In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.
Statement Suppose that xij for i,j = 1,...,n are commuting variables. Write Eij for the polarization operator
E i j = ∑ a = 1 n x i a ∂ ∂ x j a . {\displaystyle E_{ij}=\sum _{a=1}^{n}x_{ia}{\frac {\partial }{\partial x_{ja}}}.}
The Capelli identity states that the following differential operators, expressed as determinants, are equal:
| E 11 + n − 1 ⋯ E 1 , n − 1 E 1 n ⋮ ⋱ ⋮ ⋮ E n − 1 , 1 ⋯ E n − 1 , n − 1 + 1 E n − 1 , n E n 1 ⋯ E n , n − 1 E n n + 0 | = | x 11 ⋯ x 1 n ⋮ ⋱ ⋮ x n 1 ⋯ x n n | | ∂ ∂ x 11 ⋯ ∂ ∂ x 1 n ⋮ ⋱ ⋮ ∂ ∂ x n 1 ⋯ ∂ ∂ x n n | . {\displaystyle {\begin{vmatrix}E_{11}+n-1&\cdots &E_{1,n-1}&E_{1n}\\\vdots &\ddots &\vdots &\vdots \\E_{n-1,1}&\cdots &E_{n-1,n-1}+1&E_{n-1,n}\\E_{n1}&\cdots &E_{n,n-1}&E_{nn}+0\end{vmatrix}}={\begin{vmatrix}x_{11}&\cdots &x_{1n}\\\vdots &\ddots &\vdots \\x_{n1}&\cdots &x_{nn}\end{vmatrix}}{\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}.}
Both sides are differential operators. The determinant on the left has non-commuting entries, and is expanded with all terms preserving their "left to right" order. Such a determinant is often called a column-determinant, since it can be obtained by the column expansion of the determinant starting from the first column. It can be formally written as
det ( A ) = ∑ σ ∈ S n sgn ( σ ) A σ ( 1 ) , 1 A σ ( 2 ) , 2 ⋯ A σ ( n ) , n , {\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )A_{\sigma (1),1}A_{\sigma (2),2}\cdots A_{\sigma (n),n},}
where in the product first come the elements from the first column, then from the second and so on. The determinant on the far right is Cayley's omega process, and the one on the left is the Capelli determinant. The operators Eij can be written in a matrix form:
E = X D t , {\displaystyle E=XD^{t},}
where E , X , D {\displaystyle E,X,D} are matrices with elements Eij, xij, ∂ ∂ x i j {\displaystyle {\frac {\partial }{\partial x_{ij}}}} respectively. If all elements in these matrices would be commutative then clearly det ( E ) = det ( X ) det ( D t ) {\displaystyle \det(E)=\det(X)\det(D^{t})} . The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above. The only price for the noncommutativity is a small correction: ( n − i ) δ i j {\displaystyle (n-i)\delta _{ij}} on the left hand side. For generic noncommutative matrices formulas like
det ( A B ) = det ( A ) det ( B ) {\displaystyle \det(AB)=\det(A)\det(B)}
do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices. That is why the Capelli identity still holds some mystery, despite many proofs offered for it. A very short proof does not seem to exist. Direct verification of the statement can be given as an exercise for n = 2, but is already long for n = 3.
