Abstract
In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\). We present basis-free formulas for all characteristic polynomial coefficients in the cases \(n\le 6\), alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case \(n=4\), and one of the formulas in the case \(n=5\). We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases \(n\le 5\). The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.
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Notes
The Clifford conjugation is a superposition of the grade involution \(\widehat{~~}\) and the reversion \(\widetilde{~~}\). Note that some authors [19] denote the Clifford conjugation by \({\mathop {\quad }\limits ^{\overline{\quad }}}\). We do not use separate notation for the Clifford conjugation in this paper and write the combination of the two symbols \(\widehat{~~}\) and \(\widetilde{~~}\).
Note that the property (2.12) is also follows from the definition of \(\bullet \) and the fact that grade projections commute with operations of conjugation.
The commutator and anticommutator of two arbitrary elements \(U,V \in \mathcal {G}_{p,q}\) are denoted by \([U,V]=UV-VU\) and \(\{U,V\}=UV+VU\) respectively.
We remind that we use the simplified notation \(U_k:=\langle U \rangle _k\) in this paper.
Note that alternatively one could get directly \(D_{(k)}(\lambda )=\frac{1}{(N-k)!}\frac{ \partial ^{N-k} D_{(N)}(\lambda )}{ \partial \lambda ^{N-k}}\).
References
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Acknowledgements
The main results of this paper were reported at the International Conference “Computer Graphics International 2021 (CGI2021)” (online, Geneva, Switzerland, September 2021). The authors are grateful to the organizers and the participants of this conference for fruitful discussions. The authors are grateful to the anonymous Reviewers for their careful reading of the paper and helpful comments on how to improve the presentation. The publication was prepared within the framework of the Academic Fund Program at the HSE University in 2022 (grant 22-00-001).
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This article is part of the Topical Collection on ENGAGE 2021 edited by Andreas Aristidou.
Appendix A: Basis-Free Formulas in the Case \(n=6\)
Appendix A: Basis-Free Formulas in the Case \(n=6\)
In the case \(n=6\), we have the following basis-free formulas for the characteristic polynomial coefficients \(C_{(3)}, C_{(4)}, C_{(5)}, C_{(6)}\in \mathcal {G}_{p,q}\). The remaining characteristic polynomial coefficients \(C_{(1)}, C_{(2)}, C_{(7)}, C_{(8)}\in \mathcal {G}_{p,q}\) are presented in Sect. 8.
$$\begin{aligned} C_{(3)}= & {} (\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}+U(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup } +U\widetilde{U}\widehat{U}^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}+U\widetilde{U}\widetilde{U}^{\bigtriangleup } + U\widetilde{U}U^{\bigtriangleup }\\&+ U\widetilde{U}\widehat{U}+\widetilde{U}(U\widetilde{U})^{\bigtriangleup }+U(U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}) +\frac{1}{3}((\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+(\widehat{U}U\widetilde{U})^{\bigtriangleup } +(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup } +\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup } +\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup } +\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\\&+\widehat{U}(U\widetilde{U})^{\bigtriangleup }+\widehat{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup } +\widehat{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+ \widehat{U}(\widehat{U}\widetilde{U})^{\bigtriangleup } +\widehat{U}(\widehat{U}U)^{\bigtriangleup } +\widehat{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup } + \widehat{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup }\\&+\widehat{U}\widehat{\widetilde{U}}U^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup } +\widetilde{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widetilde{U}(\widehat{U}\widetilde{U})^{\bigtriangleup } +\widetilde{U}(\widehat{U}U)^{\bigtriangleup }\\&+\widetilde{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup } +\widetilde{U}\widehat{\widetilde{U}}U^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup } +\widetilde{U}\widehat{U}\widetilde{U}^{\bigtriangleup }+\widetilde{U}\widehat{U}U^{\bigtriangleup } +\widetilde{U}\widehat{U}\widehat{\widetilde{U}}^{\bigtriangleup }\\&+\widetilde{U}\widehat{U}\widehat{U}^{\bigtriangleup } +U(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U(\widehat{\widetilde{U}}U)^{\bigtriangleup }+U(\widehat{U}\widetilde{U})^{\bigtriangleup }+U(\widehat{U}U)^{\bigtriangleup } +U\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup } +U\widehat{\widetilde{U}}U^{\bigtriangleup }\\&+U\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+U\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup } +U\widehat{U}\widetilde{U}^{\bigtriangleup }+ U\widehat{U}U^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}^{\bigtriangleup }+U\widehat{U}\widehat{U}^{\bigtriangleup }) +\frac{2}{3}(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }\\&+\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\\&+(\widehat{U}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U)^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }\\&+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup }+\widetilde{U}\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+\widetilde{U}\widehat{U}^{\bigtriangleup }U^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }U)^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }U)^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }\\&+\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+U\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+U\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup }+U\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+U\widehat{U}^{\bigtriangleup }U^{\bigtriangleup } +U(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }\\&+U(\widehat{\widetilde{U}}^{\bigtriangleup }U)^{\bigtriangleup } +U(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+ U(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+U(\widehat{U}^{\bigtriangleup }U)^{\bigtriangleup }+U(\widehat{U}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }\widehat{U})^{\bigtriangleup }), \end{aligned}$$
$$\begin{aligned} C_{(4)}= & {} -( (U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}+U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}(U\widetilde{U})^{\bigtriangleup }) +\frac{1}{3}((\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }\\&+\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widehat{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widehat{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }\\&+\widehat{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ \widehat{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ \widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ \widetilde{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}\widehat{U}(U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{U}\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{U}U)^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup }\\&+ \widetilde{U}\widehat{U}\widehat{\widetilde{U}}U^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup }+U(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup } +U(\widehat{U}U\widetilde{U})^{\bigtriangleup }\\&+U(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+U\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup } +U\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+U\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup } +U\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widehat{U}(U\widetilde{U})^{\bigtriangleup }+U\widehat{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+U\widehat{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup } +U\widehat{U}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widehat{U}(\widehat{U}U)^{\bigtriangleup }+ U\widehat{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup }\\&+U\widehat{U}\widehat{\widetilde{U}}U^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{U}U)^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup } + U\widetilde{U}\widehat{\widetilde{U}}U^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }\\&+U\widetilde{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup }+U\widetilde{U}\widehat{U}\widetilde{U}^{\bigtriangleup } +U\widetilde{U}\widehat{U}U^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}^{\bigtriangleup }+U\widetilde{U}\widehat{U}\widehat{U}^{\bigtriangleup }) \\&+\frac{2}{3}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }\!