{"history":[],"article":{"id":"bb8c0468-2aec-4186-ba00-3dfccec9703a","title":"Capelli's identity","slug":"capelli-s-identity","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **Capelli's identity**, named after, 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](https://bliptext.com/articles/lie-algebra-representation).\nIt can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is [Cayley's Ω process](https://bliptext.com/articles/cayley-s-process).\n\n# Statement\nSuppose that xij for i,j = 1,...,n are commuting variables.\nWrite Eij for the polarization operator\nThe Capelli identity states that the following differential operators, expressed as determinants, are equal:\nBoth sides are differential operators.\nThe determinant on the left has non-commuting entries, and is expanded with all terms preserving their \"left to right\" order.\nSuch 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.\nIt can be formally written as\nwhere in the product first come the elements from the first column, then from the second and so on.\nThe determinant on the far right is [Cayley's omega process](https://bliptext.com/articles/cayley-s-omega-process), and the one on the left is the Capelli determinant.\nThe operators Eij can be written in a matrix form:\nwhere E, X, D are matrices with elements Eij, xij, respectively.\nIf all elements in these matrices would be commutative then clearly.\nThe Capelli identity shows that despite noncommutativity there exists a \"quantization\" of the formula above.\nThe only price for the noncommutativity is a small correction: on the left hand side.\nFor generic noncommutative matrices formulas like\ndo not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.\nThat is why the Capelli identity still holds some mystery, despite many proofs offered for it.\nA very short proof does not seem to exist.\nDirect verification of the statement can be given as an exercise for n = 2, but is already long for n = 3.\n\n# Relations with representation theory\nConsider the following slightly more general context.\nSuppose that n and m are two integers and x_{ij} for, be commuting variables.\nRedefine E_{ij} by almost the same formula:\nwith the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:\nHere [a,b] denotes the [commutator](https://bliptext.com/articles/commutator) ab-ba.\nThese are the same commutation relations which are satisfied by the matrices e_{ij} which have zeros everywhere except the position (i,j), where 1 stands.\n(e_{ij} are sometimes called matrix units).\nHence we conclude that the correspondence defines a [representation of the Lie algebra](https://bliptext.com/articles/lie-algebra-representation) in the vector space of polynomials of x_{ij}.\n\n## Case m = 1 and representation Sk Cn\nIt is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:\nIn particular, for the polynomials of the first degree it is seen that:\nHence the action of E_{ij} restricted to the space of first-order polynomials is exactly the same as the action of matrix units e_{ij} on vectors in.\nSo, from the representation theory point of view, the subspace of polynomials of first degree is a [subrepresentation](https://bliptext.com/articles/representation-theory) of the Lie algebra, which we identified with the standard representation in.\nGoing further, it is seen that the differential operators E_{ij} preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a [subrepresentation](https://bliptext.com/articles/representation-theory) of the Lie algebra.\nOne can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation \\mathbb C^n.\nOne can also easily identify the [highest weight](https://bliptext.com/articles/highest-weight) structure of these representations.\nThe monomial x^k_1 is a [highest weight vector](https://bliptext.com/articles/highest-weight-vector), indeed: for i < j.\nIts highest weight equals to (k, 0, ... ,0), indeed:.\nSuch representation is sometimes called bosonic representation of.\nSimilar formulas define the so-called fermionic representation, here \\psi_{i} are anti-commuting variables.\nAgain polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to i.e. anti-symmetric tensor power of.\nHighest weight of such representation is (0, ..., 0, 1, 0, ..., 0).\nThese representations for k = 1, ..., n are [fundamental representations](https://bliptext.com/articles/fundamental-representation) of.\n\n### Capelli identity for m = 1\nLet us return to the Capelli identity.\nOne can prove the following:\nthe motivation for this equality is the following: consider for some commuting variables x_i, p_j.\nThe matrix E^{c} is of rank one and hence its determinant is equal to zero.\nElements of matrix E are defined by the similar formulas, however, its elements do not commute.\nThe Capelli identity shows that the commutative identity: can be preserved for the small price of correcting matrix E by.\nLet us also mention that similar identity can be given for the characteristic polynomial:\nwhere.\nThe commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients.\nConsider an example for n = 2.