8a. Block modifications

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We discuss in this chapter some extensions and unifications of our results from chapter 7. As before with the usual or block-transposed Wishart matrices, there will be some non-trivial combinatorics here, that we will fully understand only later, in chapters 9-12, when doing free probability. Thus, the material below will be an introduction to this.

Let us begin with some general block modification considerations, following and the more recent papers , . We have the following construction:

Definition

Given a complex Wishart [math]dn\times dn[/math] matrix, appearing as

[[math]] W=YY^*\in M_{dn}(L^\infty(X)) [[/math]]

with [math]Y[/math] being a complex Gaussian [math]dn\times dm[/math] matrix, and a linear map

[[math]] \varphi:M_n(\mathbb C)\to M_n(\mathbb C) [[/math]]

we consider the following matrix, obtained by applying [math]\varphi[/math] to the [math]n\times n[/math] blocks of [math]W[/math],

[[math]] \widetilde{W}=(id\otimes\varphi)W\in M_{dn}(L^\infty(X)) [[/math]]

and call it block-modified Wishart matrix.

Here we are using some standard tensor product identifications, the details being as follows. Let [math]Y[/math] be a complex Gaussian [math]dn\times dm[/math] matrix, as above:

[[math]] Y\in M_{dn\times dm}(L^\infty(X)) [[/math]]

We can then form the corresponding complex Wishart matrix, as follows:

[[math]] W=YY^*\in M_{dn}(L^\infty(X)) [[/math]]

The size of this matrix being a composite number, [math]N=dn[/math], we can regard this matrix as being a [math]n\times n[/math] matrix, with random [math]d\times d[/math] matrices as entries. Equivalently, by using standard tensor product notations, this amounts in regarding [math]W[/math] as follows:

[[math]] W\in M_d(L^\infty(X))\otimes M_n(\mathbb C) [[/math]]

With this done, we can come up with our linear map, namely:

[[math]] \varphi:M_n(\mathbb C)\to M_n(\mathbb C) [[/math]]

We can apply [math]\varphi[/math] to the tensors on the right, and we obtain a matrix as follows:

[[math]] \widetilde{W}=(id\otimes\varphi)W\in M_d(L^\infty(X))\otimes M_n(\mathbb C) [[/math]]

Finally, we can forget now about tensors, and as a conclusion to all this, we have constructed a matrix as follows, that we can call block-modified Wishart matrix:

[[math]] \widetilde{W}\in M_{dn}(L^\infty(X)) [[/math]]

In practice now, what we mostly need for fully understanding Definition 8.1 are examples. Following Aubrun , and the series of papers by Collins and Nechita , , , we have the following basic examples, for our general construction:

Definition

We have the following examples of block-modified Wishart matrices [math]\widetilde{W}=(id\otimes\varphi)W[/math], coming from various linear maps [math]\varphi:M_n(\mathbb C)\to M_n(\mathbb C)[/math]:

• Wishart matrices: [math]\widetilde{W}=W[/math], obtained via [math]\varphi=id[/math].
• Aubrun matrices: [math]\widetilde{W}=(id\otimes t)W[/math], with [math]t[/math] being the transposition.
• Collins-Nechita one: [math]\widetilde{W}=(id\otimes\varphi)W[/math], with [math]\varphi=tr(.)1[/math].
• Collins-Nechita two: [math]\widetilde{W}=(id\otimes\varphi)W[/math], with [math]\varphi[/math] erasing the off-diagonal part.

These examples, whose construction is something very elementary, appear in a wide context of interesting situations, for the most in connection with various questions in quantum physics , , , , . They will actually serve as a main motivation for what we will be doing, in what follows. More on this later.

Getting back now to the general case, that of Definition 8.1 as stated, the linear map [math]\varphi:M_n(\mathbb C)\to M_n(\mathbb C)[/math] there is certainly useful for understanding the construction of the block-modified Wishart matrix [math]\widetilde{W}=(id\otimes\varphi)W[/math], as illustrated by the above examples. In practice, however, we would like to have as block-modification “data” something more concrete, such as a usual matrix. To be more precise, we would like to use:

Proposition

We have a correspondence between linear maps

[[math]] \varphi:M_n(\mathbb C)\to M_n(\mathbb C) [[/math]]

and square matrices [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math], given by the formula

[[math]] \Lambda_{ab,cd}=\varphi(e_{ac})_{bd} [[/math]]

where [math]e_{ab}\in M_n(\mathbb C)[/math] are the standard generators of the matrix algebra [math]M_n(\mathbb C)[/math], given by the formula [math]e_{ab}:e_b\to e_a[/math], with [math]\{e_1,\ldots,e_n\}[/math] being the standard basis of [math]\mathbb C^n[/math].

