Essential dimension of involutions and subalgebras
type: Item
instance of: scholarly article
title: Essential dimension of involutions and subalgebras
zbMATH Open document ID: 1259.14050
author: Roland Lötscher
published in: Israel Journal of Mathematics
publication date: 2013-01-21T00:00:00Z
review text: 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\(_{p}\) \(G=p^{2n}-p^{n+a-b} -p^{n-a+b}+p^{n-a-b}.\)
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14L40
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 6127528
MaRDI profile type: Publication
full work available at URL: s11856-012-0037-9
OpenAlex ID: W1981719461
cites work: Open problems on central simple algebras.
cites work: Essential dimension: A functorial point of view (after A. Merkurjev)
cites work: Essential dimension of central simple algebras.
cites work: Q4345956
cites work: On the notion of canonical dimension for algebraic groups
cites work: Essential dimension of moduli of curves and other algebraic stacks
cites work: Essential dimensions of \(A_7\) and \(S_7\)
cites work: Tori and essential dimension
cites work: A lower bound on the essential dimension of a connected linear group
cites work: Division algebras over Henselian fields.
cites work: Q4488712
cites work: Essential dimension of finite \(p\)-groups.
cites work: Q4210545
cites work: Zero cycles on homogeneous varieties
cites work: Essential dimension of algebraic groups and integral representations of Weyl groups
cites work: Essential dimension of algebraic tori
cites work: FIELDS OF DEFINITION FOR DIVISION ALGEBRAS
cites work: Cohomological invariants of Jordan algebras with frames
cites work: Q5852631
cites work: Essential 𝑝-dimension of 𝐏𝐆𝐋(𝐩²)
cites work: A lower bound on the essential dimension of simple algebras.
cites work: The essential dimension of the normalizer of a maximal torus in the projective linear group
cites work: An upper bound on the essential dimension of a central simple algebra.
cites work: On the notion of essential dimension for algebraic groups
cites work: Essential \(p\)-dimension of \(\text{PGL}_n\).
cites work: Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties
cites work: Q4152672
cites work: Q3884236
DOI: 10.1007/S11856-012-0037-9
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description: scientific article; zbMATH DE number 6127528