TRACE IDENTITIES FROM IDENTITIES FOR DETERMINANTS 3 Magnus

We present new identities for determinants of matrices (A i,j) with entries A i,j equal to a i,j or a i,0 a 0,j − a i,j , where the a i,j 's are indeterminates. We show that these identities are behind trace identities for SL(2, C) matrices found earlier by Magnus in his study of trace algebras. 1. Introduction. In this paper we derive an infinite family of new trace identities for 2 × 2 matrices by using an infinite family of new determinantal identities. These trace identities generalise a certain trace identity due to Magnus. Trace identities for 2 × 2 matrices have been studied for over 100 years, one of the original motivations being the investigation of Teichmüller space via representations of surface groups as (certain equivalence classes of) subgroups of SL(2, C). This approach originated with Fricke and Klein [1] and there have been many subsequent attempts at ways of giving real analytic trace coordinates for Teichmüller space (see for example [2, 3, 5, 7] and references therein). Actions of groups on trace algebras have been investigated by Vogt [10] and more recently by Magnus [4]. Vogt was interested in studying invariants of differential equations , while Magnus was concerned with automorphisms and outer automorphisms of free groups. Physicists have also taken an interest in trace relations [6]. One thus sees the variety of applications that these ideas have.