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On low-degree representations of the symmetric group. (English) Zbl 1360.20007
The problem of understanding the irreducible modular representations of the symmetric group \(\mathcal S_n\) is the subject of a lot of research, and a solution seems almost as far out of reach now as it was when Gordon James laid the foundations for this field in the 1970s. Even finding the dimensions of the irreducibles is at present a very difficult task. A fruitful approach has been to consider representations that are small in some sense: a small value of \(n\), small \(p\)-defect, or (as in the paper under review) small dimension. The main theorem here is a classification of the irreducible representations of \(\mathcal S_n\) of dimension at most \(n^3\). For large \(n\), the main result states that the dimension of the irreducible module \(D^\mu\) is at most \(n^3\) only if either \(\mu\) or its image under the Mullineux map has first part at least \(n-3\). For small values of \(n\), tables are given for exceptional cases.
A variety of techniques is used, most importantly the LLT algorithm for computing (via Ariki’s theorem) decomposition numbers for the corresponding Iwahori-Hecke algebras, the modular branching rules which describe (to some extent) the restriction of an irreducible to \(\mathcal S_{n-1}\), and the Mullineux map, which describes the effect of tensoring an irreducible module with the sign module. In addition, the author brings his considerable computational expertise to the problem.
The paper is very well written, and everything is explained in great detail; in fact, I think this paper has the longest introduction of any paper I have read. This paper would serve as an excellent introduction to a variety of aspects of modular representation theory of symmetric groups, and the long list of references is highly appropriate and useful. I only found a couple of minor mistakes; most notably the first open problem stated in the introduction asks for a proof that when \(p=3\) there are no Jantzen-Seitz partitions \(\mu\) with \(\mu_1=\mu_2\). Obviously there are such partitions (for example, the partition \((m,m)\) for any \(m\)); the author means to specify further conditions on the partition as described in Remark 5.4.

MSC:
20C30 Representations of finite symmetric groups
20C20 Modular representations and characters
20C08 Hecke algebras and their representations
20C40 Computational methods (representations of groups) (MSC2010)
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