Essential dimension of involutions and subalgebras
Let \(F\) be a field, \(E\) an étale \(F\)-algebra, and \(A\) an Azumaya \(E\)-algebra. This work is a computation of both the essential dimension (ed \(G)\) and the essential \(p\)-dimension (ed\(_{p}\) \(G),\) \(p\) prime, of various subgroups \(G\) of \(R_{E/F}\left( \mathrm{GL}_{1}\left( A\right) \right) ,\) where \(R_{E/F}\) is the Weil restriction functor. A primary tool is a result of Karpenko on incompressibility. To provide the reader with some examples, let \(\left( A,\sigma\right) \) be a central simple \(F\)-algebra with involution, where \(A\) is a division algebra of degree \(2^{r},\;r\geq1\). Let \(\mathrm{Sim}\left( A,\sigma\right) \) (resp. \(\mathrm{SimGL}\left( A,\sigma\right)\) be the similitudes (resp. isometries) of \(A\). Then it is shown that \[ \text{ed} \mathrm{Sim}\left( A,\sigma\right) = \text{ed}_{2} \mathrm{Sim}\left( A,\sigma\right) =\dim_{F}A-\dim \mathrm{Sim}\left( A,\sigma\right), \] which is either \(2^{2r} -1,\;2^{r-1}\left( 2^{r}+1\right) -1,\) or \(2^{r-1}\left( 2^{r}-1\right) -1\;\)if \(\sigma\) is unitary, orthogonal, or symplectic respectively. Additionally, \[ \text{ed} \mathrm{Iso}\left( A,\sigma\right) =\text{ed}_{2} \mathrm{Iso}\left( A,\sigma\right) =\dim_{F}A-\dim \mathrm{Iso}\left( A,\sigma\right) , \] which is either \(2^{r-1}\left( 2^{r}+1\right) \) or \(2^{r-1}\left( 2^{r}-1\right) \) if \(\sigma\) is orthogonal or symplectic respectively. Further results for \(\mathrm{GO}\left( A,\sigma\right) \) and \(\mathrm{GO}^{+}\left( A,\sigma\right) \) are computed, the former in the case \(p=2,\) the latter for \(p>2.\) Another set of computations concern a normalizer. Let \(G=N_{\mathrm{GL}_{1}\left( A\right) }\left(\mathrm{GL}_{1}\left( B\right) \right),\) where \(A\) is a division \(F\)-algebra of degree \(p^{n}\) and \(B\subset A\) satisfying certain conditions involving non-negative integers \(a,b\) with \(a+b\leq n.\) Here it is shown that ed \(G=\)ed\(_{p}\) \(G=p^{2n}-p^{n+a-b} -p^{n-a+b}+p^{n-a-b}.\)
scientific article; zbMATH DE number 6127528
scientific article