From joeb@matematik.su.se Sat May 24 21:06:40 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
Received: (joeb@localhost) by nasse.matematik.su.se (8.8.5/8.6.9) id VAA29857; Sat, 24 May 1997 21:06:37 +0200
Date: Sat, 24 May 1997 21:06:37 +0200
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To: emil@matematik.su.se, joeb@matematik.su.se, lambe@matematik.su.se,
        ralff@matematik.su.se
Subject: Re: help!
Cc: bww@maths.nott.ac.uk
Status: RO

Dear Larry,

There may be a (minor) problem about not using standard lex order.
There is however a major problem in the fact that your ideal is not
homogeneous. If you can homogenise the problem in some reasonable
sense, then please send me the homogenised problem, and I'll try it!
If not, then perhaps some other program might help you. (You could
e.g. look up Felix, which I believe has its own home page at the
University of Leipzig, and see what it can do.)

Best regards, Joergen.

From bww@maths.nott.ac.uk Sat May 24 21:24:06 1997
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From: Bruce W Westbury <bww@maths.nott.ac.uk>
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Subject: Re: help!
To: joeb@matematik.su.se (Jorgen Backelin)
Date: Sat, 24 May 1997 20:24:02 +0100 (BST)
Cc: bww@maths.nott.ac.uk (Bruce W Westbury), lambe@matematik.su.se
In-Reply-To: <199705241906.VAA29857@nasse.matematik.su.se> from "Jorgen Backelin" at May 24, 97 09:06:37 pm
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Dear Joergen,

Thanks for the speedy reply!  We've homogenized the relations and hope
that it works now.  Here they are

The algebra has generators c, e1, e2, g1, g2 and relations

c e1 = e1 c
c e2 = e2 c
c g1 = g1 c
c g2 = g2 c

e1 g1 = 0
e2 g2 = 0

e1 e1 = c e1
e2 e2 = c e2
g1 g1 = c g1
g2 g2 = c g2

16 e1 g2 g1 = 9 e1 e2 g1
16 g1 g2 e1 = 9 g1 e2 e1
16 e2 g1 g2 = 9 e2 e1 g2
16 g2 g1 e2 = 9 g2 e1 e2

12 (e2 g1 e2 - e1 g2 e1) + 9/2 (e2 e1 e2 - e1 e2 e1) + (c c e1 - c c e2) = 0

24/7 (g2 e1 g2 - g1 e2 g1) + 16/3 (g2 g1 g2 - g1 g2 g1) + (c c g1 - c c g2) = 0

This problem comes from some things that Bruce Westbury is considering
and he believes that if we have it right, the above quotient should have
dimension 20.  It should be semisimple and the dimension of the irreducibles
should be 1, 2, 1, 3, 2, and 1.  If you get a chance to run this, would you
please send the input file and the output file.  The next case is given by
simply increasing all of the subscripts above by one, adding two new variables
e3 and g3 and the further relations such that e3 and g3 commute with e1 and
g1.  We have an order in mind.  The standard lex might not be good here.  Is
that a problem?

Thanks again,

Best regards,
                Bruce and Larry


From joeb@matematik.su.se Sun May 25 15:52:42 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
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Date: Sun, 25 May 1997 15:52:40 +0200 (MET DST)
Message-Id: <199705251352.PAA27915@pavidus.matematik.su.se>
To: bww@maths.nott.ac.uk
Subject: Does this help?
Cc: joeb@matematik.su.se
Status: RO

I run your homogenised problem version, in lex.
I am not quite sure of what information you want; but I send you
copies of my script and my input, and of the four output files
produced by them. The script and the bergman command NCPBHGROEBNER
produces a log file, a groebner basis output, an output of the
beginning of the double Poincare-Betti series of the associated
monomial quotient ring, and an output of the beginning of the
Hilbert series. If this is what you want, OK. If it isn't, then
please indicate what you want and whatyou don't want. Output is
a fairly flexible thing in bergman, since I discovered that this
is something many people wish to customise.