\nUsing\nwe see that this is equal to:\n\n## The universal enveloping algebra U(\\mathfrak{gl}_n) and its center\nAn interesting property of the Capelli determinant is that it commutes with all operators Eij, that is, the [commutator](https://bliptext.com/articles/commutator) is equal to zero.\nIt can be generalized:\nConsider any elements Eij in any ring, such that they satisfy the commutation relation, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:\nwhere\nthen:\ni.e. they are sums of principal minors of the matrix E, modulo the Capelli correction.\nIn particular element C0 is the Capelli determinant considered above.\nThese 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.\nThe [universal enveloping algebra](https://bliptext.com/articles/universal-enveloping-algebra) can be defined as an algebra generated by Eij subject to the relations\nalone.\nThe proposition above shows that elements Ck belong to the [center](https://bliptext.com/articles/center-algebra) of.\nIt can be shown that they actually are free generators of the center of.\nThey are sometimes called Capelli generators.\nThe Capelli identities for them will be discussed below.\nConsider an example for n = 2.\nIt is immediate to check that element commute with E_{ij}.\n(It corresponds to an obvious fact that the identity matrix commute with all other matrices).\nMore instructive is to check commutativity of the second element with E_{ij}.\nLet us do it for E_{12}:\nWe see that the naive determinant will not commute with E_{12} and the Capelli's correction +E_{22} is essential to ensure the centrality.\n\n## General m and dual pairs\nLet us return to the general case:\nfor arbitrary n and m.\nDefinition of operators Eij can be written in a matrix form: E = X D^t, where E is n \\times n matrix with elements E_{ij}; X is n \\times m matrix with elements x_{ij}; D is n \\times m matrix with elements.\n**Capelli–Cauchy–Binet identities**\nFor general m matrix E is given as product of the two rectangular matrices: X and transpose to D.\nIf 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](https://bliptext.com/articles/cauchy-binet-formula) via [minors](https://bliptext.com/articles/major-and-minor) of X and D.\nAn analogue of this formula also exists for matrix E again for the same mild price of the correction :\nIn particular (similar to the commutative case): if m < n, then ; if m = n we return to the identity above.\nLet us also mention that similar to the commutative case (see [Cauchy–Binet for minors](https://bliptext.com/articles/minor-linear-algebra)), one can express not only the determinant of E, but also its minors via minors of X and D:\nHere K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usually M_{KL} denotes a submatrix of M formed by the elements Mk alb .\nPay attention that the Capelli correction now contains s, not n as in previous formula.\nNote 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.\nLet 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.\nAs a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention the following:\nwhere.\nThis formula is similar to the commutative case, modula at the left hand side and t[n] instead of tn at the right hand side.\n**Relation to dual pairs**\nModern interest in these identities has been much stimulated by [Roger Howe](https://bliptext.com/articles/roger-evans-howe) who considered them in his theory of [reductive dual pairs](https://bliptext.com/articles/howe-duality) (also known as Howe duality).\nTo make the first contact with these ideas, let us look more precisely on operators E_{ij}.\nSuch operators preserve the degree of polynomials.\nLet us look at the polynomials of degree 1:, we see that index l is preserved.\nOne can see that from the representation theory point of view polynomials of the first degree can be identified with direct sum of the representations, here l-th subspace (l=1...m) is spanned by x_{il}, i = 1, ..., n.\nLet us give another look on this vector space:\nSuch point of view gives the first hint of symmetry between m and n.\nTo deepen this idea consider:\nThese operators are given by the same formulas as E_{ij} modula renumeration, hence by the same arguments we can deduce that form a [representation of the Lie algebra](https://bliptext.com/articles/lie-algebra-representation) in the vector space of polynomials of xij.\nBefore going further we can mention the following property: differential operators commute with differential operators E_{kl}.\nThe [Lie group](https://bliptext.com/articles/lie-group) acts on the vector space in a natural way.\nOne can show that the corresponding action of Lie algebra is given by the differential operators E_{ij}~ and respectively.\nThis explains the commutativity of these operators.\nThe following deeper properties actually hold true:\nThe summands are indexed by the [Young diagrams](https://bliptext.com/articles/young-diagram) D, and representations \\rho^D are mutually non-isomorphic.\nAnd diagram {D} determine {D'} and vice versa.\nOne easily observe the strong similarity to [Schur–Weyl duality](https://bliptext.com/articles/schur-weyl-duality).\n\n# Generalizations\nMuch work have been done on the identity and its generalizations.