Relations with representation theory Consider the following slightly more general context. Suppose that n {\displaystyle n} and m {\displaystyle m} are two integers and x i j {\displaystyle x_{ij}} for i = 1 , … , n , j = 1 , … , m {\displaystyle i=1,\dots ,n,\ j=1,\dots ,m} , be commuting variables. Redefine E i j {\displaystyle E_{ij}} by almost the same formula:
E i j = ∑ a = 1 m x i a ∂ ∂ x j a . {\displaystyle E_{ij}=\sum _{a=1}^{m}x_{ia}{\frac {\partial }{\partial x_{ja}}}.}
with the only difference that summation index a {\displaystyle a} ranges from 1 {\displaystyle 1} to m {\displaystyle m} . One can easily see that such operators satisfy the commutation relations:
[ E i j , E k l ] = δ j k E i l − δ i l E k j . {\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}.~~~~~~~~~}
Here [ a , b ] {\displaystyle [a,b]} denotes the commutator a b − b a {\displaystyle ab-ba} . These are the same commutation relations which are satisfied by the matrices e i j {\displaystyle e_{ij}} which have zeros everywhere except the position ( i , j ) {\displaystyle (i,j)} , where 1 stands. ( e i j {\displaystyle e_{ij}} are sometimes called matrix units). Hence we conclude that the correspondence π : e i j ↦ E i j {\displaystyle \pi :e_{ij}\mapsto E_{ij}} defines a representation of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} in the vector space of polynomials of x i j {\displaystyle x_{ij}} .
Case m = 1 and representation Sk Cn It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:
E i j = x i ∂ ∂ x j . {\displaystyle E_{ij}=x_{i}{\frac {\partial }{\partial x_{j}}}.}
In particular, for the polynomials of the first degree it is seen that:
E i j x k = δ j k x i . {\displaystyle E_{ij}x_{k}=\delta _{jk}x_{i}.~~~~~~~~~~~~~~}
Hence the action of E i j {\displaystyle E_{ij}} restricted to the space of first-order polynomials is exactly the same as the action of matrix units e i j {\displaystyle e_{ij}} on vectors in C n {\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 g l n {\displaystyle {\mathfrak {gl}}_{n}} , which we identified with the standard representation in C n {\displaystyle \mathbb {C} ^{n}} . Going further, it is seen that the differential operators E i j {\displaystyle E_{ij}} preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a subrepresentation of the Lie algebra g l n {\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 S k C n {\displaystyle S^{k}\mathbb {C} ^{n}} of the standard representation C n {\displaystyle \mathbb {C} ^{n}} . One can also easily identify the highest weight structure of these representations. The monomial x 1 k {\displaystyle x_{1}^{k}} is a highest weight vector, indeed: E i j x 1 k = 0 {\displaystyle E_{ij}x_{1}^{k}=0} for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: E i i x 1 k = k δ i 1 x 1 k {\displaystyle E_{ii}x_{1}^{k}=k\delta _{i1}x_{1}^{k}} . Such representation is sometimes called bosonic representation of g l n {\displaystyle {\mathfrak {gl}}_{n}} . Similar formulas E i j = ψ i ∂ ∂ ψ j {\displaystyle E_{ij}=\psi _{i}{\frac {\partial }{\partial \psi _{j}}}} define the so-called fermionic representation, here ψ i {\displaystyle \psi _{i}} are anti-commuting variables. Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to Λ k C n {\displaystyle \Lambda ^{k}\mathbb {C} ^{n}} i.e. anti-symmetric tensor power of C n {\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 g l n {\displaystyle {\mathfrak {gl}}_{n}} .
Capelli identity for m = 1 Let us return to the Capelli identity. One can prove the following:
det ( E + ( n − i ) δ i j ) = 0 , n > 1 {\displaystyle \det(E+(n-i)\delta _{ij})=0,\qquad n>1}
the motivation for this equality is the following: consider E i j c = x i p j {\displaystyle E_{ij}^{c}=x_{i}p_{j}} for some commuting variables x i , p j {\displaystyle x_{i},p_{j}} . The matrix E c {\displaystyle E^{c}} is of rank one and hence its determinant is equal to zero. Elements of matrix E {\displaystyle E} are defined by the similar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity: det ( E c ) = 0 {\displaystyle \det(E^{c})=0} can be preserved for the small price of correcting matrix E {\displaystyle E} by ( n − i ) δ i j {\displaystyle (n-i)\delta _{ij}} . Let us also mention that similar identity can be given for the characteristic polynomial:
det ( t + E + ( n − i ) δ i j ) = t [ n ] + T r ( E ) t [ n − 1 ] , {\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\mathrm {Tr} (E)t^{[n-1]},~~~~}
where t [ k ] = t ( t + 1 ) ⋯ ( t + k − 1 ) {\displaystyle t^{[k]}=t(t+1)\cdots (t+k-1)} . The commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients. Consider an example for n = 2.
