Shuffle Product for Multiple Dedekind Zeta Values over Imaginary Quadratic Fields

Multiple Dedekind zeta values were recently defined by the second author. In a separate paper, the second author constructed double shuffle relations in some cases as a response to questions asked by Richard Hain and Alexander Goncharov. In this paper, we develop the technique for obtaining more shuffle relations and produce many examples of shuffle products over an imaginary quadratic field. We also define the notion of self shuffle of a (multiple) Dedekind zeta value and use it at many instances. We define a refinement of the multiple Dedekind zeta values. Our key examples are self shuffles of the Dedekind zeta at 2 and at 3, the shuffle product of the Dedekind zeta of 2 times itself, and the shuffle product of the Dedekind zeta at 2 times the Dedekind zeta at 3. We obtain one unexpected result that the self shuffle of multiple Dedekind zeta at (1,2) minus the self shuffle of the twisted (with a permutation) multiple Dedekind zeta value at (1,2) is a very simple expression in terms of the refined multiple Dedekind zeta values.