\nApproximately two dozens of mathematicians and physicists contributed to the subject, to name a few: [R. Howe](https://bliptext.com/articles/roger-evans-howe), [B. Kostant](https://bliptext.com/articles/bertram-kostant) [Fields medalist](https://bliptext.com/articles/fields-medal) [A. Okounkov](https://bliptext.com/articles/andrei-okounkov) [A. Sokal](https://bliptext.com/articles/alan-sokal), [D. Zeilberger](https://bliptext.com/articles/zeilberger).\nIt seems historically the first generalizations were obtained by [Herbert Westren Turnbull](https://bliptext.com/articles/herbert-westren-turnbull) in 1948, who found the generalization for the case of symmetric matrices (see for modern treatments).\nThe other generalizations can be divided into several patterns.\nMost of them are based on the Lie algebra point of view.\nSuch generalizations consist of changing Lie algebra to [simple Lie algebras](https://bliptext.com/articles/simple-lie-group) and their [super](https://bliptext.com/articles/lie-superalgebra) [(q)](https://bliptext.com/articles/quantum-group), and current versions.\nAs well as identity can be generalized for different [reductive dual pairs](https://bliptext.com/articles/howe-duality).\nAnd finally one can consider not only the determinant of the matrix E, but its permanent, trace of its powers and immanants.\nLet us mention few more papers;\nstill the list of references is incomplete.\nIt has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras.\nSurprisingly a new purely algebraic generalization of the identity have been found in 2008 by S. Caracciolo, A. Sportiello, A. D. Sokal which has nothing to do with any Lie algebras.\n\n## Turnbull's identity for symmetric matrices\nConsider symmetric matrices\n[Herbert Westren Turnbull](https://bliptext.com/articles/herbert-westren-turnbull) in 1948 discovered the following identity:\nCombinatorial proof can be found in the paper, another proof and amusing generalizations in the paper, see also discussion below.\n\n## The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices\nConsider antisymmetric matrices\nThen\n\n## The Caracciolo–Sportiello–Sokal identity for Manin matrices\nConsider two matrices M and Y over some associative ring which satisfy the following condition\nfor some elements Qil.\nOr ”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 Mik and Ykl depends only on i, l, but does not depend on k.\nAssume that M is a [Manin matrix](https://bliptext.com/articles/manin-matrix) (the simplest example is the matrix with commuting elements).\nThen for the square matrix case\nHere Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal.\nSee proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.\nObviously the original Cappeli's identity the particular case of this identity.\nMoreover from this identity one can see that in the original Capelli's identity one can consider elements\nfor arbitrary functions fij and the identity still will be true.\n\n## The Mukhin–Tarasov–Varchenko identity and the Gaudin model\n\n### Statement\nConsider matrices X and D as in Capelli's identity, i.e. with elements x_{ij} and at position (ij).\nLet z be another formal variable (commuting with x).\nLet A and B be some matrices which elements are complex numbers.\nHere the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries.\nThe 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.\n(Such recipe is called a [Wick ordering](https://bliptext.com/articles/wick-ordering) in the [quantum mechanics](https://bliptext.com/articles/quantum-mechanics)).\n\n### The Gaudin quantum integrable system and Talalaev's theorem\nThe matrix\nis a [Lax matrix](https://bliptext.com/articles/lax-pair) for the Gaudin quantum integrable spin chain system.\nD. 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.\nConsider\nThen for all i,j,z,w\ni.e. Hi(z) are generating functions in z for the differential operators in x which all commute.\nSo they provide quantum commuting conservation laws for the Gaudin model.\n\n## Permanents, immanants, traces – \"higher Capelli identities\"\nThe original Capelli identity is a statement about determinants.\nLater, analogous identities were found for [permanents](https://bliptext.com/articles/permanent-mathematics), [immanants](https://bliptext.com/articles/immanant) and traces.\nBased on the combinatorial approach paper by S.G. Williamson\nwas one of the first results in this direction.\n\n### Turnbull's identity for permanents of antisymmetric matrices\nConsider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of the HUKS identity above.\nThen\nLet us cite: \"...is stated without proof at the end of Turnbull’s paper\".\nThe authors themselves follow Turnbull – at the very end of their\npaper they write:\n\"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.\".\nThe identity is deeply analyzed in paper\n.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-26T11:33:48.256Z","updated_at":"2025-01-26T11:33:48.256Z","search_vector":"'capelli':1 'ident':3","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), **Capelli's identity**, named after, 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](https://bliptext.com/articles/lie-algebra-representation).