| t + E 11 + 1 E 12 E 21 t + E 22 | = | t + x 1 ∂ 1 + 1 x 1 ∂ 2 x 2 ∂ 1 t + x 2 ∂ 2 | = ( t + x 1 ∂ 1 + 1 ) ( t + x 2 ∂ 2 ) − x 2 ∂ 1 x 1 ∂ 2 = t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) + x 1 ∂ 1 x 2 ∂ 2 + x 2 ∂ 2 − x 2 ∂ 1 x 1 ∂ 2 {\displaystyle {\begin{aligned}&{\begin{vmatrix}t+E_{11}+1&E_{12}\\E_{21}&t+E_{22}\end{vmatrix}}={\begin{vmatrix}t+x_{1}\partial _{1}+1&x_{1}\partial _{2}\\x_{2}\partial _{1}&t+x_{2}\partial _{2}\end{vmatrix}}\\[8pt]&=(t+x_{1}\partial _{1}+1)(t+x_{2}\partial _{2})-x_{2}\partial _{1}x_{1}\partial _{2}\\[6pt]&=t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})+x_{1}\partial _{1}x_{2}\partial _{2}+x_{2}\partial _{2}-x_{2}\partial _{1}x_{1}\partial _{2}\end{aligned}}}
Using
∂ 1 x 1 = x 1 ∂ 1 + 1 , ∂ 1 x 2 = x 2 ∂ 1 , x 1 x 2 = x 2 x 1 {\displaystyle \partial _{1}x_{1}=x_{1}\partial _{1}+1,\partial _{1}x_{2}=x_{2}\partial _{1},x_{1}x_{2}=x_{2}x_{1}}
we see that this is equal to:
t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) + x 2 x 1 ∂ 1 ∂ 2 + x 2 ∂ 2 − x 2 x 1 ∂ 1 ∂ 2 − x 2 ∂ 2 = t ( t + 1 ) + t ( x 1 ∂ 1 + x 2 ∂ 2 ) = t [ 2 ] + t T r ( E ) . {\displaystyle {\begin{aligned}&{}\quad t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})+x_{2}x_{1}\partial _{1}\partial _{2}+x_{2}\partial _{2}-x_{2}x_{1}\partial _{1}\partial _{2}-x_{2}\partial _{2}\\[8pt]&=t(t+1)+t(x_{1}\partial _{1}+x_{2}\partial _{2})=t^{[2]}+t\,\mathrm {Tr} (E).\end{aligned}}}
The universal enveloping algebra U ( g l n ) {\displaystyle U({\mathfrak {gl}}_{n})} and its center An interesting property of the Capelli determinant is that it commutes with all operators Eij, that is, the commutator [ E i j , det ( E + ( n − i ) δ i j ) ] = 0 {\displaystyle [E_{ij},\det(E+(n-i)\delta _{ij})]=0} is equal to zero. It can be generalized: Consider any elements Eij in any ring, such that they satisfy the commutation relation [ E i j , E k l ] = δ j k E i l − δ i l E k j {\displaystyle [E_{ij},E_{kl}]=\delta _{jk}E_{il}-\delta _{il}E_{kj}} , (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:
det ( t + E + ( n − i ) δ i j ) = t [ n ] + ∑ k = n − 1 , … , 0 t [ k ] C k , {\displaystyle \det(t+E+(n-i)\delta _{ij})=t^{[n]}+\sum _{k=n-1,\dots ,0}t^{[k]}C_{k},~~~~~}
where t [ k ] = t ( t + 1 ) ⋯ ( t + k − 1 ) ,