The script:
--------
mv log20~ log20~~
mv log20 log20~
mv g20out g20out~
mv pb20out pb20out~
mv h20out h20out~
/usr/local/matte/bergman/Newbergman/bin/sun4/bergman <<%EOI >log20 &
(set-heap-size 16000000)
(system "echo Running on $HOST")
(NONCOMMIFY)
(SETQ ALGOUTMODE 'MAPLE)
(NCPBHGROEBNER "inp2" "g20out" "pb20out" "h20out")
(INTERPBACKTRACE)
(QUIT)
%EOI
--------

The input file inp2:
--------
(ALGFORMINPUT)
setinvars c, e1, e2, g1, g2;

c*e1 - e1*c,
c*e2 - e2*c,
c*g1 - g1*c,
c*g2 - g2*c,

e1*g1,
e2*g2,

e1*e1 - c*e1,
e2*e2 - c*e2,
g1*g1 - c*g1,
g2*g2 - c*g2,

16*e1*g2*g1 - 9*e1*e2*g1,
16*g1*g2*e1 - 9*g1*e2*e1,
16*e2*g1*g2 - 9*e2*e1*g2,
16*g2*g1*e2 - 9*g2*e1*e2,

24*e2*g1*e2 - 24*e1*g2*e1 + 9*e2*e1*e2 - 9*e1*e2*e1 + 2*c*c*e1 - 2*c*c*e2,

72*g2*e1*g2 - 72*g1*e2*g1 + 112*g2*g1*g2
 - 112*g1*g2*g1 + 21*c*c*g1 - 21*c*c*g2;
--------

The log file log20 (0 denotes the characteristic):
--------
Bergman 0.92, 14-Apr-97
1 lisp> *** Garbage collection starting
*** GC 1: 25-May-97 15:27:16 (~ 238 ms cpu time, gc : 14 %)
*** time 34 ms, 15801 occupied, 1332 recovered, 1684199 free
*** GC 2: 25-May-97 15:27:16 (~ 272 ms cpu time, gc : 25 %)
*** time 34 ms, 15801 occupied, 0 recovered, 1684199 free
*** Heap space has been enlarged by 14300000 items
*** GC 2: 25-May-97 15:27:16 (~ 272 ms cpu time, gc : 25 %)
*** time 34 ms, 15807 occupied, 0 recovered, 15984193 free
16000000
2 lisp> Running on pavidus
0
3 lisp> NIL
4 lisp> MAPLE
5 lisp> NIL
*** Function `DEGREEENDDISPLAY' has been redefined
SetupGlobals
 ... done

% No. of Spolynomials calculated until degree 2: 0
% No. of ReducePol(0) demanded until degree 2: 0
% Time: 391
*** Garbage collection starting
*** GC 3: 25-May-97 15:27:17 (~ 425 ms cpu time, gc : 24 %)
*** time 34 ms, 18257 occupied, 9240 recovered, 15981743 free

% No. of Spolynomials calculated until degree 3: 14
% No. of ReducePol(0) demanded until degree 3: 0
% Time: 442
*** Garbage collection starting
*** GC 4: 25-May-97 15:27:17 (~ 476 ms cpu time, gc : 28 %)
*** time 34 ms, 19589 occupied, 1896 recovered, 15980411 free

% No. of Spolynomials calculated until degree 4: 42
% No. of ReducePol(0) demanded until degree 4: 0
% Time: 510
*** Garbage collection starting
*** GC 5: 25-May-97 15:27:17 (~ 544 ms cpu time, gc : 28 %)
*** time 17 ms, 24197 occupied, 5804 recovered, 15975803 free

% No. of Spolynomials calculated until degree 5: 84
% No. of ReducePol(0) demanded until degree 5: 0
% Time: 629
*** Garbage collection starting
*** GC 6: 25-May-97 15:27:17 (~ 697 ms cpu time, gc : 29 %)
*** time 51 ms, 34457 occupied, 14764 recovered, 15965543 free

