4 research outputs found
Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities
Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」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\u27s 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
Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
The classic Cayley identity states that
where is an matrix of indeterminates and 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 “product-parametrized” and “border-parametrized” rectangular Cayley identities
Notes on the combinatorial fundamentals of algebra
This is a detailed survey -- with rigorous and self-contained proofs -- of
some of the basics of elementary combinatorics and algebra, including the
properties of finite sums, binomial coefficients, permutations and
determinants. It is entirely expository (and written to a large extent as a
repository for folklore proofs); no new results (and few, if any, new proofs)
appear.Comment: 1360 pages. v2 corrects typos and adds Exercises 6.62--6.64. Not a
textbook; rather a repository of proofs I could cite. Posted here for easier
referencing (and long-term archival). This project is tracked on
https://github.com/darijgr/detnote