\nIt can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is [Cayley's Ω process](https://bliptext.com/articles/cayley-s-process).\n\n# Statement\nSuppose that xij for i,j = 1,...,n are commuting variables.\nWrite Eij for the polarization operator\nThe Capelli identity states that the following differential operators, expressed as determinants, are equal:\nBoth sides are differential operators.\nThe determinant on the left has non-commuting entries, and is expanded with all terms preserving their \"left to right\" order.\nSuch 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.\nIt can be formally written as\nwhere in the product first come the elements from the first column, then from the second and so on.\nThe determinant on the far right is [Cayley's omega process](https://bliptext.com/articles/cayley-s-omega-process), and the one on the left is the Capelli determinant.\nThe operators Eij can be written in a matrix form:\nwhere E, X, D are matrices with elements Eij, xij, respectively.\nIf all elements in these matrices would be commutative then clearly.\nThe Capelli identity shows that despite noncommutativity there exists a \"quantization\" of the formula above.\nThe only price for the noncommutativity is a small correction: on the left hand side.\nFor generic noncommutative matrices formulas like\ndo not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.\nThat is why the Capelli identity still holds some mystery, despite many proofs offered for it.\nA very short proof does not seem to exist.\nDirect verification of the statement can be given as an exercise for n = 2, but is already long for n = 3.\n\n# Relations with representation theory\nConsider the following slightly more general context.\nSuppose that n and m are two integers and x_{ij} for, be commuting variables.\nRedefine E_{ij} by almost the same formula:\nwith the only difference that summation index a ranges from 1 to m. One can easily see that such operators satisfy the commutation relations:\nHere [a,b] denotes the [commutator](https://bliptext.com/articles/commutator) ab-ba.\nThese are the same commutation relations which are satisfied by the matrices e_{ij} which have zeros everywhere except the position (i,j), where 1 stands.\n(e_{ij} are sometimes called matrix units).\nHence we conclude that the correspondence defines a [representation of the Lie algebra](https://bliptext.com/articles/lie-algebra-representation) in the vector space of polynomials of x_{ij}.\n\n## Case m = 1 and representation Sk Cn\nIt is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:\nIn particular, for the polynomials of the first degree it is seen that:\nHence the action of E_{ij} restricted to the space of first-order polynomials is exactly the same as the action of matrix units e_{ij} on vectors in.\nSo, from the representation theory point of view, the subspace of polynomials of first degree is a [subrepresentation](https://bliptext.com/articles/representation-theory) of the Lie algebra, which we identified with the standard representation in.\nGoing further, it is seen that the differential operators E_{ij} preserve the degree of the polynomials, and hence the polynomials of each fixed degree form a [subrepresentation](https://bliptext.com/articles/representation-theory) of the Lie algebra.\nOne can see further that the space of homogeneous polynomials of degree k can be identified with the symmetric tensor power of the standard representation \\mathbb C^n.\nOne can also easily identify the [highest weight](https://bliptext.com/articles/highest-weight) structure of these representations.\nThe monomial x^k_1 is a [highest weight vector](https://bliptext.com/articles/highest-weight-vector), indeed: for i < j.\nIts highest weight equals to (k, 0, ... ,0), indeed:.\nSuch representation is sometimes called bosonic representation of.\nSimilar formulas define the so-called fermionic representation, here \\psi_{i} are anti-commuting variables.\nAgain polynomials of k-th degree form an irreducible subrepresentation which is isomorphic to i.e. anti-symmetric tensor power of.\nHighest weight of such representation is (0, ..., 0, 1, 0, ..., 0).\nThese representations for k = 1, ..., n are [fundamental representations](https://bliptext.com/articles/fundamental-representation) of.\n\n### Capelli identity for m = 1\nLet us return to the Capelli identity.\nOne can prove the following:\nthe motivation for this equality is the following: consider for some commuting variables x_i, p_j.\nThe matrix E^{c} is of rank one and hence its determinant is equal to zero.\nElements of matrix E are defined by the similar formulas, however, its elements do not commute.\nThe Capelli identity shows that the commutative identity: can be preserved for the small price of correcting matrix E by.\nLet us also mention that similar identity can be given for the characteristic polynomial:\nwhere.\nThe commutative counterpart of this is a simple fact that for rank = 1 matrices the characteristic polynomial contains only the first and the second coefficients.\nConsider an example for n = 2.\nUsing\nwe see that this is equal to:\n\n## The universal enveloping algebra U(\\mathfrak{gl}_n) and its center\nAn interesting property of the Capelli determinant is that it commutes with all operators Eij, that is, the [commutator](https://bliptext.com/articles/commutator) is equal to zero.