% No. of Spolynomials calculated until degree 6: 153
% No. of ReducePol(0) demanded until degree 6: 0
% Time: 833
*** Garbage collection starting
*** GC 7: 25-May-97 15:27:18 (~ 901 ms cpu time, gc : 28 %)
*** time 51 ms, 47407 occupied, 32962 recovered, 15952593 free

% No. of Spolynomials calculated until degree 7: 188
% No. of ReducePol(0) demanded until degree 7: 0
% Time: 1003
*** Garbage collection starting
*** GC 8: 25-May-97 15:27:18 (~ 1054 ms cpu time, gc : 29 %)
*** time 51 ms, 57121 occupied, 19514 recovered, 15942879 free

% No. of Spolynomials calculated until degree 8: 200
% No. of ReducePol(0) demanded until degree 8: 0
% Time: 1105
*** Garbage collection starting
*** GC 9: 25-May-97 15:27:18 (~ 1173 ms cpu time, gc : 31 %)
*** time 68 ms, 63205 occupied, 8556 recovered, 15936795 free

% No. of Spolynomials calculated until degree 9: 201
% No. of ReducePol(0) demanded until degree 9: 0
% Time: 1190
*** Garbage collection starting
*** GC 10: 25-May-97 15:27:18 (~ 1258 ms cpu time, gc : 35 %)
*** time 68 ms, 64985 occupied, 3854 recovered, 15935015 free
*** Function `DEGREEENDDISPLAY' has been redefined
NIL
6 lisp> Backtrace, including interpreter functions, from top of stack:
INTERPBACKTRACE STANDARDLISP 
NIL
7 lisp> 
Quitting
--------

The groebner basic file g20out:
--------
% 2
-e1*c+c*e1,
   e1*e1-c*e1,
   e1*g1,
   -e2*c+c*e2,
   e2*e2-c*e2,
   e2*g2,
   -g1*c+c*g1,
   g1*g1-c*g1,
   -g2*c+c*g2,
   g2*g2-c*g2,
   
% 3
16*e1*g2*g1-9*e1*e2*g1,
   24*e2*g1*e2+9*e2*e1*e2-24*e1*g2*e1-9*e1*e2*e1-2*c*c*e2+2*c*c*e1,
   16*e2*g1*g2-9*e2*e1*g2,
   16*g1*g2*e1-9*g1*e2*e1,
   16*g2*g1*e2-9*g2*e1*e2,
   112*g2*g1*g2+72*g2*e1*g2-112*g1*g2*g1-72*g1*e2*g1-21*c*c*g2+21*c*c*g1,
   
% 4
9*e1*e2*e1*g2-16*c*c*e1*g2,
   24*e1*g2*e1*e2+9*e1*e2*e1*e2-24*c*e1*g2*e1-9*c*e1*e2*e1-2*c*c*e1*e2+2*c*c*c*
e1,
   -12*e1*g2*e1*g2-7*c*c*e1*g2,
   9*e2*e1*e2*g1-16*c*c*e2*g1,
   24*e2*e1*g2*e1+9*e2*e1*e2*e1-24*c*e1*g2*e1-9*c*e1*e2*e1-2*c*c*e2*e1+2*c*c*c*
e1,
   1152*g2*e1*g2*e1+567*g2*e1*e2*e1-1792*g1*g2*g1*e1-1152*g1*e2*g1*e1-336*c*c*g2
*e1+336*c*c*g1*e1,
   
