Dynkin diagrams, support spaces and representation type. (Q2856425)

Let \(k\) be an algebraically closed field of characteristic \(p>0\). Let \(\mathcal G\) be a finite algebraic group, and let \(k\mathcal G\) be its algebra of measures. Then \(k\mathcal G\) is a finite dimensional cocommutative \(k\)-Hopf algebra, and there is a unique block -- the ``principal block'' -- denoted \(\mathcal B_0(\mathcal G)\) of \(k\mathcal G\) such that \(\varepsilon(\mathcal B_0(\mathcal G))\neq 0\), where \(\varepsilon\colon k\mathcal G\to k\) is the counit. For \(\Lambda\) an associative \(k\)-algebra, the group \(\text{GL}_d(k)\) acts on the variety of \(d\)-dimensional \(\Lambda\)-modules, and we let \(C_d\) be a closed subset of minimal dimension such that \(\text{GL}_d(k)\cdot C_d\) contains the set of indecomposable \(\Lambda\)-modules of dimension \(d\). If \(C_d=0\) for all \(d\) then \(\Lambda\) is said to be representation finite; if this is not the case and \(C_d\) has dimension at most \(1\) for all \(d\) then \(\Lambda\) is said to be tame; otherwise, \(\Lambda\) is said to be wild. Information about the principal block \(\mathcal B_0(\mathcal G)\) can give results about the representation type of \(\mathcal G\): for example, \(\mathcal B_0(\mathcal G)\) is simple if and only if \(k\mathcal G\) is semisimple, in which case \(\mathcal G\) is representation-finite.NEWLINENEWLINE As the classification of the indecomposable modules in the wild case is (from the paper) ``deemed hopeless'', in this survey article (based on a series of lectures at the Advances in Group Theory and its Applications conference in 2011), the author investigates conditions for when \(\mathcal B_0(\mathcal G)\) is representation-finite or tame. The objective of this work is to give an overview of these conditions, consequently most of the results are provided without proof (although a citation is provided for each unproven result).