\nIt can be generalized:\nConsider any elements Eij in any ring, such that they satisfy the commutation relation, (so they can be differential operators above, matrix units eij or any other elements) define elements Ck as follows:\nwhere\nthen:\ni.e. they are sums of principal minors of the matrix E, modulo the Capelli correction.\nIn particular element C0 is the Capelli determinant considered above.\nThese 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.\nThe [universal enveloping algebra](https://bliptext.com/articles/universal-enveloping-algebra) can be defined as an algebra generated by Eij subject to the relations\nalone.\nThe proposition above shows that elements Ck belong to the [center](https://bliptext.com/articles/center-algebra) of.\nIt can be shown that they actually are free generators of the center of.\nThey are sometimes called Capelli generators.\nThe Capelli identities for them will be discussed below.\nConsider an example for n = 2.\nIt is immediate to check that element commute with E_{ij}.\n(It corresponds to an obvious fact that the identity matrix commute with all other matrices).\nMore instructive is to check commutativity of the second element with E_{ij}.\nLet us do it for E_{12}:\nWe see that the naive determinant will not commute with E_{12} and the Capelli's correction +E_{22} is essential to ensure the centrality.\n\n## General m and dual pairs\nLet us return to the general case:\nfor arbitrary n and m.\nDefinition of operators Eij can be written in a matrix form: E = X D^t, where E is n \\times n matrix with elements E_{ij}; X is n \\times m matrix with elements x_{ij}; D is n \\times m matrix with elements.\n**Capelli–Cauchy–Binet identities**\nFor general m matrix E is given as product of the two rectangular matrices: X and transpose to D.\nIf 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](https://bliptext.com/articles/cauchy-binet-formula) via [minors](https://bliptext.com/articles/major-and-minor) of X and D.\nAn analogue of this formula also exists for matrix E again for the same mild price of the correction :\nIn particular (similar to the commutative case): if m < n, then ; if m = n we return to the identity above.\nLet us also mention that similar to the commutative case (see [Cauchy–Binet for minors](https://bliptext.com/articles/minor-linear-algebra)), one can express not only the determinant of E, but also its minors via minors of X and D:\nHere K = (k1 < k2 < ... < ks), L = (l1 < l2 < ... < ls), are arbitrary multi-indexes; as usually M_{KL} denotes a submatrix of M formed by the elements Mk alb .\nPay attention that the Capelli correction now contains s, not n as in previous formula.\nNote 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.\nLet 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.\nAs a corollary of this formula and the one for the characteristic polynomial in the previous section let us mention the following:\nwhere.\nThis formula is similar to the commutative case, modula at the left hand side and t[n] instead of tn at the right hand side.\n**Relation to dual pairs**\nModern interest in these identities has been much stimulated by [Roger Howe](https://bliptext.com/articles/roger-evans-howe) who considered them in his theory of [reductive dual pairs](https://bliptext.com/articles/howe-duality) (also known as Howe duality).\nTo make the first contact with these ideas, let us look more precisely on operators E_{ij}.\nSuch operators preserve the degree of polynomials.\nLet us look at the polynomials of degree 1:, we see that index l is preserved.\nOne can see that from the representation theory point of view polynomials of the first degree can be identified with direct sum of the representations, here l-th subspace (l=1...m) is spanned by x_{il}, i = 1, ..., n.\nLet us give another look on this vector space:\nSuch point of view gives the first hint of symmetry between m and n.\nTo deepen this idea consider:\nThese operators are given by the same formulas as E_{ij} modula renumeration, hence by the same arguments we can deduce that form a [representation of the Lie algebra](https://bliptext.com/articles/lie-algebra-representation) in the vector space of polynomials of xij.\nBefore going further we can mention the following property: differential operators commute with differential operators E_{kl}.\nThe [Lie group](https://bliptext.com/articles/lie-group) acts on the vector space in a natural way.\nOne can show that the corresponding action of Lie algebra is given by the differential operators E_{ij}~ and respectively.\nThis explains the commutativity of these operators.\nThe following deeper properties actually hold true:\nThe summands are indexed by the [Young diagrams](https://bliptext.com/articles/young-diagram) D, and representations \\rho^D are mutually non-isomorphic.\nAnd diagram {D} determine {D'} and vice versa.\nOne easily observe the strong similarity to [Schur–Weyl duality](https://bliptext.com/articles/schur-weyl-duality).\n\n# Generalizations\nMuch work have been done on the identity and its generalizations.