% 5
-27*e1*e2*e1*e2*e1-56*c*c*e1*g2*e1+33*c*c*e1*e2*e1-6*c*c*c*c*e1,
   -81*e2*e1*e2*e1*e2+162*c*c*e2*e1*e2-168*c*c*e1*g2*e1-63*c*c*e1*e2*e1-32*c*c*c
*c*e2+14*c*c*c*c*e1,
   24*e2*g1*e1*g2*e1+9*e2*g1*e1*e2*e1-2*c*c*e2*g1*e1,
   -1215*g1*e2*e1*e2*e1+28672*c*g1*g2*g1*e1+18432*c*g1*e2*g1*e1-10368*c*g1*e1*g2
*e1-3888*c*g1*e1*e2*e1+2160*c*c*g1*e2*e1-4512*c*c*c*g1*e1,
   336*g1*g2*g1*e1*e2+216*g1*e2*g1*e1*e2-45*c*g1*e2*e1*e2-384*c*g1*e1*g2*e1-144*
c*g1*e1*e2*e1-63*c*c*g1*e1*e2+80*c*c*c*g1*e2+32*c*c*c*g1*e1,
   -112*g1*g2*g1*e1*g2-72*g1*e2*g1*e1*g2+21*c*c*g1*e1*g2,
   405*g2*e1*e2*e1*e2-405*c*g2*e1*e2*e1+5376*c*g1*g2*g1*e1+3456*c*g1*e2*g1*e1-
720*c*g1*e2*e1*e2-6144*c*g1*e1*g2*e1-2304*c*g1*e1*e2*e1-720*c*c*g2*e1*e2+720*c*c
*c*g2*e1+1280*c*c*c*g1*e2-496*c*c*c*g1*e1,
   6144*g2*g1*e1*g2*e1+2304*g2*g1*e1*e2*e1+405*c*g2*e1*e2*e1-5376*c*g1*g2*g1*e1-
3456*c*g1*e2*g1*e1-512*c*c*g2*g1*e1-720*c*c*c*g2*e1+1008*c*c*c*g1*e1,
   
% 6
306*c*c*g1*e2*e1*e2+2136*c*c*g1*e1*g2*e1+801*c*c*g1*e1*e2*e1-544*c*c*c*c*g1*e2-
178*c*c*c*c*g1*e1,
   60928*c*c*g1*g2*g1*e1+39168*c*c*g1*e2*g1*e1-7128*c*c*g1*e1*g2*e1-2673*c*c*g1*
e1*e2*e1-10830*c*c*c*c*g1*e1,
   -306*c*c*g2*e1*e2*e1-2136*c*c*g1*e1*g2*e1-801*c*c*g1*e1*e2*e1+544*c*c*c*c*g2*
e1+178*c*c*c*c*g1*e1,
   
% 7
24*c*c*c*g1*e1*g2*e1+9*c*c*c*g1*e1*e2*e1-2*c*c*c*c*c*g1*e1,
   
Done
--------

The ass. mon. quot. ring pbseries file pb20out:
--------
+t^2*(10*z^2)
+t^3*(6*z^2+14*z^3)
+t^4*(6*z^2+28*z^3+18*z^4)
+t^5*(8*z^2+42*z^3+72*z^4+22*z^5)
+t^6*(3*z^2+67*z^3+152*z^4+143*z^5+26*z^6)
+t^7*(z^2+33*z^3+303*z^4+412*z^5+247*z^6+30*z^7)
+t^8*(12*z^3+285*z^4+1007*z^5+943*z^6+390*z^7+34*z^8)
+t^9*(z^3+195*z^4+1499*z^5+2766*z^6+1913*z^7+578*z^8+38*z^9)
--------

The hseries file h20out:
--------
+15*z^2
+33*z^3
+59*z^4
+89*z^5
+128*z^6
+183*z^7
+257*z^8
+355*z^9
--------

Please check the input file inp2 carefully, i order to see that my
conversion from equations to ideal generators (with integer
coefficients and on "maple form") was correct!

As you see from the log file, the run was fairly fast: less than
1200 milliseconds. Thus, if the results are useful for you, I believe
that we should succeed also with the next two or three sizes of the
problem.

