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The images of multilinear polynomials evaluated on $3\times 3$ matrices


Authors: Alexey Kanel-Belov, Sergey Malev and Louis Rowen
Journal: Proc. Amer. Math. Soc. 144 (2016), 7-19
MSC (2010): Primary 16R99, 15A24, 17B60; Secondary 16R30
DOI: https://doi.org/10.1090/proc/12478
Published electronically: September 11, 2015
MathSciNet review: 3415572
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Abstract:

Let $p$ be a multilinear polynomial in several noncommuting variables, with coefficients in an algebraically closed field $K$ of arbitrary characteristic. In this paper we classify the possible images of $p$ evaluated on $3\times 3$ matrices. The image is one of the following:

  • {0},

  • the set of scalar matrices,

  • a (Zariski-)dense subset of $\operatorname {sl}_3(K)$, the matrices of trace 0,

  • a dense subset of $M_3(K)$,

  • the set of $3$-scalar matrices (i.e., matrices having eigenvalues $( \beta , \beta \varepsilon , \beta \varepsilon ^2)$ where $\varepsilon$ is a cube root of 1), or

  • the set of scalars plus $3$-scalar matrices.


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    Additional Information

    Alexey Kanel-Belov
    Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
    MR Author ID: 251623
    ORCID: 0000-0002-1371-7479
    Email: beloval@math.biu.ac.il

    Sergey Malev
    Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
    Email: malevs@math.biu.ac.il

    Louis Rowen
    Affiliation: Department of Mathematics, Bar Ilan University, Ramat Gan, Israel
    MR Author ID: 151270
    Email: rowen@math.biu.ac.il

    Keywords: Noncommutative polynomial, image, multilinear, matrices
    Received by editor(s): June 30, 2013
    Received by editor(s) in revised form: December 29, 2013
    Published electronically: September 11, 2015
    Additional Notes: This work was supported by the Israel Science Foundation (grant no. 1207/12)
    The second named author was supported by an Israeli Ministry of Immigrant Absorbtion scholarship.
    Communicated by: Birge Huisgen-Zimmermann
    Article copyright: © Copyright 2015 American Mathematical Society