2011/05/31 by Sergio Caracciolo, Alan D. Sokal, Andrea Sportiello · 1 citation
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math-ph #math.AG #math.CO #math.MP #msc:05A19 #msc:05E15 #msc:05E99 #msc:11S90 #msc:13A50 #msc:13N10 #msc:14F10 #msc:15A15 #msc:15A23 #msc:15A24 #msc:15A33 #msc:15A72 #msc:15A75 #msc:16S32 #msc:20G05 #msc:20G20 #msc:32C38 #msc:43A85 #msc:81T18 #msc:82B20
paper · pdf · doi:10.1016/j.aam.2012.12.001
published as Advances in Applied Mathematics 50, 474--594 (2013) · LaTeX2e, 144 pages. Version 2 has a slightly changed title and abstract, and includes new remarks in Sections 1, 2.6 and (especially) 9 concerning prehomogeneous vector spaces and citing the prior work of Sugiyama (2011). To be published in Advances in Applied Mathematics
arxiv created 2012/12/31 · openalex publication_date 2013/01/26 · crossref created 2013/01/26 · crossref issued 2013/04/01 · crossref published 2013/04/01 · crossref published-print 2013/04/01 · arxiv updated 2013/07/29 · openalex created_date 2016/06/24 · crossref deposited 2025/10/01 · crossref indexed 2026/03/28 · openalex updated_date 2026/07/28
The classic Cayley identity states that det(∂) (det X)s = s(s+1)...(s+n-1) (det X)s-1 where X=(xij) is an n-by-n matrix of indeterminates and ∂=(∂/∂ xij) is the corresponding matrix of partial derivatives. In this paper we present straightforward 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.