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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)
##### Software:
CTblLib; GAP Character Table Library; Spinsym
Full Text:
##### References:
 [1] Ariki, S., On the decomposition numbers of the Hecke algebra of $$G(m, 1, n)$$, J. Math. Kyoto Univ., 36, 4, 789-808, (1996) · Zbl 0888.20011 [2] Benson, D., Spin modules for symmetric groups, J. Lond. Math. Soc. (2), 38, 2, 250-262, (1988) · Zbl 0669.20005 [3] Benson, D., Some remarks on the decomposition numbers for the symmetric groups, Proc. Sympos. Pure Math., 47, 381-394, (1987) [4] Bessenrodt, C.; Olsson, J., On residue symbols and the Mullineux conjecture, J. Algebraic Combin., 7, 3, 227-251, (1998) · Zbl 0906.05077 [5] Breuer, T., GAP-package ctbllib—the GAP character table library, (2013), Version 1.2.2 [6] Brundan, J.; Kleshchev, A., Representation theory of symmetric groups and their double covers, (Groups, Combinatorics & Geometry, Durham, (2001)), 31-53 · Zbl 1043.20005 [7] Brundan, J.; Kleshchev, A., Representations of the symmetric group which are irreducible over subgroups, J. Reine Angew. Math., 530, 145-190, (2001) · Zbl 1059.20016 [8] Danz, S., Vertices of low-dimensional simple modules for symmetric groups, Comm. Algebra, 36, 12, 4521-4539, (2008) · Zbl 1226.20004 [9] Dipper, R.; James, G., Representations of Hecke algebras of general linear groups, Proc. Lond. Math. Soc., 52, 3, 20-52, (1986) · Zbl 0587.20007 [10] Fawcett, J.; O’Brien, E.; Saxl, J., Regular orbits of symmetric and alternating groups, J. Algebra, 458, 21-52, (2016) · Zbl 1395.20006 [11] Fayers, M., James’s conjecture holds for weight four blocks of Iwahori-Hecke algebras, J. Algebra, 317, 2, 593-633, (2007) · Zbl 1155.20007 [12] Fayers, M., Decomposition numbers for weight three blocks of symmetric groups and Iwahori-Hecke algebras, Trans. Amer. Math. Soc., 360, 3, 1341-1376, (2008) · Zbl 1178.20005 [13] Feit, W., The representation theory of finite groups, (1982), North-Holland · Zbl 0493.20007 [14] Ford, B., Irreducible restrictions of representations of the symmetric groups, Bull. Lond. Math. Soc., 27, 5, 453-459, (1995) · Zbl 0836.20009 [15] Ford, B.; Kleshchev, A., A proof of the Mullineux conjecture, Math. Z., 226, 2, 267-308, (1997) · Zbl 0958.20018 [16] Frame, J.; Robinson, G.; Thrall, R., The hook graphs of the symmetric groups, Canad. J. Math., 6, 316-324, (1954) · Zbl 0055.25404 [17] The GAP group, GAP—groups, algorithms, programming—a system for computational discrete algebra, (2015), Version 4.7.7 [18] Geck, M., Brauer trees of Hecke algebras, Comm. Algebra, 20, 10, 2937-2973, (1992) · Zbl 0770.20020 [19] Hiss, G.; Malle, G., Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math., LMS J. Comput. Math., 5, 95-126, (2002), Corrigenda: · Zbl 1053.20504 [20] James, G., On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc., 94, 3, 417-424, (1983) · Zbl 0544.20011 [21] James, G., On the decomposition matrices of the symmetric groups III, J. Algebra, 71, 1, 115-122, (1981) · Zbl 0465.20010 [22] James, G., The representation theory of the symmetric groups, Lecture Notes in Math., vol. 682, (1978), Springer · Zbl 0393.20009 [23] James, G., On the decomposition matrices of the symmetric groups I, J. Algebra, 43, 1, 42-44, (1976) · Zbl 0347.20005 [24] James, G., Representations of the symmetric groups over the field of order 2, J. Algebra, 38, 2, 280-308, (1976) · Zbl 0328.20013 [25] James, G.; Kerber, A., The representation theory