Yours, Joergen

From bww@maths.nott.ac.uk Wed May 28 16:48:17 1997
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From: Bruce W Westbury <bww@maths.nott.ac.uk>
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Subject: Presentation
To: joeb@matematik.su.se (Jorgen Backelin)
Date: Wed, 28 May 1997 15:48:12 +0100 (BST)
Cc: bww@maths.nott.ac.uk (Bruce W Westbury)
In-Reply-To: <199705251352.PAA27915@pavidus.matematik.su.se> from "Jorgen Backelin" at May 25, 97 03:52:40 pm
X-Mailer: ELM [version 2.4 PL21]
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Status: RO

Dear Jorgen
	The output did not come back as I expected which probably means
I made a mistake. The last relations were written out by hand. I am sending
you some new relations. These have been produced by the machine. They are
more complicated but I hope they are accurate. Here they are:

-------------------------------------------------------------------------


(ALGFORMINPUT)
setinvars c, e1, e2, g1, g2;

c*e1 - e1*c,
c*e2 - e2*c,
c*g1 - g1*c,
c*g2 - g2*c,

e1*g1,
e2*g2,
g1*e1,
g2*e2,

e1*e1 - c*e1,
e2*e2 - c*e2,
g1*g1 - c*g1,
g2*g2 - c*g2,


512*g2*g1*g2 + 576*g2*g1*e2 - 432*c*g2*e2 + 
576*g2*e1*g2 + 648*g2*e1*e2 - 96*c*g2 - 
512*g1*g2*g1 - 576*g1*g2*e1 - 576*g1*e2*g1 - 
648*g1*e2*e1 + 432*g1*e1 + 96*c*g1 - 432*c*e2*g2 + 
576*e2*g1*g2 + 648*e2*g1*e2 + 648*e2*e1*g2 + 
729*e2*e1*e2 - 162*c*c*e2 - 576*e1*g2*g1 - 
648*e1*g2*e1 + 432*c*e1*g1 - 648*e1*e2*g1 - 
729*e1*e2*e1 + 162*c*c*e1,
 

 - 768*g2*g1*g2 - 864*g2*g1*e2 + 648*c*g2*e2 - 
864*g2*e1*g2 - 972*g2*e1*e2 + 72*c*g2*e1 + 
144*c*c*g2 + 768*g1*g2*g1 + 768*g1*g2*e1 + 
864*g1*e2*g1 + 864*g1*e2*e1 - 576*c*g1*e1 - 
144*c*c*g1 + 576*c*e2*g2 - 768*e2*g1*g2 -
864*e2*g1*e2 - 
72*c*e2*g1 - 864*e2*e1*g2 - 972*e2*e1*e2 + 
216*c*c*e2 + 864*e1*g2*g1 + 864*e1*g2*e1 - 
648*c*e1*g1 + 972*e1*e2*g1 + 972*e1*e2*e1 - 
216*c*c*e1,
 

256*g2*g1*g2 + 288*g2*g1*e2 - 216*c*g2*e2 + 
288*g2*e1*g2 + 324*g2*e1*e2 - 72*c*g2*e1 - 
48*c*c*g2 - 256*g1*g2*g1 - 192*g1*g2*e1 - 
288*g1*e2*g1 - 216*g1*e2*e1 + 144*c*g1*e1 + 
48*c*c*g1 - 144*c*e2*g2 + 192*e2*g1*g2 +
216*e2*g1*e2 + 
72*c*e2*g1 + 216*e2*e1*g2 + 243*e2*e1*e2 - 
54*c*c*e2 - 288*e1*g2*g1 - 216*e1*g2*e1 + 
216*c*e1*g1 - 324*e1*e2*g1 - 243*e1*e2*e1 + 
54*c*c*e1,
 

 - 768*g2*g1*g2 - 768*g2*g1*e2 + 576*c*g2*e2 - 
864*g2*e1*g2 - 864*g2*e1*e2 + 144*c*c*g2 + 
768*g1*g2*g1 + 864*g1*g2*e1 + 864*g1*e2*g1 + 
972*g1*e2*e1 - 72*c*g1*e2 - 648*c*g1*e1 - 
144*c*c*g1 + 648*c*e2*g2 - 864*e2*g1*g2 -
864*e2*g1*e2 - 
972*e2*e1*g2 - 972*e2*e1*e2 + 216*c*c*e2 + 
768*e1*g2*g1 + 864*e1*g2*e1 + 72*c*e1*g2 - 
576*c*e1*g1 + 864*e1*e2*g1 + 972*e1*e2*e1 - 
216*c*c*e1,
 

