4c. Diagrams, easiness

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In view of the above results, no matter on what we want to do with our group, we must compute the spaces [math]Fix(v^{\otimes k})[/math]. It is technically convenient to slightly enlarge the class of spaces to be computed, by talking about Tannakian categories, as follows:

Definition

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

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

where the representations [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 representations.

Let us make a summary of what we have so far, regarding these spaces [math]C_G(k,l)[/math]. In order to formulate our result, let us start with the following definition:

Definition

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

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

satisfying the following conditions:

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

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

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

With these conventions, we have the following result, summarizing our knowledge on the subject, coming from the results established in the above:

Theorem

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

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

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

Show Proof

We know that the fundamental representation [math]v[/math] acts on the Hilbert space [math]H=\mathbb C^N[/math], and that its conjugate [math]\bar{v}[/math] acts on the Hilbert space [math]\bar{H}=\mathbb C^N[/math]. Now by multiplicativity we conclude that any Peter-Weyl representation [math]v^{\otimes k}[/math] acts on the Hilbert space [math]H^{\otimes k}[/math], and so that we have embeddings as in Definition 4.20, as follows:

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

Regarding now the fact that the axioms (1-6) in Definition 4.20 are indeed satisfied, this is something that we basically already know. To be more precise, (1-4) are clear, and (5) follows from the fact that each element [math]g\in G[/math] is a unitary, which gives:

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

As for (6), this is something trivial, coming from the fact that the matrix coefficients [math]g\to g_{ij}[/math] and their complex conjugates [math]g\to\bar{g}_{ij}[/math] commute with each other.

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Our purpose now will be that of showing that any closed subgroup [math]G\subset U_N[/math] is uniquely determined by its Tannakian category [math]C_G=(C_G(k,l))[/math]. This result, known as Tannakian duality, is something quite deep, and extremely useful. Indeed, the idea is that what we would have here is a “linearization” of [math]G[/math], allowing us to do combinatorics, and to ultimately reach to concrete and powerful results, regarding [math]G[/math] itself. We first have:

Theorem

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

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

produces a closed subgroup [math]G_C\subset U_N[/math].

Show Proof

This is something elementary, with the fact that the closed subset [math]G_C\subset U_N[/math] constructed in the statement is indeed stable under the multiplication, unit and inversion operation for the unitary matrices [math]g\in U_N[/math] being clear from definitions.

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We can now formulate the Tannakian duality result, as follows:

Theorem

The above Tannakian constructions

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

are bijective, and inverse to each other.

Show Proof

This is something quite technical, obtained by doing some abstract algebra, and for details here, we refer to the Tannakian duality literature. The whole subject is actually, in modern times, for the most part of quantum algebra, and you can consult here , , both quantum group papers, for details on the above.

■

In order to reach now to more concrete things, following Brauer's philosophy in , and more specifically the more modern paper , based on it, we have:

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].

There are many examples of such categories, as for instance the category of all pairings [math]P_2[/math], or of all matching pairings [math]\mathcal P_2[/math]. We will be back to examples in a moment.

Let us formulate as well the following definition, also from :

Definition

Given a partition [math]\pi\in P(k,l)[/math] and an integer [math]N\in\mathbb N[/math], we can construct a linear map between tensor powers of [math]\mathbb C^N[/math],

[[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,

[[math]] \delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}\in\{0,1\} [[/math]]

whose values depend on whether the indices fit or not.

To be more precise, we put the indices of [math]i,j[/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]. The relation with the Tannakian categories comes from:

Proposition

The assignement [math]\pi\to T_\pi[/math] is categorical, in the sense that

[[math]] T_\pi\otimes T_\nu=T_{[\pi\nu]}\quad,\quad T_\pi T_\nu=N^{c(\pi,\nu)}T_{[^\nu_\pi]}\quad,\quad T_\pi^*=T_{\pi^*} [[/math]]

where [math]c(\pi,\nu)[/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 can be checked as follows:

[[math]] \begin{eqnarray*} &&(T_\pi\otimes T_\nu)(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_\nu\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\nu]}\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\nu]}(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 can be checked as follows:

[[math]] \begin{eqnarray*} &&T_\pi T_\nu(e_{i_1}\otimes\ldots\otimes e_{i_p})\\ &=&\sum_{j_1\ldots j_q}\delta_\nu\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,\nu)}\delta_{[^\nu_\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,\nu)}T_{[^\nu_\pi]}(e_{i_1}\otimes\ldots\otimes e_{i_p}) \end{eqnarray*} [[/math]]

(3) Finally, the involution axiom can be checked as follows:

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

Theorem

Each category of partitions [math]D=(D(k,l))[/math] produces a family of compact groups [math]G=(G_N)[/math], with [math]G_N\subset_vU_N[/math], via the formula

[[math]] Hom(v^{\otimes k},v^{\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 4.25, and then the collection of linear spaces in the statement, namely:

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

According to Proposition 4.26, and to our axioms for the categories of partitions, from Definition 4.24, this collection of spaces [math]C=(C(k,l))[/math] satisfies the axioms for the Tannakian categories, from Definition 4.20. Thus the Tannakian duality result, Theorem 4.23, applies, and provides us with a closed subgroup [math]G_N\subset_vU_N[/math] such that:

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

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

■

We can now formulate a key definition, as follows:

Definition

A closed subgroup [math]G\subset_vU_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].

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:

• [math]U_N[/math] is easy, coming from the category of 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 is something very standard, the idea being as follows:

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

■

Beyond this, a first natural question is that of computing the easy group associated to the category [math]P[/math] itself, and we have here the following Brauer type theorem:

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 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 partition [math]\mu\in P(2,1)[/math], we have:

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

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

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

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

[[math]] T_\mu\in Hom(v^{\otimes 2},v)\iff v_{ij}v_{ik}=\delta_{jk}v_{ij},\forall i,j,k [[/math]]

In other words, the elements [math]v_{ij}[/math] must be projections, and these projections must be pairwise orthogonal on the rows of [math]v=(v_{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 claimed.

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In fact, we have the following general easiness result, from , regarding the series of complex reflection groups [math]H_N^s\subset U_N[/math], that we introduced in chapter 2:

Theorem

The 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 [math]\#\circ=\#\bullet(s)[/math] in each block. In particular:

• [math]S_N[/math] is easy, coming from the category [math]P[/math].
• [math]H_N[/math] is easy, coming from the category [math]P_{even}[/math].
• [math]K_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[/math], from Theorem 4.30. 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 [math]P^s[/math] in the statement, implements the relations producing the subgroup [math]H_N^s\subset U_N[/math]. As for the last assertions, these follow from the following observations:

(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.

■

Let us go back now to probability questions, with the aim of applying the above abstract theory, to questions regarding characters. The situation here is as follows:

(1) Given a closed subgroup [math]G\subset_vU_N[/math], we know from Peter-Weyl that the moments of the main character count the fixed points of the representations [math]v^{\otimes k}[/math].

(2) On the other hand, assuming that our group [math]G\subset_vU_N[/math] is easy, coming from a category of partitions [math]D=(D(k,l))[/math], the space formed by these fixed points is spanned by the following vectors, indexed by partitions [math]\pi[/math] belonging to the set [math]D(k)=D(0,k)[/math]:

[[math]] \xi_\pi=\sum_{i_1\ldots i_k}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\end{pmatrix}e_{i_1}\otimes\ldots\otimes e_{i_k} [[/math]]

(3) Thus, we are left with investigating linear independence questions for the vectors [math]\xi_\pi[/math], and once these questions solved, to compute the moments of [math]\chi[/math].

In order to investigate linear independence questions for the vectors [math]\xi_\pi[/math], we will use the Gram matrix of these vectors. Let us begin with some standard definitions:

Definition

Let [math]P(k)[/math] be the set of partitions of [math]\{1,\ldots,k\}[/math], and let [math]\pi,\nu\in P(k)[/math].

• We write [math]\pi\leq\nu[/math] if each block of [math]\pi[/math] is contained in a block of [math]\nu[/math].
• We let [math]\pi\vee\nu\in P(k)[/math] be the partition obtained by superposing [math]\pi,\nu[/math].