of the symmetric group, Encyclopedia Math. Appl., vol. 16, (1981), Addison-Wesley [26] James, G.; Williams, A., Decomposition numbers of symmetric groups by induction, J. Algebra, 228, 1, 119-142, (2000) · Zbl 0961.20013 [27] Jansen, C.; Lux, K.; Parker, R.; Wilson, R., An atlas of Brauer characters, (1995), Clarendon Press Oxford · Zbl 0831.20001 [28] Jantzen, J.; Seitz, G., On the representation theory of the symmetric groups, Proc. Lond. Math. Soc. (3), 65, 3, 475-504, (1992) · Zbl 0779.20004 [29] Kashiwara, M., On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J., 63, 2, 465-516, (1991) · Zbl 0739.17005 [30] Kleshchev, A., Linear and projective representations of symmetric groups, Cambridge Tracts in Math., vol. 163, (2005), Cambridge University Press · Zbl 1080.20011 [31] Kleshchev, A., On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. Lond. Math. Soc. (3), 75, 3, 497-558, (1997) · Zbl 0907.20023 [32] Kleshchev, A., Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineux, J. Lond. Math. Soc. (2), 54, 1, 25-38, (1996) · Zbl 0854.20014 [33] Kleshchev, A., Branching rules for modular representations of symmetric groups II, J. Reine Angew. Math., 459, 163-212, (1995) · Zbl 0817.20009 [34] Kleshchev, A., Branching rules for modular representations of symmetric groups I, J. Algebra, 178, 2, 493-511, (1995) · Zbl 0854.20013 [35] Kleshchev, A., On restrictions of irreducible modular representations of semisimple algebraic groups and symmetric groups to some natural subgroups I, Proc. Lond. Math. Soc. (3), 69, 3, 515-540, (1994) · Zbl 0808.20039 [36] Lascoux, A.; Leclerc, B.; Thibon, J., Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys., 181, 1, 205-263, (1996) · Zbl 0874.17009 [37] Lübeck, F., Small degree representations of finite Chevalley groups in defining characteristic, LMS J. Comput. Math., 4, 135-169, (2001) · Zbl 1053.20008 [38] Maas, L., Modular spin characters of symmetric groups, (2011), University of Essen, Ph.D. thesis [39] Mathas, A., SPECHT—decomposition matrices for the Hecke algebras of type A, (1997), University of Sydney [40] Mathas, A., Iwahori-Hecke algebras and Schur algebras of the symmetric group, Univ. Lecture Ser., vol. 15, (1999), American Mathematical Society · Zbl 0940.20018 [41] Michel, J., CHEVIE—development version of the GAP part of CHEVIE [42] Müller, J., The 2-modular decomposition matrices of the symmetric groups $$S_{15}$$, $$S_{16}$$, and $$S_{17}$$, Comm. Algebra, 28, 4997-5005, (2000) · Zbl 0982.20008 [43] Mullineux, G., Bijections of p-regular partitions and p-modular irreducibles of the symmetric groups, J. Lond. Math. Soc. (2), 20, 1, 60-66, (1979) · Zbl 0401.05011 [44] Peel, M., Hook representations of the symmetric groups, Glasg. Math. J., 12, 136-149, (1971) · Zbl 0235.20012 [45] Richards, M., Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc., 119, 3, 383-402, (1996) · Zbl 0855.20011 [46] Scopes, J., Symmetric group blocks of defect two, Quart. J. Math. Oxford Ser. (2), 46, 182, 201-234, (1995) · Zbl 0835.20022 [47] Varagnolo, M.; Vasserot, E., On the decomposition matrices of the quantized Schur algebra, Duke Math. J., 100, 2, 267-297, (1999) · Zbl 0962.17006 [48] Wales, D., Some projective representations of $$S_n$$, J. Algebra, 61, 1, 37-57, (1979) · Zbl 0433.20010 [49] Wilson, R.; Thackray, J.; Parker, R.; Noeske, F.; Müller, J.; Lübeck, F.; Jansen, C.; Hiss, G.; Breuer, T., The modular atlas project
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