The invariant rings of the Sylow groups of GU(3,q²), GU(4,q²), Sp(4,q) and O⁺(4,q) in the natural characteristic

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• 作者：Jorge N.M. Ferreira^a ; ¹ ; ^{jorge.nelio.ferreira@staff.uma.pt" class="auth_mail" title="E-mail the corresponding author} ; Peter Fleischmann^b
• 关键词：Invariant rings ; SAGBI bases ; Modular invariant theory ; Sylow subgroups ; Finite classical groups
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P2
• 页码：356-371
• 全文大小：386 K

文摘

Let m>Gm> be a Sylow m>p  m>-subgroup of the unitary groups an id="mmlsi1" class="mathmlsrc">an 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,q²)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si1.gif" overflow="scroll"><mi>Gmi><mi>Umi><mo stretchy="false">(mo><mn>3mn><mo>,mo><msup><mrow><mi>qmi>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">)mo>math>an>an>an>, an id="mmlsi2" class="mathmlsrc">an 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,q²)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si2.gif" overflow="scroll"><mi>Gmi><mi>Umi><mo stretchy="false">(mo><mn>4mn><mo>,mo><msup><mrow><mi>qmi>mrow><mrow><mn>2mn>mrow>msup><mo stretchy="false">)mo>math>an>an>an>, the symplectic group an id="mmlsi3" class="mathmlsrc">an 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)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Smi><mi>pmi><mo stretchy="false">(mo><mn>4mn><mo>,mo><mi>qmi><mo stretchy="false">)mo>math>an>an>an> and, for m>q  m> odd, the orthogonal group an id="mmlsi4" class="mathmlsrc">an 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⁺(4,q)an>an class="mathContainer hidden">an class="mathCode"><math altimg="si4.gif" overflow="scroll"><msup><mrow><mi>Omi>mrow><mrow><mo>+mo>mrow>msup><mo stretchy="false">(mo><mn>4mn><mo>,mo><mi>qmi><mo stretchy="false">)mo>math>an>an>an>. In this paper we construct a presentation for the invariant ring of m>Gm> 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 m>SAGBIm> basis and the invariant ring for m>Gm> is a complete intersection.