As an illustration here, at [math]k=2[/math] we have [math]P(2)=\{||,\sqcap\}[/math], and the order is:

[[math]] ||\leq\sqcap [[/math]]

At [math]k=3[/math] we have [math]P(3)=\{|||,\sqcap|,\sqcap\hskip-3.2mm{\ }_|\,,|\sqcap,\sqcap\hskip-0.7mm\sqcap\}[/math], and the order relation is as follows:

[[math]] |||\leq\sqcap|,\sqcap\hskip-3.2mm{\ }_|\,,|\sqcap\leq\sqcap\hskip-0.7mm\sqcap [[/math]]

Observe also that we have [math]\pi,\nu\leq\pi\vee\nu[/math]. In fact, [math]\pi\vee\nu[/math] is the smallest partition with this property, called supremum of [math]\pi,\nu[/math]. Now back to the easy groups, we have:

Proposition

The Gram matrix [math]G_{kN}(\pi,\nu)= \lt \xi_\pi,\xi_\nu \gt [/math] is given by

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

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

Show Proof

According to our formula of the vectors [math]\xi_\pi[/math], we have:

[[math]] \begin{eqnarray*} \lt \xi_\pi,\xi_\nu \gt &=&\sum_{i_1\ldots i_k}\delta_\pi(i_1,\ldots,i_k)\delta_\nu(i_1,\ldots,i_k)\\ &=&\sum_{i_1\ldots i_k}\delta_{\pi\vee\nu}(i_1,\ldots,i_k)\\ &=&N^{|\pi\vee\nu|} \end{eqnarray*} [[/math]]

Thus, we have obtained the formula in the statement.

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In order to study the Gram matrix, and more specifically to compute its determinant, we will need several standard facts about the partitions. We first have:

Definition

The Möbius function of any lattice, and so of [math]P[/math], is given by

[[math]] \mu(\pi,\nu)=\begin{cases} 1&{\rm if}\ \pi=\nu\\ -\sum_{\pi\leq\tau \lt \nu}\mu(\pi,\tau)&{\rm if}\ \pi \lt \nu\\ 0&{\rm if}\ \pi\not\leq\nu \end{cases} [[/math]]

with the construction being performed by recurrence.

As an illustration here, let us go back to the set of 2-point partitions, [math]P(2)=\{||,\sqcap\}[/math]. Here we have by definition:

[[math]] \mu(||,||)=\mu(\sqcap,\sqcap)=1 [[/math]]

Also, we know that we have [math]|| \lt \sqcap[/math], with no intermediate partition in between, and so the above recurrence procedure gives the following formular:

[[math]] \mu(||,\sqcap)=-\mu(||,||)=-1 [[/math]]

Finally, we have [math]\sqcap\not\leq||[/math], which gives [math]\mu(\sqcap,||)=0[/math]. Thus, as a conclusion, the Möbius matrix [math]M_{\pi\nu}=\mu(\pi,\nu)[/math] of the lattice [math]P(2)=\{||,\sqcap\}[/math] is as follows:

[[math]] M=\begin{pmatrix}1&-1\\ 0&1\end{pmatrix} [[/math]]

The interest in the Möbius function comes from the Möbius inversion formula:

[[math]] f(\nu)=\sum_{\pi\leq\nu}g(\pi)\implies g(\nu)=\sum_{\pi\leq\nu}\mu(\pi,\nu)f(\pi) [[/math]]

In linear algebra terms, the statement and proof of this formula are as follows:

Theorem

The inverse of the adjacency matrix of [math]P[/math], given by

[[math]] A_{\pi\nu}=\begin{cases} 1&{\rm if}\ \pi\leq\nu\\ 0&{\rm if}\ \pi\not\leq\nu \end{cases} [[/math]]

is the Möbius matrix of [math]P[/math], given by [math]M_{\pi\nu}=\mu(\pi,\nu)[/math].

Show Proof

This is well-known, coming for instance from the fact that [math]A[/math] is upper triangular. Thus, when inverting, we are led into the recurrence from Definition 4.34.

