Explicit Capelli Identities for Skew Symmetric Matrices

Kinoshita, Kenji; Wakayama, Masato

doi:10.1017/s0013091500001176

Cited by 11 publications

(11 citation statements)

References 5 publications

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“…The symmetric analogues (2.4)/(2.5) are due to Gårding in 1948 [41]; see also [ [4]) is equivalent to a pfaffian, and see also [61,Corollary 3.13].…”

Section: Historical Remarksmentioning

confidence: 99%

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

Caracciolo

Sokal

Sportiello

2013

Advances in Applied Mathematics

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The classic Cayley identity states that det(∂) (det X) s = s(s + 1) · · · (s + n − 1) (det X) s−1 where X = (x ij ) is an n × n matrix of indeterminates and ∂ = (∂/∂x ij ) is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of "diagonal-parametrized" Cayley identities, a pair of "Laplacian-parametrized" Cayley identities, and the "productparametrized" and "border-parametrized" rectangular Cayley identities. Contents15 See also Turnbull [105] for a similar result, but expressed in difficult-to-follow notation. 16 After making the translation of conventions s → s + 1: see footnote 5 above. See also F. Sato and Sugiyama [85, Section 3.1 and Lemma 4.2] for an alternate approach to (2.1), (2.7) and (2.18). 17 We are grateful to Nero Budur for explaining to us the connection between our results and the theory of prehomogeneous vector spaces, and for drawing our attention to the work of Sugiyama [100].

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“…The symmetric analogues (2.4)/(2.5) are due to Gårding in 1948 [41]; see also [ [4]) is equivalent to a pfaffian, and see also [61,Corollary 3.13].…”

Section: Historical Remarksmentioning

confidence: 99%

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

Caracciolo

Sokal

Sportiello

2013

Advances in Applied Mathematics

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“…Recently, applications of the minor summation formula presented in [8] have been made in several directions, e.g., to study a certain limit law for shifted Schur measures in [33], to find an explicit description of the skew-Capelli identity in [17], and to generalize further the so-called Littlewood formulas, for instance in [11] (see also [2,9,10,13,14,21,27]), etc. Moreover, the formula has been generalized to the case of hyperpfaffians in [22] (see also [23]).…”

Section: Introductionmentioning

confidence: 99%

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

Ishikawa

Wakayama

2006

Journal of Combinatorial Theory, Series A

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The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.

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“…. , n − 1)] (16) which are the Capelli identities [2][3][4][5] of classical invariant theory [6][7][8], a field of research that, in more than a century, has remained active up to recent days (a forcerly incomplete selection of papers on the subject includes [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). Because of this example, the correction term due to the presence of the matrix Q which appears in the non-commutative case is sometimes called the "quantum" correction with respect to the formula in the commutative case (2).…”

Section: The Cauchy-binet Theoremmentioning

confidence: 99%

“…where B is a m × m matrix whose elements are supposed to commute with everything. Remark that, whenever B is invertible, 3 from (18) by multiplication of B −1 js and sum over j we get [(XB −1 ) is , Y kℓ ] = −A iℓ δ ks (19) which is of the form (17), and similarly by multiplication of B −1 sk and sum over k (20) and, as if X is row-pseudo-commutative also XB −1 is such, while if Y is columnpseudo-commutative also B −1 Y is such. Thus, quite trivially, Proposition 1.1 can be used to express, for example in the case (a)…”

Section: The Cauchy-binet Theoremmentioning

confidence: 99%

Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation

Caracciolo¹,

Sportiello²

2014

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We prove that, for X, Y , A and B matrices with entries in a non-commutative ring such that [X ij , Y kℓ ] = −A iℓ B kj , satisfying suitable commutation relations (in particular, X is a Manin matrix), the following identity holdsNotations: 0|, |0 , are respectively the bra and the ket of the ground state, a † and a the creation and annihilation operators of a quantum harmonic oscillator, whilē ψ i and ψ i are Grassmann variables in a Berezin integral. These results should be seen as a generalization of the classical Cauchy-Binet formula, in which A and B are null matrices, and of the non-commutative generalization, the Capelli identity, in which A and B are identity matrices and [X ij , X kℓ ] = [Y ij , Y kℓ ] = 0.2

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Explicit Capelli Identities for Skew Symmetric Matrices

Cited by 11 publications

References 5 publications

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

Noncommutative determinants, Cauchy–Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation

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