Reverse decomposition of unipotents

The talk surveys an unexpected advance in the structure theory of reductive groups over commutative rings. Decomposition of unipotents, developed by Alexei Stepanov, the author, and others over the last decades, can be viewed as an effective version of the normality of the elementary subgroup of a reductive group in its group of points. In its simplest form, it gives explicit polynomial formulae expressing a conjugate of an elementary unipotent as a product of elementary generators. This year, Raimund Preusser observed that for classical groups essentially the same calculations work also the other way, and give effective versions of description of normal subgroups. Quite surprisingly, in many cases we can get explicit polynomial bounds on the length of expressions of the elementary generators, in terms of the elementary conjugates of an arbitrary group element. In the talk I describe variants of this method for exceptional groups, as well as some applications of this idea, for instance, in description of various classes of subgroups.