■

As an illustration here, for [math]P(2)[/math] the formula [math]M=A^{-1}[/math] appears as follows:

[[math]] \begin{pmatrix}1&-1\\ 0&1\end{pmatrix}= \begin{pmatrix}1&1\\ 0&1\end{pmatrix}^{-1} [[/math]]

Now back to our Gram matrix considerations, we have the following result:

Proposition

The Gram matrix is given by [math]G_{kN}=AL[/math], where

[[math]] L(\pi,\nu)= \begin{cases} N(N-1)\ldots(N-|\pi|+1)&{\rm if}\ \nu\leq\pi\\ 0&{\rm otherwise} \end{cases} [[/math]]

and where [math]A=M^{-1}[/math] is the adjacency matrix of [math]P(k)[/math].

Show Proof

We have the following computation:

[[math]] \begin{eqnarray*} N^{|\pi\vee\nu|} &=&\#\left\{i_1,\ldots,i_k\in\{1,\ldots,N\}\Big|\ker i\geq\pi\vee\nu\right\}\\ &=&\sum_{\tau\geq\pi\vee\nu}\#\left\{i_1,\ldots,i_k\in\{1,\ldots,N\}\Big|\ker i=\tau\right\}\\ &=&\sum_{\tau\geq\pi\vee\nu}N(N-1)\ldots(N-|\tau|+1) \end{eqnarray*} [[/math]]

According to Proposition 4.33 and to the definition of [math]A,L[/math], this formula reads:

[[math]] (G_{kN})_{\pi\nu} =\sum_{\tau\geq\pi}L_{\tau\nu} =\sum_\tau A_{\pi\tau}L_{\tau\nu} =(AL)_{\pi\nu} [[/math]]

Thus, we obtain the formula in the statement.

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With the above result in hand, we can now investigate the linear independence properties of the vectors [math]\xi_\pi[/math]. To be more precise, we have the following result:

Theorem

The determinant of the Gram matrix [math]G_{kN}[/math] is given by

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

and in particular, for [math]N\geq k[/math], the vectors [math]\{\xi_\pi|\pi\in P(k)\}[/math] are linearly independent.

Show Proof

According to the formula in Proposition 4.36, we have:

[[math]] \det(G_{kN})=\det(A)\det(L) [[/math]]

Now if we order [math]P(k)[/math] as usual, with respect to the number of blocks, and then lexicographically, we see that [math]A[/math] is upper triangular, and that [math]L[/math] is lower triangular. Thus [math]\det(A)[/math] can be computed simply by making the product on the diagonal, and we obtain [math]1[/math]. As for [math]\det(L)[/math], this can computed as well by making the product on the diagonal, and we obtain the number in the statement, with the technical remark that in the case [math]N \lt k[/math] the convention is that we obtain a vanishing determinant.

■

We refer to , , for more on all this, and we will be back to this interesting topic later on in this book. Now back to the laws of characters, we can formulate:

Proposition

For an easy group [math]G=(G_N)[/math], coming from a category of partitions [math]D=(D(k,l))[/math], the asymptotic moments of the main character are given by

[[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 follows indeed from the Peter-Weyl theory, by using the linear independence result for the vectors [math]\xi_\pi[/math] coming from Theorem 4.37.

■

With these preliminaries in hand, we can now state and prove:

Theorem

In the [math]N\to\infty[/math] limit, the laws of the main character for the main easy groups, real and complex, and discrete and continuous, are as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ K_N\ar[r]&U_N\\ H_N\ar[u]\ar[r]&O_N\ar[u]}\qquad \item[a]ymatrix@R=25pt@C=50pt{\\:} \qquad \item[a]ymatrix@R=50pt@C=50pt{ B_1\ar[r]&G_1\\ b_1\ar[u]\ar[r]&g_1\ar[u]} [[/math]]

with these laws, namely the real and complex Gaussian and Bessel laws, being the main limiting laws in real and complex, and discrete and continuous probability.

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=16mm{ \mathcal P_{even}\ar[d]&\mathcal P_2\ar[l]\ar[d]\\ P_{even}&P_2\ar[l]} [[/math]]

Thus, we can use Proposition 4.38, are we are led into counting partitions, and then recovering the measures via their moments, and this leads to the result.

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General references

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