Abstract
We prove an Amitsur–Levitzki-type theorem for Grassmann algebras, stating that the minimal degree of a standard identity that is a polynomial identity of the ring of n × n matrices over the m-generated Grassmann algebra is at least \(2\lfloor\frac{m}{2}\rfloor+4n-4\) for all n, m ≥ 2 and this bound is sharp for m = 2,3 and any n ≥ 2. The arguments are purely combinatorial, based on computing sums of signs corresponding to Eulerian trails in directed graphs.
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The research reported in this paper has been supported by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program). The research reported in this paper was supported by the BMEArtificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC).
This research was partially supported by National Research, Development and Innovation Office, NKFIH K 119934.
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Balázs, B.A., Mészáros, S. The minimal degree standard identity on MnE2 and MnE3. Isr. J. Math. 238, 279–312 (2020). https://doi.org/10.1007/s11856-020-2025-9
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DOI: https://doi.org/10.1007/s11856-020-2025-9