Fwd: sage undergrad
William Stein
Paul Smith (http://www.math.washington.edu/~smith/personal.html) is
interested in hiring a student to write some Sage code this quarter.
If you're interested, please respond to this email and/or email Paul
at sm...@math.washington.edu directly.
Best regards,
William
---------- Forwarded message ----------
From: Paul Smith <sm...@math.washington.edu>
Date: Tue, Sep 29, 2009 at 12:11 PM
Subject: sage undergrad
To: William Stein <wst...@math.washington.edu>
William
I would like to get an undergrad student to do some sage programming for me.
The problem involves calculations in one particular non-commutative algebra.
I want to be able to multiply/add/subtract elements and write the answers in a
normal form. Explicitly the algebra is generated by 7 elements that I will
call a,b,c,..., g. The following equalities hold in the algebra.
gf=fg + bc - cb + de - ed
ge = eg + ac - ca - df + fd
gd = dg - ab + ba + ef - fe
gc = cg + ea - ae + fb - bf
gb = bg + ad - da + cf - fc
ga = ag + db - bd + ce - ec
fa = af + eb - be + dc - cd.
These equalities should be viewed as rewriting rules. E.G., whenever
the program encounters a word
like ggaa it should rewrite it using the above 7 "rewriting rules" etc
etc. For example,
I have computed by hand that the two possible ways of rewriting gfa
(first rewrite gf vs.
first rewrite fa, and repeat whenever whenever a subword ga,gb,gc,gd,
ge,gf, or fa appears) give the same expression. It follows from this
that a basis for the
algebra is given by all words in the letters a,...,g that do not
contain ga,gb,gc,gd,
ge,gf, or fa as a subword. So ggaa would involve applying the
rewriting rules several
times (in any order---the end result is the same because of what I
checked for gfa).
I would want the output elements written with the terms in "dictionary
order", e.g.
aaba + abba, not abba+aaba.
I will work over the integers, i.e., words have integer coefficients.
Oh, probably
better to use rational cofficients.
I want to look at questions like is gw=wg where w is some specific linear
combination of words, or is gw a rational multiple of wg.
Paul
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org