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13.
On the split structure of lifted groups
Malnič, Aleksander ; Požar, Rok, 1986-
Naj bo ▫$\wp \colon \tilde{X} \to X$▫ regularna krovna projekcija povezanih grafov, grupa krovnih transformacij ▫$\rm{CT}_\wp$▫ pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov ▫$G \le ...\rm{Aut} X$▫ dvigne vzdolž ▫$\wp$▫ do grupe ▫$\tilde{G} \le \rm{Aut} \tilde{X}$▫, podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje ▫$\rm{id} \to \rm{CT}_\wp \to \tilde{G} \to G \to \rm{id}$▫ razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natančno. V gornjem kombinatoričnem sestavu je razširitev podana samo implicitno: podani niso ne ▫$\tilde{G}$▫ ne delovanje ▫$G\to \rm{Aut} \rm{CT}_\wp$▫ ne 2-kocikel ▫$G \times G \to \rm{CT}_\wp$▫. Eksplicitno konstruiranje krova ▫$\tilde{X}$▫ ter ▫$\rm{CT}_\wp$▫ in ▫$\tilde{G}$▫ kot permutacijskih grup na ▫$\tilde{X}$▫ je časovno in prostorsko zahtevno vselej, kadar je ▫$\rm{CT}_\wp$▫ velik; tako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejšo informacijo o delovanju in 2-kociklu učinkovito izluščimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo); zdaj bi bilo mogoče uporabiti standardno metodo reduciranja problema na reševanje sistema linearnih enačb nad celimi števili. Vendar tukaj uberemo malce drugačen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Časovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je ▫$\rm{CT}_\wp$▫ elementarna abelova.
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