Request for info on the space of commuting matrices preserving a flag. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T11:43:52Z http://mathoverflow.net/feeds/question/422 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/422/request-for-info-on-the-space-of-commuting-matrices-preserving-a-flag Request for info on the space of commuting matrices preserving a flag. Ben Webster 2009-10-13T14:19:03Z 2009-10-13T15:22:25Z <p>Fix a flag of subspaces V<sub>1</sub> in V<sub>2</sub> in V<sub>3</sub>, etc. all in C<sup>n</sup>.</p> <p>Consider the space of pairs of commuting linear transformations A and B such that:<br> A preserves the flag (i.e. A(V<sub>i</sub>) is in V<sub>i</sub>), and<br> B <em>strictly</em> preserves the flag (i.e. B(V<sub>i</sub>) is in V<sub>i-1</sub>).</p> <p>Does anyone know anything about this space? Is there any literature on it? Is it smooth?</p> <p>Just as an example: if the flag is trivial, B=0 and A can be anything. So that's smooth.<br></p> http://mathoverflow.net/questions/422/request-for-info-on-the-space-of-commuting-matrices-preserving-a-flag/425#425 Answer by Ian Shipman for Request for info on the space of commuting matrices preserving a flag. Ian Shipman 2009-10-13T14:51:28Z 2009-10-13T15:22:25Z <p>for the case where n = 2 and the complete flag the variety in question is isomorphic to A^2 x Spec C[x,y]/(xy). so the space is not nec. smooth.</p>