Analogue of vertex operator triality for ternary codes (Q1892304)

For each self-orthogonal ternary code of length \(k\) which contains \((1, \dots, 1)\) the author constructs an integral even lattice \(L_0\) and a lattice \(L = L_{-1} \cup L_0 \cup L_1\) which is a union of cosets for \(L_0\). For each coset the author constructs one untwisted space \(V_{L_1}\) and two twisted spaces \(V_{T_{L_1}}\) and \(V_{\widetilde T_{L_1}}\); let us denote these nine spaces by \(V_0, \dots, V_8\). The author also constructs the corresponding untwisted and twisted vertex operators on \(V_{L_0} = V_0\) with coefficients in \(\text{End} V_i\), say \(Y_{V_i} (-,z)\). All nine untwisted and twisted spaces \(V_i\) are irreducible for the corresponding vertex operators and they provide different realizations of the affine Lie algebra \(\text{sl} (3, \mathbb{C})^\wedge\) [\textit{X. Xu}, Algebra Colloq. 2, 241-268 (1995; Zbl 0838.17030; see the preceding review)]. The main result of the paper under review is a construction of an order 4 linear automorphism \(\sigma\) of \(V_0 \oplus \cdots \oplus V_8\), where \(\sigma (V_0) = V_0\) and \(\sigma (V_i) = V_j\) \((j\) depends on \(i)\), such that \(\sigma Y_{V_i} (u,z) \sigma^{-1} = Y_{\sigma^{-1} V_i} (\sigma u,z)\). This is an analogue of the vertex operator triality found by \textit{I. B. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman} [Vertex operator algebras and the Monster, Pure Appl. Math. 134, Academic Press (1988; Zbl 0674.17001)].