A note on Bruhat decomposition of GL(n) over local principal ideal rings

Onn, Uri; Prasad, Amritanshu; Vaserstein, Leonid

A note on Bruhat decomposition of GL(n) over local principal ideal rings

Authors

Onn, Uri

Prasad, Amritanshu

Vaserstein, Leonid

Description

Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A^n. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend on the ring in question. For n=3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.
Final version, 9 pages, to appear in Communications in Algebra