384*g2*g1*g2 + 288*g2*g1*e2 - 504*c*g2*e2 + 
528*g2*e1*g2 + 432*g2*e1*e2 - 132*c*g2*e1 - 
72*c*c*g2 - 384*g1*g2*g1 - 288*g1*g2*e1 - 
528*g1*e2*g1 - 432*g1*e2*e1 + 132*c*g1*e2 + 
504*c*g1*e1 + 72*c*c*g1 - 504*c*e2*g2 +
288*e2*g1*g2 + 
180*e2*g1*e2 + 132*c*e2*g1 + 432*e2*e1*g2 + 
324*e2*e1*e2 - 72*c*c*e2 - 288*e1*g2*g1 - 
180*e1*g2*e1 - 132*c*e1*g2 + 504*c*e1*g1 - 
432*e1*e2*g1 - 324*e1*e2*e1 + 72*c*c*e1,
 


768*g2*g1*g2 + 912*g2*g1*e2 - 252*c*g2*e2 + 
720*g2*e1*g2 + 864*g2*e1*e2 + 60*c*g2*e1 - 
144*c*c*g2 - 768*g1*g2*g1 - 864*g1*g2*e1 - 
720*g1*e2*g1 - 828*g1*e2*e1 - 72*c*g1*e2 + 
264*c*g1*e1 + 144*c*c*g1 - 264*c*e2*g2 +
864*e2*g1*g2 + 
1008*e2*g1*e2 - 60*c*e2*g1 + 828*e2*e1*g2 + 
972*e2*e1*e2 - 216*c*c*e2 - 912*e1*g2*g1 - 
1008*e1*g2*e1 + 72*c*e1*g2 + 252*c*e1*g1 - 
864*e1*e2*g1 - 972*e1*e2*e1 + 216*c*c*e1,
 

256*g2*g1*g2 + 192*g2*g1*e2 - 144*c*g2*e2 + 
288*g2*e1*g2 + 216*g2*e1*e2 - 48*c*c*g2 - 
256*g1*g2*g1 - 288*g1*g2*e1 - 288*g1*e2*g1 - 
324*g1*e2*e1 + 72*c*g1*e2 + 216*c*g1*e1 + 
48*c*c*g1 - 216*c*e2*g2 + 288*e2*g1*g2 +
216*e2*g1*e2 + 
324*e2*e1*g2 + 243*e2*e1*e2 - 54*c*c*e2 - 
192*e1*g2*g1 - 216*e1*g2*e1 - 72*c*e1*g2 + 
144*c*e1*g1 - 216*e1*e2*g1 - 243*e1*e2*e1 + 
54*c*c*e1,
 

768*g2*g1*g2 + 864*g2*g1*e2 - 264*c*g2*e2 + 
720*g2*e1*g2 + 828*g2*e1*e2 + 72*c*g2*e1 - 
144*c*c*g2 - 768*g1*g2*g1 - 912*g1*g2*e1 - 
720*g1*e2*g1 - 864*g1*e2*e1 - 60*c*g1*e2 + 
252*g1*e1 + 144*g1 - 252*c*e2*g2 + 912*e2*g1*g2 + 
1008*e2*g1*e2 - 72*c*e2*g1 + 864*e2*e1*g2 + 
972*e2*e1*e2 - 216*c*c*e2 - 864*e1*g2*g1 - 
1008*e1*g2*e1 + 60*c*e1*g2 + 264*c*e1*g1 - 
828*e1*e2*g1 - 972*e1*e2*e1 + 216*c*c*e1,
 

