Generators and relations for the unitary group of a skew hermitian form\n over a local ring

2017/10/31 by James Cruickshank, Cruickshank, James, Fernando Szechtman +1
Mathematics · Chemistry · #Advanced Algebra and Geometry #Axial and Atropisomeric Chirality Synthesis #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.1710.11574

Abstract

Let (S,*) be an involutive local ring and let U(2m,S) be the unitary\ngroup associated to a nondegenerate skew hermitian form defined on a free\nS-module of rank 2m. A presentation of U(2m,S) is given in terms of\nBruhat generators and their relations. This presentation is used to construct\nan explicit Weil representation of the symplectic group Sp(2m,R) when S=R\nis commutative and * is the identity.\n When S is commutative but * is arbitrary with fixed ring R, an\nelementary proof that the special unitary group SU(2m,S) is generated by\nunitary transvections is given. This is used to prove that the reduction\nhomomorphisms SU(2m,S)\→ SU(2m,\S) and U(2m,S)\→ U(2m,\S)\nare surjective for any factor ring \S of S. The corresponding\nresults for the symplectic group Sp(2m,R) are obtained as corollaries when\n* is the identity.\n

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