On the ring of simultaneous invariants for the Gleason-MacWilliams group
Invariants of the irreducible complex reflection group \(G\) of order~192, which appears as number~9 in the list of Shephard and Todd, are of special interest in coding theory [see \textit{A. M. Gleason}, Actes Congr. Internat. Math., Nice 1970, 3, 211-215 (1971; Zbl 0287.05010)]. They also play a role in the construction of modular forms and Jacobi forms. In this paper the authors study vector invariants of \(G\), i.e., invariants from \({\mathbb C}[V ^l] ^G\), where \(V\) is the natural 2-dimensional representation and \(V ^l\) is the direct sum of~\(l\) copies thereof. By the theory of polarization, it suffices to study multilinear invariants. The authors give generators for the spaces \(R _l := {\mathbb C}[V ^l] ^G _{1,\dots,1}\) of multilinear invariants. This is achieved by decomposing the action of the symmetric group \(S _l\) on \(R _l\) into irreducibles. The paper contains interesting techniques, although the resulting generating set appears somewhat technical.
scientific article; zbMATH DE number 1382654
scientific article