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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
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
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
Email: rowen@math.biu.ac.il

DOI: https://doi.org/10.1090/proc/12478
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