tag:blogger.com,1999:blog-5195188167565410449.post7168515559265758671..comments2015-10-31T08:21:47.984+00:00Comments on Haskell for Maths: Introducing the Group AlgebraDavidAhttp://www.blogger.com/profile/16359932006803389458noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-5195188167565410449.post-78900183491096433092012-11-05T22:17:41.655+00:002012-11-05T22:17:41.655+00:00Good question.
Note that the definition of group a...Good question.<br />Note that the definition of group algebra that I'm using allows only *finite* sums of group elements. (Definitions allowing infinite sums are possible, but I don't see straight away how we would represent them in a computer program.)<br />It is possible to work over an infinite group. Indeed, the code above is really doing so. It is working over the infinite group of finite permutations of the natural numbers. Given any finite sum of finite permutations, the group generated by the permutations is finite.<br />If we allow elements of infinite order, then I suspect that their inverses would typically involve infinite sums, and therefore be disallowed by the above definition. For example, if x has infinite order, then (1+x)^-1 = 1-x+x^2-x^3+... (assuming some kind of convergence of the right hand side).<br />DavidAhttp://www.blogger.com/profile/16359932006803389458noreply@blogger.comtag:blogger.com,1999:blog-5195188167565410449.post-38776635977848963042012-11-02T09:07:31.961+00:002012-11-02T09:07:31.961+00:00This comment has been removed by the author.DavidAhttp://www.blogger.com/profile/16359932006803389458noreply@blogger.comtag:blogger.com,1999:blog-5195188167565410449.post-41891118524486566102012-10-25T15:02:35.696+01:002012-10-25T15:02:35.696+01:00When calculating the inverse of
x = 1+2*p[[1,2]]+3...When calculating the inverse of<br />x = 1+2*p[[1,2]]+3*p[[1,2,3]]<br />you look for the inverse in the subalgebra generated by 1, p[[1,2]]] and 3[[1,2,3]]. More generally, the subalgebra generated by all group elements occuring in the sum presentation of x with non-zero coefficient.<br />Then you solve a system of equations.<br />Is there any (idea for an) algorithm that computes an inverse or at least checks if x is invertible in the case of a group algebra for non-finite groups?<br />yellhttp://www.blogger.com/profile/14455252009479151390noreply@blogger.com