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The Waring problem for Lie groups and Chevalley groups

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Abstract

The classical Waring problem deals with expressing every natural number as a sum of g(k) k th powers. Similar problems were recently studied in group theory, where we aim to present group elements as short products of values of a given word w ≠ 1. In this paper we study this problem for Lie groups and Chevalley groups over infinite fields.

We show that for a fixed word w ≠ 1 and for a classical connected real compact Lie group G of sufficiently large rank we have w(G)2 = G, namely every element of G is a product of 2 values of w.

We prove a similar result for non-compact Lie groups of arbitrary rank, arising from Chevalley groups over ℝ or over a p-adic field. We also study this problem for Chevalley groups over arbitrary infinite fields, and show in particular that every element in such a group is a product of two squares.

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Author information

Authors and Affiliations

  1. Mathematics Research Unit, University of Luxembourg, 6 rue Richard Coudenhove-Kalergi, L-1359, Luxembourg, Luxembourg

    Chun Yin Hui

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Michael Larsen

  3. Einstein Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

    Aner Shalev

Authors
  1. Chun Yin Hui
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  2. Michael Larsen
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  3. Aner Shalev
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Corresponding author

Correspondence to Chun Yin Hui.

Additional information

The first author was supported by ERC Advanced Grant no. 247034.

The second author was partially supported by the Simons Foundation, the MSRI, NSF Grant DMS-1101424, and BSF Grant no. 2008194.

The third author was partially supported by ERC Advanced Grant no. 247034, ISF grant no. 1117/13, BSF Grant no. 2008194 and the Vinik Chair of Mathematics which he holds

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Hui, C.Y., Larsen, M. & Shalev, A. The Waring problem for Lie groups and Chevalley groups. Isr. J. Math. 210, 81–100 (2015). https://doi.org/10.1007/s11856-015-1246-9

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  • DOI: https://doi.org/10.1007/s11856-015-1246-9

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