,r\in\mathbb Z[/math],

[[math]] f_{kr}=\binom{2k}{k-r}-\binom{2k}{k-r-1} [[/math]]

with the convention [math]f_{kr}=0[/math] for [math]k\notin\mathbb Z[/math], and where [math]a_k=\sum_{\pi\in \mathcal P(k)}(2|\pi|-k)[/math].

Show Proof

This follows indeed from Theorem 13.34, by using Proposition 13.32.

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We refer to , and related papers, for more on the above.

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].