On groups associated with the affine subgroups of \(Sp_{2n}(2)\). (English) Zbl 07888185
Summary: The symplectic group \(Sp_{2n}(2)\) has an affine maximal subgroup of structure \(ASp_n = 2^{2n-1}: Sp_{2n-2}(2)\) which is a split extension of an elementary abelian 2-group \(N = 2^{2n-1}\) by \(G = Sp_{2n-2}(2)\). The vector space \(N = 2^{2n-1}\) and its dual \(N^\ast\) are not equivalent as \(2n-1\) dimensional \(G\)-modules over \(GF(2)\). Therefore, a split extension of the form \(\overline{G}_n = N^\ast:Sp_{2n-2}(2)\ncong N:Sp_{2n-2}(2)\) exists. In this paper, it will be shown that \(\overline{G}_n\cong \mathrm{Aut}(2^{2n-2}: Sp_{2n-2}(2)) = (2^{2n-2}:Sp_{2n-2}(2)):2\) for \(n \geq 3\). Moreover, the ordinary irreducible characters of \(\overline{G}_n\) are studied through the lens of Fischer-Clifford theory. As an example, the Fischer-Clifford matrix technique is used to construct the set \(\mathrm{Irr}(\overline{G}_5)\) of the group \(\overline{G}_5 = 2^9:Sp_8(2)\) which is associated with the affine subgroup \(ASp_5=2^9:Sp_8(2)\) of \(Sp_{10}(2)\).
MSC:
| 20C15 | Ordinary representations and characters |
| 20C40 | Computational methods (representations of groups) (MSC2010) |
Keywords:
automorphism group; affine subgroup; symplectic group; split extension; Fischer-Clifford matricesReferences:
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