## A lower bound for the dimension of a highest weight module

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- by Daniel Goldstein, Robert Guralnick and Richard Stong PDF
- Represent. Theory
**21**(2017), 611-625

## Abstract:

For each integer $t>0$ and each simple Lie algebra $\mathfrak {g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak {g}$ whose highest weight has width $t$. As a consequence, we classify all irreducible modules whose dimension equals a product of two primes. This consequence, which was in fact the driving force behind our paper, answers a question of N. Katz.## References

- Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - N. Bourbaki,
*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR**0240238** - E. Breuillard and G. Pisier,
*Remarks on random unitaries and amenable linear groups*, in preparation. - William Fulton and Joe Harris,
*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249**, DOI 10.1007/978-1-4612-0979-9 - Roe Goodman and Nolan R. Wallach,
*Representations and invariants of the classical groups*, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR**1606831** - Kathryn E. Hare,
*The size of characters of compact Lie groups*, Studia Math.**129**(1998), no. 1, 1–18. MR**1611918**, DOI 10.4064/sm-129-1-1-18 - Jens Carsten Jantzen,
*Low-dimensional representations of reductive groups are semisimple*, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 255–266. MR**1635685** - Nicholas M. Katz,
*Exponential sums and differential equations*, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR**1081536**, DOI 10.1515/9781400882434 - Nicholas M. Katz,
*Gauss sums, Kloosterman sums, and monodromy groups*, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR**955052**, DOI 10.1515/9781400882120 - Nicholas M. Katz,
*Convolution and equidistribution*, Annals of Mathematics Studies, vol. 180, Princeton University Press, Princeton, NJ, 2012. Sato-Tate theorems for finite-field Mellin transforms. MR**2850079** - Frank Lübeck,
*Small degree representations of finite Chevalley groups in defining characteristic*, LMS J. Comput. Math.**4**(2001), 135–169. MR**1901354**, DOI 10.1112/S1461157000000838 - Robert Steinberg,
*Lectures on Chevalley groups*, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR**0466335**

## Additional Information

**Daniel Goldstein**- Affiliation: Center for Communications Research, San Diego, California 92121
- MR Author ID: 709300
- Email: danielgolds@gmail.com
**Robert Guralnick**- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 78455
- Email: guralnic@usc.edu
**Richard Stong**- Affiliation: Center for Communications Research, San Diego, California 92121
- MR Author ID: 167705
- Email: stong@ccrwest.org
- Received by editor(s): April 1, 2016
- Received by editor(s) in revised form: August 11, 2017
- Published electronically: December 19, 2017
- Additional Notes: The second author was partially supported by NSF grants DMS-1302886 and DMS-1600056.
- © Copyright 2017 Institute for Defense Analyses
- Journal: Represent. Theory
**21**(2017), 611-625 - MSC (2010): Primary 17B10, 22E46
- DOI: https://doi.org/10.1090/ert/509
- MathSciNet review: 3738091