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

Abstract

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.

The Andrews–Stanley partition function and Al-Salam–Chihara polynomials

“…Though it is not possible to specialize the Schur functions to z (λ) , we show in this paper that this approach still works, i.e., we can evaluate the weighted sum ω(λ)z (λ) by using Pfaffians and minor summation formulas as tools (see [8,9]), but, as an afterthought, we also provide alternative combinatorial proofs.…”

“…(2.12) (See [8,9].) We call the formula (2.11) the Pfaffian expansion along the jth column, and the formula (2.12) the Pfaffian expansion along the ith row.…”

An elliptic generalization of Schur's Pfaffian identity

“…The key is to use the following Pfaffian version of the Desnanot-Jacobi formula. See [7,3,5] for a proof and related formulae. Proof of Theorem 1.…”

Multivariable Christoffel-Darboux Kernels and Characteristic Polynomials of Random Hermitian Matrices

“…In the special case z = (x, x), choosing We remark that the relevant results of [8], [16] and [22] are closely related to Sundquist's identities [30], see also [6] and [7]. Moreover, (11a) is related to the Izergin-Korepin determinant for the partition function of the six-vertex model [10], which, as well as the Pfaffians in (11), has applications to alternating sign matrices [13,14,23,32].…”