Geodesic
Macdonald identities and multidimensional theta-functions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 67-77

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A new proof of the combinatorial Macdonald identities is presented. It is shown that one may regard these identities as decomposition of multidimensional theta-functions into infinite products. The proof is pased on some analytical properties of theta-functions. It is discussed briefly now one may modify the proof in order to replace analytical reasonings by the formal ones involving only operations with formal series. 
@article{ZNSL_1997_240_a4,
     author = {M. A. Vsemirnov},
     title = {Macdonald identities and multidimensional theta-functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--77},
     publisher = {mathdoc},
     volume = {240},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a4/}
}
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M. A. Vsemirnov. Macdonald identities and multidimensional theta-functions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 67-77. http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a4/