The Capelli identity shows that despite noncommutativity there exists a "quantization" of the formula above.
restricted to the space of first-order polynomials is exactly the same as the action of matrix units
So, from the representation theory point of view, the subspace of polynomials of first degree is a subrepresentation of the Lie algebra
One can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power
One can also easily identify the highest weight structure of these representations.
Again polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to
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.
It can be generalized: Consider any elements Eij in any ring, such that they satisfy the commutation relation
then: i.e. they are sums of principal minors of the matrix E, modulo the Capelli correction
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 proposition above shows that elements Ck belong to the center of
(It corresponds to an obvious fact that the identity matrix commute with all other matrices).
Let us return to the general case: for arbitrary n and m. Definition of operators Eij can be written in a matrix form:
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
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 Eij, 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: where
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).
The following deeper properties actually hold true: The summands are indexed by the Young diagrams D, and representations
Approximately two dozens of mathematicians and physicists contributed to the subject, to name a few: R. Howe, B. Kostant[1][2] Fields medalist A. Okounkov[3][4] A. Sokal,[5] D.
[13] As well as identity can be generalized for different reductive dual pairs.
[14][15] And finally one can consider not only the determinant of the matrix E, but its permanent,[16] trace of its powers and immanants.
It has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras.
Consider symmetric matrices Herbert Westren Turnbull[7] in 1948 discovered the following identity: Combinatorial proof can be found in the paper,[6] another proof and amusing generalizations in the paper,[5] see also discussion below.
The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, while derivations on the right.
D. Talalaev solved the long-standing problem of the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discovering the following theorem.
Consider Then for all i,j,z,w i.e. Hi(z) are generating functions in z for the differential operators in x which all commute.
So they provide quantum commuting conservation laws for the Gaudin model.
Based on the combinatorial approach paper by S.G. Williamson [26] was one of the first results in this direction.
Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of the HUKS identity above.
The authors themselves follow Turnbull – at the very end of their paper they write: "Since the proof of this last identity is very similar to the proof of Turnbull’s symmetric analog (with a slight twist), we leave it as an instructive and pleasant exercise for the reader.".