Case m = 1 and representation S^k Cⁿ

It is especially instructive to consider the special case m = 1; in this case we have x_i1, which is abbreviated as x_i:

${\displaystyle E_{ij}=x_{i}{\frac {\partial }{\partial x_{j}}}.}$

In particular, for the polynomials of the first degree it is seen that:

${\displaystyle E_{ij}x_{k}=\delta _{jk}x_{i}.~~~~~~~~~~~~~~}$

Hence the action of ${\displaystyle E_{ij}}$ restricted to the space of first-order polynomials is exactly the same as the action of matrix units ${\displaystyle e_{ij}}$ on vectors in ${\displaystyle \mathbb {C} ^{n}}$. So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation of the Lie algebra ${\displaystyle {\mathfrak {gl}}_{n}}$, which we identified with the standard representation in ${\displaystyle \mathbb {C} ^{n}}$. Going further, it is seen that the differential operators ${\displaystyle E_{ij}}$ preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation of the Lie algebra ${\displaystyle {\mathfrak {gl}}_{n}}$. One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power ${\displaystyle S^{k}\mathbb {C} ^{n}}$ of the standard representation ${\displaystyle \mathbb {C} ^{n}}$.

One can also easily identify the highest weight structure of these representations. The monomial ${\displaystyle x_{1}^{k}}$ is a highest weight vector, indeed: ${\displaystyle E_{ij}x_{1}^{k}=0}$ for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: ${\displaystyle E_{ii}x_{1}^{k}=k\delta _{i1}x_{1}^{k}}$.

Such representation is sometimes called bosonic representation of ${\displaystyle {\mathfrak {gl}}_{n}}$. Similar formulas ${\displaystyle E_{ij}=\psi _{i}{\frac {\partial }{\partial \psi _{j}}}}$ define the so-called fermionic representation, here ${\displaystyle \psi _{i}}$ are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to ${\displaystyle \Lambda ^{k}\mathbb {C} ^{n}}$ i.e. anti-symmetric tensor power of ${\displaystyle \mathbb {C} ^{n}}$. Highest weight of such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representations of ${\displaystyle {\mathfrak {gl}}_{n}}$.

The universal enveloping algebra ${\displaystyle U({\mathfrak {gl}}_{n})}$ and its center

An interesting property of the Capelli determinant is that it commutes with all operators E_ij, that is, the commutator ${\displaystyle [E_{ij},\det(E+(n-i)\delta _{ij})]=0}$ is equal to zero. It can be generalized:

Consider any elements E_ij in any ring, such that they satisfy the commutation relation ${\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}}$, (so they can be differential operators above, matrix units e_ij or any other elements) define elements C_k as follows:

${\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\sum _{k=n-1,\dots ,0}t^{[k]}C_{k},~~~~~}$

where ${\displaystyle t^{[k]}=t(t+1)\cdots (t+k-1),}$

then:

• elements C_k commute with all elements E_ij
• elements C_k can be given by the formulas similar to the commutative case:

${\displaystyle C_{k}=\sum _{I=(i_{1}<i_{2}<\cdots <i_{k})}\det(E+(k-i)\delta _{ij})_{II},}$

i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction ${\displaystyle +(k-i)\delta _{ij}}$. In particular element C₀ is the Capelli determinant considered above.

These statements are interrelated with the Capelli identity, as will be discussed below, and similarly to it the direct few lines short proof does not seem to exist, despite the simplicity of the formulation.

The universal enveloping algebra ${\displaystyle U({\mathfrak {gl}}_{n})}$ can be defined as an algebra generated by E_ij subject to the relations

${\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}}$

alone. The proposition above shows that elements C_k belong to the center of ${\displaystyle U({\mathfrak {gl}}_{n})}$. It can be shown that they actually are free generators of the center of ${\displaystyle U({\mathfrak {gl}}_{n})}$. They are sometimes called Capelli generators. The Capelli identities for them will be discussed below.

Consider an example for n = 2.

