Abstract
This paper contains the classification of the orbits of elements of the tensor product spaces \({{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}\), \({r\geq 1}\), under the action of two natural groups, for all finite; real; and algebraically closed fields. For each of the orbits we determine: a canonical form; the tensor rank; the rank distribution of the contraction spaces; and a geometric description. The proof is based on the study of the contraction spaces in \({{\mathrm{PG}}({\mathbb{F}}^2\otimes {\mathbb{F}}^3)}\) and is geometric in nature. Although the main focus is on finite fields, the techniques are mostly field independent.
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The research of the M. Lavrauw was supported by the Fund for Scientific Research—Flanders (FWO) and by a Progetto di Ateneo from Università di Padova (CPDA113797/11).
J. Sheekey acknowledges the support of the Fund for Scientific Research—Flanders (FWO).
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Lavrauw, M., Sheekey, J. Classification of subspaces in \({{\mathbb{F}}^2\otimes {\mathbb{F}}^3}\) and orbits in \({{\mathbb{F}}^2\otimes{\mathbb{F}}^3 \otimes {\mathbb{F}}^r}\) . J. Geom. 108, 5–23 (2017). https://doi.org/10.1007/s00022-016-0316-4
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DOI: https://doi.org/10.1007/s00022-016-0316-4