 - 1344*g2*g1*g2 - 1344*g2*g1*e2 + 576*c*g2*e2 - 
1392*g2*e1*g2 - 1404*g2*e1*e2 + 60*c*g2*e1 + 
252*c*c*g2 + 1344*g1*g2*g1 + 1344*g1*g2*e1 + 
1392*g1*e2*g1 + 1404*g1*e2*e1 - 60*c*g1*e2 - 
576*c*g1*e1 - 252*c*c*g1 + 576*c*e2*g2 -
1344*e2*g1*g2 - 
1332*e2*g1*e2 - 60*c*e2*g1 - 1404*e2*e1*g2 - 
1404*e2*e1*e2 + 312*c*c*e2 + 1344*e1*g2*g1 + 
1332*e1*g2*e1 + 60*c*e1*g2 - 576*c*e1*g1 + 
1404*e1*e2*g1 + 1404*e1*e2*e1 - 312*c*c*e1,

-------------------------------------------------------------------------

These relations are intended to be consequences of the above relations.
I guess one way to check this would be to include them and see if it
makes any difference.

16*e1*g2*g1 - 9*e1*e2*g1,
16*g1*g2*e1 - 9*g1*e2*e1,
16*e2*g1*g2 - 9*e2*e1*g2,
16*g2*g1*e2 - 9*g2*e1*e2,

24*e2*g1*e2 - 24*e1*g2*e1 + 9*e2*e1*e2 - 9*e1*e2*e1 + 2*c*c*e1 - 2*c*c*e2,

72*g2*e1*g2 - 72*g1*e2*g1 + 112*g2*g1*g2,

-------------------------------------------------------------------------

	I am also not clear what your output is telling me. I dont know
what you mean by the Poincare-Betti series. Also I assume that the Hilbert
series is the generating function for the series of dimensions of the
homogenous subspaces. If this is right then I expect the Hilbert series
to be
	(1+4z+8z^2+7z^3)/(1-z)

	I imagine that this output depends on the ordering on monomials.

	I have put in the generator c to comply with your requirement that
all relations are homogenous. I am interested in the case c=1 in which
case the algebras are finite dimensional. I hope this is harmless.

	Ultimately I am interested in the representation theory of these
algebras. At the moment I dont know the dimension!

			Bruce Westbury

	

From joeb@matematik.su.se Wed May 28 17:09:24 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
Received: (joeb@localhost) by homsan.matematik.su.se (8.8.5/8.6.9) id RAA19594; Wed, 28 May 1997 17:09:22 +0200
Date: Wed, 28 May 1997 17:09:22 +0200
Message-Id: <199705281509.RAA19594@homsan.matematik.su.se>
To: bww@maths.nott.ac.uk
Subject: Re: Presentation
Cc: joeb@matematik.su.se
Status: RO

I'll run your new input soon; but just to be sure: The caracteristic
is assumed to be 0, isn't it?
Joergen
cjoeb

From bww@maths.nott.ac.uk Wed May 28 17:23:01 1997
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From: Bruce W Westbury <bww@maths.nott.ac.uk>
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Subject: Re: Presentation
To: joeb@matematik.su.se (Jorgen Backelin)
Date: Wed, 28 May 1997 16:22:58 +0100 (BST)
In-Reply-To: <199705281509.RAA19594@homsan.matematik.su.se> from "Jorgen Backelin" at May 28, 97 05:09:22 pm
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> 
> I'll run your new input soon; but just to be sure: The caracteristic
> is assumed to be 0, isn't it?
> Joergen
> cjoeb
> 
	Yes. However I dont expect it to make a difference unless

	2^p = 3^q for some p, q >= 0 and "small".

"small" probably means at most three or four.

		Bruce


From joeb@matematik.su.se Tue Jun  3 16:40:01 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
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To: bww@maths.nott.ac.uk, joeb@matematik.su.se
Subject: Re:  Presentation
Status: RO

Dear Bruce,

I run into trouble in the run, and due to teaching didn't have
time to pinpoint the error until now. Two of your input
polynomials are non-homogeneous:


