Permutation resolutions for Specht modules.
The Specht modules form an important family of modules for the group algebra of the symmetric group over any base ring. They have a simple combinatorial construction, and are of fundamental importance in the representation theory of the symmetric group. The aim of the paper under review is to try to construct resolutions for the Specht modules in which all the terms are permutation modules. This is a very natural aim, and (as the authors discuss in the final section of the paper) similar work has been done before. But the aim here is to provide a clean combinatorial construction to work over any base ring. The authors give a very clear summary of the background details they need; these are almost all taken from \textit{G. D. James}'s classic book [The representation theory of the symmetric groups. Lect. Notes Math. 682. Berlin-Heidelberg-New York: Springer-Verlag (1978; Zbl 0393.20009)], but actually much of the exposition here is even clearer than James's. Then the authors construct complexes of permutation modules which (they conjecture) yield resolutions for the Specht modules. Again, everything is very clear: the homomorphisms defining the complexes are defined using James's combinatorial constructions with tableaux. The main conjecture in the paper is Conjecture 3.4, which asserts that the complex defined is exact in each case, and therefore gives a resolution for the Specht module. In the remainder of the paper, the authors prove some special cases of their conjecture: they prove that each complex is exact in degrees \(-1\) and 0 for all partitions, and in all degrees for so-called tame partitions. This paper is very nicely written, and is recommended to anyone interested in the combinatorial representation theory of the symmetric group. Of course it is disappointing that the authors cannot (yet) prove their conjecture. It is also disappointing that computer calculations to verify the conjecture have only reached partitions of size 5.
scientific article; zbMATH DE number 5956814
scientific article