Representation type of Hecke algebras of type \(A\) (Q2750954)

A finite dimensional algebra \(A\) has finite representation type if and only if there are finitely many isomorphism classes of finite dimensional indecomposable modules. A finite dimensional algebra of infinite representation type is said to be tame if for each dimension \(d\) there are finitely many one parameter families of modules containing all \(d\)-dimensional indecomposable modules. The third, and only other, possibility is that the algebra has wild representation type, meaning that its representation theory is at least as complicated as that of a polynomial ring in two non-commuting indeterminates.NEWLINENEWLINENEWLINELet \({\mathcal H}_q(r)\) be the Hecke algebra of the symmetric group on \(r\) letters, and let \(q\) be an \(l\)-th root of unity for some \(l\leq r\). For each partition \(\lambda\) of \(r\), there is a simple module \(D^\lambda\), and \(D^\lambda\) and \(D^\mu\) belong to the same \(l\)-block \({\mathcal B}_\lambda\) of the Hecke algebra if and oly if \(\lambda\) and \(\mu\) have the same \(l\)-core; the latter is a certain combinatorially defined partition. The weight, \(w(\lambda)\), of the partition \(\lambda\) is an integer defined in terms of the \(l\)-core of \(\lambda\).NEWLINENEWLINENEWLINEThe main result of this paper, given in \S 1.2, is a very elegant theorem classifying the representation type of the block \({\mathcal B}_\lambda\) for \({\mathcal H}_q(r)\) in terms of \(q\), \(l\) and \(w(\lambda)\). The simplest possible case is that \({\mathcal B}_\lambda\) is semisimple, which happens if and only if \(w(\lambda)=0\). The computation of the complexity of Young modules for the Hecke algebra is a key component of the proof.