On the coordinate rings of quiver varieties associated to extended Dynkin diagrams (Q5930008)

\textit{P. B. Kronheimer} [J. Differ. Geom. 29, 665-683 (1989; Zbl 0671.53045)] has constructed quiver varieties from extended Dynkin diagrams of types \(\widetilde A_n,\widetilde D_n,\widetilde E_n\). These quiver varieties are important objects for the study of simple singularities. Let \(p\) be the quotient map of the Cartan subalgebra by the Weyl group. The semiuniversal deformations of simple singularities are constructed on the quotient space of Cartan subalgebra by the Weyl group of corresponding types. Then these quiver varieties are the pull-back of semiuniversal deformations of simple singularities by the quotient map \(p\). These quiver varieties are obtained as the symplectic quotients of symplectic vector spaces by a reductive group. So the coordinate rings are invariant subrings of polynomial rings with respect to the action of the group. In general it is difficult to find a minimal set of generators of an invariant ring and the relations between them.NEWLINENEWLINENEWLINEIn this paper the author shows that this is possible for the case of quiver varieties constructed by P. B. Kronheimer. Moreover surprisingly it can be shown that the obtained relation is unique and irreducible. In this research the invariant theory of quivers by \textit{L. Le Bruyn} and \textit{C. Procesi} [Trans. Am. Math. Soc. 317, 585-598 (1990; Zbl 0693.16018)] and of matrices of low degrees by \textit{K. Nakamoto} [J. Pure Appl. Algebra 166, 125-148 (2002; Zbl 1001.15022)] is used.