${\displaystyle {\begin{aligned}{}\quad {\begin{vmatrix}t+E_{11}+1&E_{12}\\E_{21}&t+E_{22}\end{vmatrix}}&=(t+E_{11}+1)(t+E_{22})-E_{21}E_{12}\\&=t(t+1)+t(E_{11}+E_{22})+E_{11}E_{22}-E_{21}E_{12}+E_{22}.\end{aligned}}}$

It is immediate to check that element ${\displaystyle (E_{11}+E_{22})}$ commute with ${\displaystyle E_{ij}}$. (It corresponds to an obvious fact that the identity matrix commute with all other matrices). More instructive is to check commutativity of the second element with ${\displaystyle E_{ij}}$. Let us do it for ${\displaystyle E_{12}}$:

${\displaystyle [E_{12},E_{11}E_{22}-E_{21}E_{12}+E_{22}]}$

${\displaystyle =[E_{12},E_{11}]E_{22}+E_{11}[E_{12},E_{22}]-[E_{12},E_{21}]E_{12}-E_{21}[E_{12},E_{12}]+[E_{12},E_{22}]}$

${\displaystyle =-E_{12}E_{22}+E_{11}E_{12}-(E_{11}-E_{22})E_{12}-0+E_{12}}$

${\displaystyle =-E_{12}E_{22}+E_{22}E_{12}+E_{12}=-E_{12}+E_{12}=0.}$

We see that the naive determinant ${\displaystyle E_{11}E_{22}-E_{21}E_{12}}$ will not commute with ${\displaystyle E_{12}}$ and the Capelli's correction ${\displaystyle +E_{22}}$ is essential to ensure the centrality.

General m and dual pairs

Let us return to the general case:

${\displaystyle E_{ij}=\sum _{a=1}^{m}x_{ia}{\frac {\partial }{\partial x_{ja}}},}$

for arbitrary n and m. Definition of operators E_ij can be written in a matrix form: ${\displaystyle E=XD^{t}}$, where ${\displaystyle E}$ is ${\displaystyle n\times n}$ matrix with elements ${\displaystyle E_{ij}}$; ${\displaystyle X}$ is ${\displaystyle n\times m}$ matrix with elements ${\displaystyle x_{ij}}$; ${\displaystyle D}$ is ${\displaystyle n\times m}$ matrix with elements ${\displaystyle {\frac {\partial }{\partial x_{ij}}}}$.

Capelli–Cauchy–Binet identities

For general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements of these matrices would commute then one knows that the determinant of E can be expressed by the so-called Cauchy–Binet formula via minors of X and D. An analogue of this formula also exists for matrix E again for the same mild price of the correction ${\displaystyle E\rightarrow (E+(n-i)\delta _{ij})}$:

${\displaystyle \det(E+(n-i)\delta _{ij})=\sum _{I=(1\leq i_{1}<i_{2}<\cdots <i_{n}\leq m)}\det(X_{I})\det(D_{I}^{t})}$,

In particular (similar to the commutative case): if m < n, then ${\displaystyle \det(E+(n-i)\delta _{ij})=0}$; if m = n we return to the identity above.

Let us also mention that similar to the commutative case (see Cauchy–Binet for minors), one can express not only the determinant of E, but also its minors via minors of X and D:

${\displaystyle \det(E+(s-i)\delta _{ij})_{KL}=\sum _{I=(1\leq i_{1}<i_{2}<\cdots <i_{s}\leq m)}\det(X_{KI})\det(D_{IL}^{t})}$,

Here K = (k₁ < k₂ < ... < k_s), L = (l₁ < l₂ < ... < l_s), are arbitrary multi-indexes; as usually ${\displaystyle M_{KL}}$ denotes a submatrix of M formed by the elements M _kalb. Pay attention that the Capelli correction now contains s, not n as in previous formula. Note that for s=1, the correction (s − i) disappears and we get just the definition of E as a product of X and transpose to D. Let us also mention that for generic K,L corresponding minors do not commute with all elements E_ij, so the Capelli identity exists not only for central elements.