Show Proof

This is standard linear algebra. Given a linear map [math]\varphi:M_n(\mathbb C)\to M_n(\mathbb C)[/math], we can associated to it numbers [math]\Lambda_{ab,cd}\in\mathbb C[/math] by the formula in the statement, namely:

[[math]] \Lambda_{ab,cd}=\varphi(e_{ac})_{bd} [[/math]]

Now by using these [math]n^4[/math] numbers, we can construct a [math]n^2\times n^2[/math] matrix, as follows:

[[math]] \Lambda=\sum_{abcd}\Lambda_{ab,cd}e_{ac}\otimes e_{bd}\in M_n(\mathbb C)\otimes M_n(\mathbb C) [[/math]]

Thus, we have constructed a correspondence [math]\varphi\to\Lambda[/math], and since this correspondence is injective, and the dimensions match, this correspondence is bijective, as claimed.

■

Now by getting back to the block-modified Wishart matrices, we have:

Proposition

Given a Wishart [math]dn\times dn[/math] matrix [math]W=YY^*[/math], and a linear map

[[math]] \varphi:M_n(\mathbb C)\to M_n(\mathbb C) [[/math]]

the entries of the corresponding block-modified matrix [math]\widetilde{W}=(id\otimes\varphi)W[/math] are given by

[[math]] \widetilde{W}_{ia,jb}=\sum_{cd}\Lambda_{ca,db}W_{ic,jd} [[/math]]

where [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math] is the square matrix associated to [math]\varphi[/math], as above.

Show Proof

Again, this is trivial linear algebra, coming from the following computation:

[[math]] \widetilde{W}_{ia,jb} =\sum_{cd}W_{ic,jd}\varphi(e_{cd})_{ab} =\sum_{cd}\Lambda_{ca,db}W_{ic,jd} [[/math]]

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

■

At the level of the main examples, from Definition 8.2, the very basic linear maps [math]\varphi:M_n(\mathbb C)\to M_n(\mathbb C)[/math] used there can only correspond to some basic examples of matrices [math]\Lambda\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math], via the correspondence in Proposition 8.3. This is indeed the case, and in order to clarify this, and at a rather conceptual level, let us formulate, inspired by the representation theory material from chapter 4, the following definition:

Definition

Let [math]P(k,l)[/math] be the set of partitions between an upper row of [math]k[/math] points, and a lower row of [math]l[/math] points. Associated to any [math]\pi\in P(k,l)[/math] is the linear map

[[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]]

between tensor powers of [math]\mathbb C^N[/math], called “easy”, with the Kronecker type symbol on the right being given by [math]\delta_\pi=1[/math] when the indices fit, and [math]\delta_\pi=0[/math] otherwise.

Observe the obvious connection with notion of easy group, from chapter 4, the point being that a closed subgroup [math]G\subset U_N[/math] is easy precisely when its Tannakian category [math]C_G=(C_G(k,l))[/math] with [math]C_G(k,l)\subset\mathcal L((\mathbb C^N)^k,(\mathbb C^N)^l)[/math] is spanned by easy maps.

For our purposes here, we will need a slight modification of Definition 8.5, as follows:

Definition

Associated to any partition [math]\pi\in P(2s,2s)[/math] is the linear map

[[math]] \varphi_\pi(e_{a_1\ldots a_s,c_1\ldots c_s})=\sum_{b_1\ldots b_s}\sum_{d_1\ldots d_s}\delta_\pi\begin{pmatrix}a_1&\ldots&a_s&c_1&\ldots&c_s\\ b_1&\ldots&b_s&d_1&\ldots&d_s\end{pmatrix}e_{b_1\ldots b_s,d_1\ldots d_s} [[/math]]

obtained from [math]T_\pi[/math] by contracting all the tensors, via the operation

[[math]] e_{i_1}\otimes\ldots\otimes e_{i_{2s}}\to e_{i_1\ldots i_s,i_{s+1}\ldots i_{2s}} [[/math]]

with [math]\{e_1,\ldots,e_N\}[/math] standing as usual for the standard basis of [math]\mathbb C^N[/math].

