Polynomial identities for matrices over the Grassmann algebra (Q2408028)

Let \(R\) be a commutative ring with \(1\) and let \(E^{m}\) be the Grassman algebra (exterior algebra) over the free \(R\)-module with basis \(v_{1},\dots,v_{m}\). So \(E^{m}=\bigoplus_{i=0}^{m}E_{i}^{m}\) where \(E_{i}^{m}\) is a free \(R\)-module of rank \(\binom{m}{i}\) and \(E_{0}^{m}=R\). Every \(n\times n\) matrix over \(E^{m}\) can be written \(A=\sum_{i=0}^{m}A_{i}\) where the entries of \(A_{i}\) lie in \(E_{i}^{m}\). It is shown in [\textit{L. Márki} et al., Isr. J. Math. 208, 373--384 (2015; Zbl 1344.16023)] that if \(R\) is a field of characteristic \(0\), then \(A\) satisfies a polynomial of degree at most \(n2^{m-1}\) over \(R\).\ The present paper improves this bound to \(n(\left\lceil m/2\right\rceil +1)\) for the general case. More precisely, it is shown that if \(f(x):=\det (xI-A_{0})\) (note that \(A_{0}\) is a matrix over \(R\)), then \(f(A)^{r}=0\) for \(r=\left\lceil m/2\right\rceil +1\). Moreover, if \(R\) is a field and has characteristic \(0\) or characteristic \(p>\left\lceil m/2\right\rceil \), then \(f(x)^{r}\) is the minimal polynomial for \(A\) over \(R\). In a (generally noncommutative) ring \(\mathcal{A}\) the Cappeli polynomial \(d_{k}\) is defined by \(d_{k}(x_{1},\dots,x_{k};y_{0},\dots,y_{k}):=\sum_{\pi }(\text{sgn}\pi)y_{0}x_{\pi(1)}y_{1}x_{\pi(2)}...y_{k-1}x_{\pi (k)}y_{k}\) where \(\pi\) runs over all permutations of \([1,\dots,k]\). The Capelli identity of \(x\)-degree \(k\) holds in \(\mathcal{A}\) if the value of \(d_{k}\) is zero for all \(x_{i}\) and \(y_{i}\) in \(\mathcal{A}\). It is well known that the Capelli identity of \(x\)-degree \(n^{2}+1\) holds in the ring of \(n\times n\) matrices over a commutative ring, but the identity for \(x\)-degree \(n^{2}\) does not hold. The author shows that the ring of \(n\times n\) matrices over \(E^{m}\) satisfies the Capelli identity of \(x\)-degree \(n^{2}+2\left\lfloor m/2\right\rfloor +1\) but in general for no smaller degree. Similarly, generalizing the Amitsur-Levitzki theorem, it is shown that the standard identity of degree \(2n(\left\lfloor m/2\right\rfloor +1)\) holds for these matrix rings (the paper referred to above gives a weaker result).