# Download Trace Rings of Generic 2 by 2 Matrices by Lieven Le Bruyn PDF

By Lieven Le Bruyn

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From section three we recall that R^ is generated by the elements (m, Uj) = uauji + ui2Uj2 + ( ui3uj3 u>u ujx ukl Ui2 Uj2 Uk2 Ui3 Uj3 Uk3 We will give an interpretation of these symbols in the x°. X°X°k) Finishing the proof. We say that a Young tableau is of shape a = 3 a 2 6 l c if the array consists of a rows of length 3, b rows of length 2 and c rows of length one. An interpretation of the results of the foregoing section yields. 3. : There is a one-to-one correspondence between an F-vectorspace basis of R^ and standard Young tableaux of shape a = 3 a 2 2 6 l 2 c ; a, 6,c € IN.

The normalizing (resp. central) classgroup of A are then defined to be the quotient groups : NCl{K) = ID(A)/JN(A) CCl(k) = D(A)/

As always, let A m i 2 be the classical ring of quotients of (EJm,2 (or T£m,2) i-e. the generic division algebra for m generic 2 by 2 matrices and P m ,2 is the polynomial ring -Pm,2 = F[xii[i),xi2{i),x2i{i),X22{i) : 1 < t < m] One can embed Pm,2 and A m ,2 naturally in Prn+i,2 and A rn+t> 2 respectively for any i . : TTm,2 = A m , 2 n M 2 ( P m + 2 , 2 ) PROOF: The inclusion TTrn,2 C A m i 2n^2(-fm+2,2 is obvious. , i? Xm+i) we can express it in an iMinear combination : Tr(YXm+i) = E y o y . r r ( n ) .