As a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention the following:

${\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\sum _{k=n-1,\dots ,0}t^{[k]}\sum _{I,J}\det(X_{IJ})\det(D_{JI}^{t}),}$

where ${\displaystyle I=(1\leq i_{1}<\cdots <i_{k}\leq n),}$ ${\displaystyle J=(1\leq j_{1}<\cdots <j_{k}\leq n)}$. This formula is similar to the commutative case, modula ${\displaystyle +(n-i)\delta _{ij}}$ at the left hand side and t^[n] instead of tⁿ at the right hand side.

Relation to dual pairs

Modern interest in these identities has been much stimulated by Roger Howe who considered them in his theory of reductive dual pairs (also known as Howe duality). To make the first contact with these ideas, let us look more precisely on operators ${\displaystyle E_{ij}}$. Such operators preserve the degree of polynomials. Let us look at the polynomials of degree 1: ${\displaystyle E_{ij}x_{kl}=x_{il}\delta _{jk}}$, we see that index l is preserved. One can see that from the representation theory point of view polynomials of the first degree can be identified with direct sum of the representations ${\displaystyle \mathbb {C} ^{n}\oplus \cdots \oplus \mathbb {C} ^{n}}$, here l-th subspace (l=1...m) is spanned by ${\displaystyle x_{il}}$, i = 1, ..., n. Let us give another look on this vector space:

${\displaystyle \mathbb {C} ^{n}\oplus \cdots \oplus \mathbb {C} ^{n}=\mathbb {C} ^{n}\otimes \mathbb {C} ^{m}.}$

Such point of view gives the first hint of symmetry between m and n. To deepen this idea consider:

${\displaystyle E_{ij}^{\text{dual}}=\sum _{a=1}^{n}x_{ai}{\frac {\partial }{\partial x_{aj}}}.}$

These operators are given by the same formulas as ${\displaystyle E_{ij}}$ modula renumeration ${\displaystyle i\leftrightarrow j}$, hence by the same arguments we can deduce that ${\displaystyle E_{ij}^{\text{dual}}}$ form a representation of the Lie algebra ${\displaystyle {\mathfrak {gl}}_{m}}$ in the vector space of polynomials of x_ij. Before going further we can mention the following property: differential operators ${\displaystyle E_{ij}^{\text{dual}}}$ commute with differential operators ${\displaystyle E_{kl}}$.

The Lie group ${\displaystyle GL_{n}\times GL_{m}}$ acts on the vector space ${\displaystyle \mathbb {C} ^{n}\otimes \mathbb {C} ^{m}}$ in a natural way. One can show that the corresponding action of Lie algebra ${\displaystyle {\mathfrak {gl}}_{n}\times {\mathfrak {gl}}_{m}}$ is given by the differential operators ${\displaystyle E_{ij}~~~~}$ and ${\displaystyle E_{ij}^{\text{dual}}}$ respectively. This explains the commutativity of these operators.

The following deeper properties actually hold true:

• The only differential operators which commute with ${\displaystyle E_{ij}~~~~}$ are polynomials in ${\displaystyle E_{ij}^{\text{dual}}}$, and vice versa.
• Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible representations of ${\displaystyle GL_{n}}$ and ${\displaystyle GL_{m}}$ can be given as follows:

${\displaystyle \mathbb {C} [x_{ij}]=S(\mathbb {C} ^{n}\otimes \mathbb {C} ^{m})=\sum _{D}\rho _{n}^{D}\otimes \rho _{m}^{D'}.}$

The summands are indexed by the Young diagrams D, and representations ${\displaystyle \rho ^{D}}$ are mutually non-isomorphic. And diagram ${\displaystyle {D}}$ determine ${\displaystyle {D'}}$ and vice versa.

• In particular the representation of the big group ${\displaystyle GL_{n}\times GL_{m}}$ is multiplicity free, that is each irreducible representation occurs only one time.

One easily observe the strong similarity to Schur–Weyl duality.

