Onn, Uri; Prasad, Amritanshu; Vaserstein, Leonid A note on Bruhat decomposition of \(\mathrm{GL}(n)\) over local principal ideal rings. (English) Zbl 1112.15020 Commun. Algebra 34, No. 11, 4119-4130 (2006). The double coset space \(B\setminus G/B\) is considered where \(G=\text{GL}_n (A)\) is the group of invertible \(n\times n\)-matrices over a local principal commutative ring \(A\), and \(B\) defines the stabilizer of the standard flag in \(A^n\), being a subgroup of upper triangular matrices in \(G\). It is pointed out that the space \(B\setminus G/B\) corresponds to the possible relative positions in two full flags of the primitive submodules of \(A^n\) which are isomorphic to the standard flag. Some invariants for the double coset space, called intersection invariants, are introduced. The pairs \((n,k)\), where \(k\) is the length of the ring \(A\), are determined, for which the space \(B\setminus G/B\) depends on the ring in question. For the particular case \(k=3\), a complete parametrization of the double coset space is given and the rate of growth of the number of double cosets is estimated. Reviewer: A. A. Bogush (Minsk) Cited in 7 Documents MSC: 15B33 Matrices over special rings (quaternions, finite fields, etc.) 20G35 Linear algebraic groups over adèles and other rings and schemes 13H05 Regular local rings Keywords:flags; stabilizer; parametrization; double coset space; commutative ring; Bruhat decomposition; local rings; reduction of matrices; intersection invariants PDFBibTeX XMLCite \textit{U. Onn} et al., Commun. Algebra 34, No. 11, 4119--4130 (2006; Zbl 1112.15020) Full Text: DOI arXiv References: [1] Bader U., J. Pure Appl. Algebra [2] DOI: 10.1080/00927879308824747 · Zbl 0788.20026 [3] DOI: 10.1006/jabr.1995.1143 · Zbl 0839.22021 [4] DOI: 10.1016/j.jpaa.2005.07.002 · Zbl 1147.16019 [5] DOI: 10.1016/j.jpaa.2005.02.003 · Zbl 1128.16012 [6] Simson , D. ( 2002 ). Chain categories of modules and subprojective representations of posets over uniserial algebras . In: Proceedings of the Second Honolulu Conference on Abelian Groups and Modules (Honolulu, HI, 2001 ), Vol. 32 , pp. 1627 – 1650 . · Zbl 1048.16006 [7] Zelevinsky A. V., Representations of Finite Classical Groups 869 (1981) · Zbl 0465.20009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.