In relation with our Wishart matrix considerations, the point is that the above linear map [math]\varphi_\pi[/math] can be viewed as a “block-modification” map, as follows:

[[math]] \varphi_\pi:M_{N^s}(\mathbb C)\to M_{N^s}(\mathbb C) [[/math]]

As an illustration, let us discuss the case [math]s=1[/math]. There are 15 partitions [math]\pi\in P(2,2)[/math], and among them, the most “basic” are the 4 partitions [math]\pi\in P_{even}(2,2)[/math]. We have:

Theorem

The partitions [math]\pi\in P_{even}(2,2)[/math] are as follows,

[[math]] \pi_1=\begin{bmatrix}\circ&\bullet\\ \circ&\bullet\end{bmatrix}\quad,\quad \pi_2=\begin{bmatrix}\circ&\bullet\\ \bullet&\circ\end{bmatrix}\quad,\quad \pi_3=\begin{bmatrix}\circ&\circ\\ \bullet&\bullet\end{bmatrix}\quad,\quad \pi_4=\begin{bmatrix}\circ&\circ\\ \circ&\circ\end{bmatrix} [[/math]]

with the associated linear maps [math]\varphi_\pi:M_n(\mathbb C)\to M_n(\mathbb C)[/math] being as follows,

[[math]] \varphi_1(A)=A\quad,\quad \varphi_2(A)=A^t\quad,\quad \varphi_3(A)=Tr(A)1\quad,\quad \varphi_4(A)=A^\delta [[/math]]

and the associated square matrices [math]\Lambda_\pi\in M_n(\mathbb C)\otimes M_n(\mathbb C)[/math] being as follows,

[[math]] \Lambda^1_{ab,cd}=\delta_{ab}\delta_{cd}\quad,\quad \Lambda^2_{ab,cd}=\delta_{ad}\delta_{bc}\quad,\quad \Lambda^3_{ab,cd}=\delta_{ac}\delta_{bd}\quad,\quad \Lambda^4_{ab,cd}=\delta_{abcd} [[/math]]

producing the main examples of block-modified Wishart matrices, from Definition 8.2.

Show Proof

This is something elementary, coming from the formula in Definition 8.6. Indeed, in the case [math]s=1[/math], that we are interested in here, this formula becomes:

[[math]] \varphi_\pi(e_{ac})=\sum_{bd}\delta_\pi\begin{pmatrix}a&c\\ b&d\end{pmatrix}e_{bd} [[/math]]

Now in the case of the 4 partitions in the statement, such maps are given by:

[[math]] \varphi_1(e_{ac})=e_{ac}\quad,\quad \varphi_2(e_{ac})=e_{ca}\quad,\quad \varphi_3(e_{ac})=\delta_{ac}\sum_be_{bb}\quad,\quad \varphi_4(e_{ac})=\delta_{ac}e_{aa} [[/math]]

Thus, we obtain the formulae in the statement. Regarding now the associated square matrices, appearing via [math]\Lambda_{ab,cd}=\varphi(e_{ac})_{bd}[/math], these are given by:

[[math]] \Lambda^1_{ab,cd}=\delta_{ab}\delta_{cd}\quad,\quad \Lambda^2_{ab,cd}=\delta_{ad}\delta_{bc}\quad,\quad \Lambda^3_{ab,cd}=\delta_{ac}\delta_{bd}\quad,\quad \Lambda^4_{ab,cd}=\delta_{abcd} [[/math]]

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

■

As a conclusion so far to what we did in this chapter, we have a nice definition for the block-modified Wishart matrices, and then a fine-tuning of this definition, using easy maps, which in the simplest case, that of the 4 partitions [math]\pi\in P_{even}(2,2)[/math], produces the main 4 examples of block-modified Wishart matrices. The idea in what follows will be that of doing the combinatorics, a bit as in chapter 7, as to extend the results there.

General references

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