Turnbull's identity for symmetric matrices

Consider symmetric matrices

${\displaystyle X={\begin{vmatrix}x_{11}&x_{12}&x_{13}&\cdots &x_{1n}\\x_{12}&x_{22}&x_{23}&\cdots &x_{2n}\\x_{13}&x_{23}&x_{33}&\cdots &x_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1n}&x_{2n}&x_{3n}&\cdots &x_{nn}\end{vmatrix}},D={\begin{vmatrix}2{\frac {\partial }{\partial x_{11}}}&{\frac {\partial }{\partial x_{12}}}&{\frac {\partial }{\partial x_{13}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\[6pt]{\frac {\partial }{\partial x_{12}}}&2{\frac {\partial }{\partial x_{22}}}&{\frac {\partial }{\partial x_{23}}}&\cdots &{\frac {\partial }{\partial x_{2n}}}\\[6pt]{\frac {\partial }{\partial x_{13}}}&{\frac {\partial }{\partial x_{23}}}&2{\frac {\partial }{\partial x_{33}}}&\cdots &{\frac {\partial }{\partial x_{3n}}}\\[6pt]\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{1n}}}&{\frac {\partial }{\partial x_{2n}}}&{\frac {\partial }{\partial x_{3n}}}&\cdots &2{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}}$

Herbert Westren Turnbull^[7] in 1948 discovered the following identity:

${\displaystyle \det(XD+(n-i)\delta _{ij})=\det(X)\det(D)}$

Combinatorial proof can be found in the paper,^[6] another proof and amusing generalizations in the paper,^[5] see also discussion below.

The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices

Consider antisymmetric matrices

${\displaystyle X={\begin{vmatrix}0&x_{12}&x_{13}&\cdots &x_{1n}\\-x_{12}&0&x_{23}&\cdots &x_{2n}\\-x_{13}&-x_{23}&0&\cdots &x_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\-x_{1n}&-x_{2n}&-x_{3n}&\cdots &0\end{vmatrix}},D={\begin{vmatrix}0&{\frac {\partial }{\partial x_{12}}}&{\frac {\partial }{\partial x_{13}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\[6pt]-{\frac {\partial }{\partial x_{12}}}&0&{\frac {\partial }{\partial x_{23}}}&\cdots &{\frac {\partial }{\partial x_{2n}}}\\[6pt]-{\frac {\partial }{\partial x_{13}}}&-{\frac {\partial }{\partial x_{23}}}&0&\cdots &{\frac {\partial }{\partial x_{3n}}}\\[6pt]\vdots &\vdots &\vdots &\ddots &\vdots \\[6pt]-{\frac {\partial }{\partial x_{1n}}}&-{\frac {\partial }{\partial x_{2n}}}&-{\frac {\partial }{\partial x_{3n}}}&\cdots &0\end{vmatrix}}.}$

Then

${\displaystyle \det(XD+(n-i)\delta _{ij})=\det(X)\det(D).}$

The Caracciolo–Sportiello–Sokal identity for Manin matrices

Consider two matrices M and Y over some associative ring which satisfy the following condition

${\displaystyle [M_{ij},Y_{kl}]=-\delta _{jk}Q_{il}~~~~~}$

for some elements Q_il. Or ”in words”: elements in j-th column of M commute with elements in k-th row of Y unless j = k, and in this case commutator of the elements M_ik and Y_kl depends only on i, l, but does not depend on k.

Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).

Then for the square matrix case

${\displaystyle \det(MY+Q\,\mathrm {diag} (n-1,n-2,\dots ,1,0))=\det(M)\det(Y).~~~~~~~}$

Here Q is a matrix with elements Q_il, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal.

See ^[5] proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.

Obviously the original Cappeli's identity the particular case of this identity. Moreover from this identity one can see that in the original Capelli's identity one can consider elements

${\displaystyle {\frac {\partial }{\partial x_{ij}}}+f_{ij}(x_{11},\dots ,x_{kl},\dots )}$

for arbitrary functions f_ij and the identity still will be true.

The Mukhin–Tarasov–Varchenko identity and the Gaudin model

Permanents, immanants, traces – "higher Capelli identities"

The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents, immanants and traces. Based on the combinatorial approach paper by S.G. Williamson ^[26] was one of the first results in this direction.