Hi Anne, I just joined sage-combinat-devel, so, this time, I can answer directly.
On 24 Mrz., 09:39, Anne Schilling <a...@math.ucdavis.edu> wrote: > Looking at the link, it seems this is working with monomials in > commutative variables. But this would still apply for monoids/ > algebras where the generators do not necessarily all commute? The trick is to interpret elements of a free algebra as elements of a very large commutative ring (namely with infinitely many generators). This is where the name "letterplace" comes from. Namely, each generator of the letterplace algebra is given by a pair formed by a letter (corresponding to a generator of the free algebra that one wants to study) and an integer (the place). Then, the commutative monomial x(2)*x(4)*y(1)*z(3) corresponds to the non-commutative monomial y*x*z*x. The free associative multiplication can be emulated by a shift of the indices followed by the commutative multiplication: The free associative product of x(1)*y(2) with y(1)*z(2) corresponds to x(1)*y(2)*y(3)*z(4). So, it somehow is a hack, but it works. > > On the other hand, currently, it is > > only usable with homogeneous elements. My letterplace wrapper will > > therefore refuse to add elements of different degrees. > > You mean when the relations are homogeneous? No. Currently, the Letterplace implementation in Singular has a bug. If I remember correctly, the bug is as follows. When you shift-multiply x(1)+y(1)*y(2) with z(1), you would obtain x(1)*z(3)+y(1)*y(2)*z(3). But x(1)*z(3) has an index gap, and It was forgotten to close the gap by "compressing the indices". That means trouble: If you shift-multiply monomials with index gap, then the result is not what we want. Such as the square of x(1)*y(4), yielding x(1)*y(4)*x(3)*y(6), which corresponds to the element x*x*y*y in the free algebra, not x*y*x*y. Therefore, in my to-be-submitted-as-soon-as-the-damned-documentation- correctly-builds patch, it is excluded to even *create* an inhomogeneous element. Probably that will be allowed as soon as the next Singular version is in Sage. Best regards, Simon -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To post to this group, send email to sage-combinat-devel@googlegroups.com. To unsubscribe from this group, send email to sage-combinat-devel+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.