\!(\widehat{\widetilde{U}}^{\bigtriangleup }\!\!(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }\!\!(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ ((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ ((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\\&+\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup } +\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\\&+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U)^{\bigtriangleup }+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U})^{\bigtriangleup }\\&+U\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+ U\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+ U(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+U(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup }\\&+U(\widehat{\widetilde{U}}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U)^{\bigtriangleup }\\&+U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }\\&+U\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup }+U\widetilde{U}\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }+ U\widetilde{U}\widehat{U}^{\bigtriangleup }U^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }U)^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+ U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U})^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}^{\bigtriangleup }U)^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{U})^{\bigtriangleup })), \end{aligned}$$
$$\begin{aligned} C_{(5)}= & {} \frac{1}{3}(\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widehat{U}\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup } +\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup } +\widetilde{U}\widehat{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }\\&+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+U\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+ U\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+U\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+U\widehat{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup } + U\widehat{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+U\widehat{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+U\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup } +U\widehat{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+ U\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+ U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }+U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }\\&+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}\widehat{U}(U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}U)^{\bigtriangleup }\\&+U\widetilde{U}\widehat{U}(\widehat{U}\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{U}(\widehat{U}U)^{\bigtriangleup } +U\widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widetilde{U}^{\bigtriangleup }\\&+U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}U^{\bigtriangleup }\!\!+U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widehat{\widetilde{U}}^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}\widehat{U}^{\bigtriangleup })+ \frac{2}{3}((\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ ((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup } +\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }\\&+U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+U\widetilde{U}\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U)^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{\widetilde{U}})^{\bigtriangleup }\\&+U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U})^{\bigtriangleup }), \end{aligned}$$
$$\begin{aligned} C_{(6)}= & {} -(\frac{1}{3}(\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }\\&+\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ \widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ U\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }\\&+U\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }+ U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+U\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}U\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ U\widetilde{U}\widehat{U}(\widehat{\widetilde{U}}U\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{U}(\widehat{U}U\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{U}(\widehat{U}\widehat{\widetilde{U}}U)^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(U\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{\widetilde{U}}U)^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}U)^{\bigtriangleup }+ U\widetilde{U}\widehat{U}\widehat{\widetilde{U}}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup })\\&+\frac{2}{3}(((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\!+\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }\!\!\!((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&\!+\widetilde{U}(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup } \\&+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ \widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup } \\&+U((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}(\widehat{\widetilde{U}}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }(U\widetilde{U})^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U\widetilde{U}(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}(\widehat{U}^{\bigtriangleup }((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }U\widetilde{U})^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup } \\&+U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{\widetilde{U}}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+ U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }\widetilde{U}^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }\\&+U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }(\widehat{U}^{\bigtriangleup }U^{\bigtriangleup })^{\bigtriangleup })^{\bigtriangleup }+U\widetilde{U}((\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup }\widehat{U}\widehat{\widetilde{U}})^{\bigtriangleup })) \end{aligned}$$
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Abdulkhaev, K., Shirokov, D. Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras. Adv. Appl. Clifford Algebras 32, 57 (2022). https://doi.org/10.1007/s00006-022-01232-0
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DOI: https://doi.org/10.1007/s00006-022-01232-0
Keywords
- Basis-free formula
- Characteristic polynomial
- Clifford algebra
- Geometric algebra
- Grade projection
- Operation of conjugation
- Spin group