The invariant rings of the Sylow groups of GU(3,q<sup>2sup>), GU(4,q<sup>2sup>), Sp(4,q) and O<sup>+sup>(4,q) in the natural characteristic
文摘
Let G be a Sylow p -subgroup of the unitary groups <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116000225&_mathId=si1.gif&_user=111111111&_pii=S0747717116000225&_rdoc=1&_issn=07477171&md5=dfd5516288f027b70f00888950a5ee08" title="Click to view the MathML source">GU(3,q<sup>2sup>)span><span class="mathContainer hidden"><span class="mathCode">span>span>span>, <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116000225&_mathId=si2.gif&_user=111111111&_pii=S0747717116000225&_rdoc=1&_issn=07477171&md5=8a87185a51c07ea0caa14f34e075c9aa" title="Click to view the MathML source">GU(4,q<sup>2sup>)span><span class="mathContainer hidden"><span class="mathCode">span>span>span>, the symplectic group <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116000225&_mathId=si3.gif&_user=111111111&_pii=S0747717116000225&_rdoc=1&_issn=07477171&md5=f91c5d9153a0ef685c0e38cdfd54ef53" title="Click to view the MathML source">Sp(4,q)span><span class="mathContainer hidden"><span class="mathCode">span>span>span> and, for q odd, the orthogonal group <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116000225&_mathId=si4.gif&_user=111111111&_pii=S0747717116000225&_rdoc=1&_issn=07477171&md5=2baf6b4499905367b06a2063b755aba7" title="Click to view the MathML source">O<sup>+sup>(4,q)span><span class="mathContainer hidden"><span class="mathCode">span>span>span>. In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.