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Tight relative ▫$t$▫-designs on two shells in hypercubes, and Hahn and Hermite polynomials
Bannai, Eiichi ...
Relativni ▫$t$▫-dizajni v ▫$n$▫-dimenzionalni hiperkocki ▫$\mathcal{Q}_n$▫ so ekvivalentni uteženim regularnim ▫$t$▫-kratnim uravnoteženim dizajnom, ki posplošujejo kombinatorne ▫$t-(n, k, \lambda)$▫ ...dizajne na ta način, da dovoljujejo večkratne velikosti blokov, pa tudi uteži. Motivirani tudi z nedavno študijo o tesnih evklidskih ▫$t$▫-dizajnih na dveh koncentričnih sferah, v članku obravnavamo tesne relativne ▫$t$▫-dizajne v hiperkocki ▫$\mathcal{Q}_n$▫, podprte na dveh lupinah. Pokažemo, da, pod blagim pogojem, velja, da takšen relativen ▫$t$▫-dizajn inducira strukturo koherentne konfiguracije z dvema vlaknoma. Iz te strukture sklepamo tudi, da mora imeti polinom iz družine Hahnovih hipergeometrijskih ortogonalnih polinomov samo celoštevilske enostavne ničle. Glavno orodje pri izpeljavi teh rezultatov je Terwilligerjeva algebra. Z eksplicitno oceno vedenja ničel Hahnovih polinomov, ki pri ustreznem limitnem procesu degenerirajo v Hermiteove polinome, dokažemo izrek, ki deloma dokazuje, da so netrivialni tesni relativni ▫$t$▫-dizajni v ▫$\mathcal{Q}_n$▫, podprti na dveh lupinah, za velike ▫$t$▫ redki.
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