Some Results Of Researching Hurwitz Problem On Composition Of Sums Of Squares By Twisted Group Algebras On Z₂ⁿ

The Hurwitz problem on composition of sums of squares is tackled with the help of a particular class of twisted group algebras Pn(m)which generalizes Clif-ford algebras Cl0,n,higher octonion(?)n.The constructed series of algebras pro-vide us a uniform explicit construction of Hurwitz-Radon square identities.By adjusting twisted function of concerned algebras carefully,we provide a new proof of Yuzvinsky-Lam-Smith formulas and confirm the family of admissible triples in the case when n?(mod 4)which were proposed by Yuzvinsky in 1984.Discus-sion of the construction of Yuzvinsky set pairs brings us the improvement of the Lenzhen-Morier-Genoud-Ovsienko formulas about infinite families of solutions on Hurwitz problem,and inspires us with the construction of several new infinite families of admissible triples.By constructing Yuzvinsky set pairs of new twist-ed group algebras from known algebras,we generalize the doubling construction,which is a fast and important way to generate new solutions to the Hurwitz prob-lem from any known ones.This method,which works in integral case,induces the construction of new intercalate matrix from known ones,which works in any field of characteristic not 2.Yuzvinsky conjecture,which describes the equivalence between the lower bound of non-signed intercalate matrix size and Hopf-stiefel function,is confirmed in many situations by inducing hooking lemma.We intro-duce several lemmas and prove that a series of non-signed intercalate matrices of special size do not exist.The dissertation is made up of four chapters:In the first chapter,we introduce history,development and our research progress of Hurwitz problem and Yuzvinsky conjecture.We start from ancient square i-dentities and introduce Hurwitz problem and Hurwitz-Radon theorem.Briefly,we introduce Lenzhen-Morier-Genoud-Ovsienko formulas and three families of admissible triples which were proposed by Yuzvinsky.Then we introduce some results of the upper and lower bound in Hurwitz problem.At last,we present our research progress briefly.In the second chapter,we introduce the definitions and properties about Yuzvin-sky set pairs and twisted group algebras on Z2n first.Then we introduce the higher Octonion On and Pn(m)where twisting functions are linear in the second argument.We introduce the definition of the Hurwitz sets and provide the con-struction of Hurwitz sets on On and Pn(4)respectively.Then we put the minor perturbations on twisting functions of Pn(4),On,Pn(n)to get the admissibility of triples which were proposed by Yuzvinsky.With these results,we introduce two basic methods of constructing new Yuzvinsky set pairs and the strength-ened Lenzhen-Morier-Genoud-Ovsienko formulas,and provide our new infinite families of solutions.Then we introduce the doubling lemma and prove it by constructing new Yuzvinsky set pairs in integral case.Finally,we provide a gen-eralization of doubling construction in any field of characteristic not 2 and prove it by constructing new intercalate matrices from known ones.In the third chapter,we introduce the definitions and properties about Hopf-stiefel function and non-signed intercalate matrix briefly.Then we analyse several special examples of non-signed intercalate matrix.After the introduction of the method of group-orbits classification and the hooking lemma,we include four interesting lemmas and use them to prove the nonexistence of non-signed inter-calate matrix of size ?9,16k-2,16k-9?.Then we provide the discussion of non-signed intercalate matrix of size ?2a +1,2b+ 1,2a + 2b?with Prof.K.Y.Lam.In the final chapter,we conclude the experience in reasearch progress and some further research directions of Hurwitz problem and related issues.