Diagonally and antidiagonally symmetric alternating sign matrices of odd order
Behrend, Roger E. ; Fischer, Ilse ; Konvalinka, Matjaž
Preučujemo preštevanje diagonalno in antidiagonalno simetričnih matrik z alternirajočim predznakom (DADASM) fiksnega lihega reda, tako da vpeljemo model šestih vozlišč, katerega konfiguracije so v ...bijekciji s takimi matrikami. Model vsebuje grid graf na trikotniku, z notranjimi in robnimi utežmi, ki zadoščajo Yang-Baxterjevi enačbi in zrcalni enačbi. Dobimo splošen izraz za particijsko funkcijo tega modela kot vsoto dveh determinant in dokažemo, da se vsak od teh izrazov specializira v Schurovo funkcijo. S pomočja tega dokažemo Robbinsovo domnevo iz sredine osemdesetih let prejšnjega stoletja, da je skupno število ▫$(2n + 1) \times (2n + 1)$▫ DADASM-jev enako ▫$\prod_{i=0}^n \frac{(3i)!}{(n+i)!}$▫ in domnevo Stroganova iz leta 2008, da je razmerje med ▫$(2n + 1) \times (2n + 1)$▫ DASASM-ji s centralnim elementom -1 in 1 enako ▫$n/(n + 1)$▫. Med produktnimi formulami za število matrik z alternirajočim predznakom, ki so bile zapisane kot domneve v osemdesetih, je ta zadnja, ki je bila dokazana.
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