\nApproximately two dozens of mathematicians and physicists contributed to the subject, to name a few: [R. Howe](https://bliptext.com/articles/roger-evans-howe), [B. Kostant](https://bliptext.com/articles/bertram-kostant) [Fields medalist](https://bliptext.com/articles/fields-medal) [A. Okounkov](https://bliptext.com/articles/andrei-okounkov) [A. Sokal](https://bliptext.com/articles/alan-sokal), [D. Zeilberger](https://bliptext.com/articles/zeilberger).\nIt seems historically the first generalizations were obtained by [Herbert Westren Turnbull](https://bliptext.com/articles/herbert-westren-turnbull) in 1948, who found the generalization for the case of symmetric matrices (see for modern treatments).\nThe other generalizations can be divided into several patterns.\nMost of them are based on the Lie algebra point of view.\nSuch generalizations consist of changing Lie algebra to [simple Lie algebras](https://bliptext.com/articles/simple-lie-group) and their [super](https://bliptext.com/articles/lie-superalgebra) [(q)](https://bliptext.com/articles/quantum-group), and current versions.\nAs well as identity can be generalized for different [reductive dual pairs](https://bliptext.com/articles/howe-duality).\nAnd finally one can consider not only the determinant of the matrix E, but its permanent, trace of its powers and immanants.\nLet us mention few more papers;\nstill the list of references is incomplete.\nIt has been believed for quite a long time that the identity is intimately related with semi-simple Lie algebras.\nSurprisingly a new purely algebraic generalization of the identity have been found in 2008 by S. Caracciolo, A. Sportiello, A. D. Sokal which has nothing to do with any Lie algebras.\n\n## Turnbull's identity for symmetric matrices\nConsider symmetric matrices\n[Herbert Westren Turnbull](https://bliptext.com/articles/herbert-westren-turnbull) in 1948 discovered the following identity:\nCombinatorial proof can be found in the paper, another proof and amusing generalizations in the paper, see also discussion below.\n\n## The Howe–Umeda–Kostant–Sahi identity for antisymmetric matrices\nConsider antisymmetric matrices\nThen\n\n## The Caracciolo–Sportiello–Sokal identity for Manin matrices\nConsider two matrices M and Y over some associative ring which satisfy the following condition\nfor some elements Qil.\nOr ”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 Mik and Ykl depends only on i, l, but does not depend on k.\nAssume that M is a [Manin matrix](https://bliptext.com/articles/manin-matrix) (the simplest example is the matrix with commuting elements).\nThen for the square matrix case\nHere Q is a matrix with elements Qil, and diag(n − 1, n − 2, ..., 1, 0) means the diagonal matrix with the elements n − 1, n − 2, ..., 1, 0 on the diagonal.\nSee proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.\nObviously the original Cappeli's identity the particular case of this identity.\nMoreover from this identity one can see that in the original Capelli's identity one can consider elements\nfor arbitrary functions fij and the identity still will be true.\n\n## The Mukhin–Tarasov–Varchenko identity and the Gaudin model\n\n### Statement\nConsider matrices X and D as in Capelli's identity, i.e. with elements x_{ij} and at position (ij).\nLet z be another formal variable (commuting with x).\nLet A and B be some matrices which elements are complex numbers.\nHere the first determinant is understood (as always) as column-determinant of a matrix with non-commutative entries.\nThe 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.\n(Such recipe is called a [Wick ordering](https://bliptext.com/articles/wick-ordering) in the [quantum mechanics](https://bliptext.com/articles/quantum-mechanics)).\n\n### The Gaudin quantum integrable system and Talalaev's theorem\nThe matrix\nis a [Lax matrix](https://bliptext.com/articles/lax-pair) for the Gaudin quantum integrable spin chain system.\nD. 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.\nConsider\nThen for all i,j,z,w\ni.e. Hi(z) are generating functions in z for the differential operators in x which all commute.\nSo they provide quantum commuting conservation laws for the Gaudin model.\n\n## Permanents, immanants, traces – \"higher Capelli identities\"\nThe original Capelli identity is a statement about determinants.\nLater, analogous identities were found for [permanents](https://bliptext.com/articles/permanent-mathematics), [immanants](https://bliptext.com/articles/immanant) and traces.\nBased on the combinatorial approach paper by S.G. Williamson\nwas one of the first results in this direction.\n\n### Turnbull's identity for permanents of antisymmetric matrices\nConsider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of the HUKS identity above.\nThen\nLet us cite: \"...is stated without proof at the end of Turnbull’s paper\".\nThe authors themselves follow Turnbull – at the very end of their\npaper they write:\n\"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.\".\nThe identity is deeply analyzed in paper\n.","revisionCount":0},"hasMore":false,"nextCursor":null}

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