2-dimensional orders and integral Hecke orders. (With an appendix by Marcos Soriano) (Q2759671)

The article provides a survey on recent work on orders over a two-dimensional regular integral domain \(R\) with a particular emphasis on Hecke orders. After a brief introduction to 2-dimensional orders including the Higman ideal and the Tits deformation theorem, blocks with cyclic defect and Brauer trees are adapted to Hecke orders. A third section is devoted to the classification of Cohen-Macaulay modules over 2-dimensional tree orders. Sections 4 and 5 deal with extension groups of Cohen-Macaulay Specht modules of Hecke orders of a symmetric group. An appendix by M. Soriano completes the latter topic.NEWLINENEWLINENEWLINEThe author's favorite base ring \(R\) is \(\mathbb{Z}[q]\) since this ring specializes to all rings of integers in algebraic number fields as well as to polynomial rings over finite fields. Therefore, extending a principal idea in R.~Brauer's classical (i.e. one-dimensional) approach, two-dimensional orders over \(\mathbb{Z}[q]\) should form a starting point for new connections between integral and modular representation theory. Some instances for this view are collected in the article.NEWLINENEWLINENEWLINEFor the representation theory of two-dimensional tree orders, \(R\) is assumed to be complete. This guarantees that representations satisfy the Krull-Schmidt theorem. The \(R\)-order of a tree is constructed similarly to a classical Brauer tree order, using an appropriate two-dimensional generalization of a hereditary order. In the ``prime defect'' case, representation theory reduces to the particular case of a tree with two nodes and one connecting edge. Such orders are not representation-finite but representation-bounded (i.e. the rational ranks of indecomposables are bounded). Thanks to the Cohen-Macaulay property, a complete list of indecomposables can be given.NEWLINENEWLINENEWLINELet \(W\) be a finite Coxeter group, and let \(\mathcal H\) be the completion of the Hecke order at the maximal ideal \((p,q-1)\), where \(p\) is a rational prime. For a dihedral group \(W\) of order \(2\cdot p^n\) with \(p\) odd, or a Coxeter group \(W\) with splitting field \(\mathbb{Q}\), the blocks \(B\) of \(\mathbb{Z}_pW\) can be lifted to blocks \(\mathcal B\) of \(\mathcal H\), and there are strong analogies between \(B\) and \(\mathcal B\). For the symmetric group \(W=S_p\), a structural result for the principal block \(\mathcal B_0\) of \(\mathcal H\) is given. Using a result of M.~Soriano, the \(\text{Ext}\)-groups of Specht modules over \(\mathcal H\) can be described.NEWLINENEWLINENEWLINEA more detailed account on the structure of extension groups between Specht modules is provided in the appendix of M.~Soriano. This appendix contains a brief history of Hecke algebras with some interesting remarks on old and new applications.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].