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Extra resources for Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman
Proof of Theorem 2. 7) together with the cyclic symmetry just proved implies that Tn (a, b, c; x, y, z) is invariant with respect to cyclic permutations of the three pairs of variables (a,x), (b,y), and (c,z). 8) that Tn(a, b, c; x, y, z) is an even function of b and of c. From the cyclic invariance it follows that the three variables a, b, and c play equal roles, so it is also an even function of a. 8) we see that Tn is unchanged if we interchange (b, y) and (c, z) and simultaneously replace a by -a, so we obtain also the invariance of Tn under odd permutations of (a, x), (b, y), and (c, z).
The equations L *wa = zw a, and 8ko wa = - (L kRo) ~ wa hold. Remark 2. Our definition of the mcKP differs from that in '[2J and [3J where Uo = 0 and A = I.
Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. (Math. ) 104 (1994), 57-75. [Z2] D. Zagier, Some combinatorial identities occurring in the theory of modular forms, in preparation. Paula Beazley Cohen URA Geometrie-Analyse-Topologie, UFR de Mathematiques, Universite des Sciences et Technologies de Lille 59655 Villeneuve d'Ascq cedex, France Yuri Manin and Don Zagier Max-Planck Institut fur Mathematik Gottfried-Claren str. 26, 53225 Bonn, Germany Received October 1995: additions December 1995 On T -FUnctions of Zakharov-Shabat and Other Matrix Hierarchies of Integrable Equations L.