2006/02/03 by Takashi Hashimoto, Hashimoto, Takashi
Mathematics · #15A15 #17B35 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:15A15 #msc:17B35
paper · pdf · doi:10.48550/arxiv.math/0602055
version 1.1(corrected typos and added a reference), 17 pages, no figures
openalex publication_date 2006/02/03 · arxiv created 2006/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the universal enveloping algebra of the orthogonal Lie algebra of size even has a central element expressed in terms of Pfaffian of a certain matrix alternating along the anti-diagonal (which we call anti-alternating for short) whose entries are in the univerasal enveloping algebra. In this paper, we establish minor summation formulae of Pfaffian for the noncommutative anti-alternating matrix, as well as for commutative anti-alternating matrix. As an application, we show that the eigenvalues on the highest weight modules of the central element given by the Pfaffian can be easily computed.