Amitsur–Levitzki theorem

The standard polynomial of degree n is

${\displaystyle S_{n}(x_{1},\dots ,x_{n})=\sum _{\sigma \in S_{n}}{\text{sgn}}(\sigma )x_{\sigma (1)}\cdots x_{\sigma (n)}}$

in non-commuting variables x₁, ..., x_n, where the sum is taken over all n! elements of the symmetric group S_n.

The Amitsur–Levitzki theorem states that for n × n matrices A₁, ..., A_2n whose entries are taken from a commutative ring then

${\displaystyle S_{2n}(A_{1},\dots ,A_{2n})=0.}$

Amitsur and Levitzki (1950) gave the first proof.

Kostant (1958) deduced the Amitsur–Levitzki theorem from the Koszul–Samelson theorem about primitive cohomology of Lie algebras.

Swan (1963) and Swan (1969) gave a simple combinatorial proof as follows. By linearity it is enough to prove the theorem when each matrix has only one nonzero entry, which is 1. In this case each matrix can be encoded as a directed edge of a graph with n vertices. So all matrices together give a graph on n vertices with 2n directed edges. The identity holds provided that for any two vertices A and B of the graph, the number of odd Eulerian paths from A to B is the same as the number of even ones. (Here a path is called odd or even depending on whether its edges taken in order give an odd or even permutation of the 2n edges.) Swan showed that this was the case provided the number of edges in the graph is at least 2n, thus proving the Amitsur–Levitzki theorem.

Razmyslov (1974) gave a proof related to the Cayley–Hamilton theorem.

Rosset (1976) gave a short proof using the exterior algebra of a vector space of dimension 2n.

Procesi (2015) gave another proof, showing that the Amitsur–Levitzki theorem is the Cayley–Hamilton identity for the generic Grassman matrix.

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• Swan, Richard G. (1969), "Correction to "An application of graph theory to algebra"" (PDF), Proceedings of the American Mathematical Society, 21 (2): 379–380, doi:10.2307/2037008, ISSN 0002-9939, JSTOR 2037008, MR 0255439
• Procesi, Claudio (2015), "On the theorem of Amitsur—Levitzki", Israel Journal of Mathematics, 207: 151–154, arXiv:1308.2421, Bibcode:2013arXiv1308.2421P, doi:10.1007/s11856-014-1118-8