512*g2*g1*g2 + 576*g2*g1*e2 - 432*c*g2*e2 + 
576*g2*e1*g2 + 648*g2*e1*e2 - >> 96*c*g2 - <<
512*g1*g2*g1 - 576*g1*g2*e1 - 576*g1*e2*g1 - 
648*g1*e2*e1 + >> 432*g1*e1 + 96*c*g1 << - 432*c*e2*g2 + 
576*e2*g1*g2 + 648*e2*g1*e2 + 648*e2*e1*g2 + 
729*e2*e1*e2 - 162*c*c*e2 - 576*e1*g2*g1 - 
648*e1*g2*e1 + 432*c*e1*g1 - 648*e1*e2*g1 - 
729*e1*e2*e1 + 162*c*c*e1,

and

768*g2*g1*g2 + 864*g2*g1*e2 - 264*c*g2*e2 + 
720*g2*e1*g2 + 828*g2*e1*e2 + 72*c*g2*e1 - 
144*c*c*g2 - 768*g1*g2*g1 - 912*g1*g2*e1 - 
720*g1*e2*g1 - 864*g1*e2*e1 - 60*c*g1*e2 + 
>> 252*g1*e1 + 144*g1 << - 252*c*e2*g2 + 912*e2*g1*g2 + 
1008*e2*g1*e2 - 72*c*e2*g1 + 864*e2*e1*g2 + 
972*e2*e1*e2 - 216*c*c*e2 - 864*e1*g2*g1 - 
1008*e1*g2*e1 + 60*c*e1*g2 + 264*c*e1*g1 - 
828*e1*e2*g1 - 972*e1*e2*e1 + 216*c*c*e1,

Now my guess is that this just was an oversight at homogenisation,
and that the erroneous terms are supposed to be multiplied with
the appropriate powers of c to get qubics. Would you please check
this? I'll try this correction and rerun the problem, if you don't
inform me that the error is another one.

	Joergen Backelin

P.S. Up to what degree would you like the Hilbert series (and
yes, that is the generating function for the series of dimensions of the
homogenous subspaces) calculated? (If you do not know what the
Poicare-Betti series of the associated monomial ring is, then
you are probably not interested in it.)

From joeb@matematik.su.se Tue Jun  3 17:11:36 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
Received: (joeb@localhost) by prosit.matematik.su.se (8.8.5/8.6.9) id RAA23790; Tue, 3 Jun 1997 17:11:34 +0200 (MET DST)
Date: Tue, 3 Jun 1997 17:11:34 +0200 (MET DST)
Message-Id: <199706031511.RAA23790@prosit.matematik.su.se>
To: bww@maths.nott.ac.uk
Subject: Not expected results.
Cc: joeb@matematik.su.se
Status: RO

I run my proposed correction of your input, without and with
your proposed additional forms. I got different results.

Without the extras, the Hilbert series output was

+13*z^2
+22*z^3
+22*z^4
+20*z^5
+20*z^6

(Here the initial terms 1+5*z are tacitly understood.) This
could be the beginning of the expansion of

 (1+4*z+8*z^2+9*z^3-2*z^5)/(1-z) .

With the additional forms, I got the output

+13*z^2
+17*z^3
+9*z^4
+6*z^5
+6*z^6

which clearly is much smaller.

	Joergen Backelin

From bww@maths.nott.ac.uk Tue Jun  3 19:43:22 1997
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Subject: Re: Not expected results.
To: joeb@matematik.su.se (Jorgen Backelin)
Date: Tue, 3 Jun 1997 18:43:18 +0100 (BST)
Cc: bww@maths.nott.ac.uk (Bruce W Westbury)
In-Reply-To: <199706031511.RAA23790@prosit.matematik.su.se> from "Jorgen Backelin" at Jun 3, 97 05:11:34 pm
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Dear Jorgen
	Yes I did make errors in homogenising as you thought.
	I found four corrections needed to make the relations homogeneous.
You only pointed out two. I hope you spotted the other two.

	I was expecting (1-z)*H(z) to be 

1 + 4*z + 8*z^2 + 7*z^3

If you had come up with a polynomial with positive coefficients whose
sum was 20 then I would have put the difference down to the choice of
ordering.

	I dont understand how you got

1 + 4*z + 8*z^2 + 9*z^3 + 0*z^4 - 2*z^5

The only explanation I can think of is that this has something to do
with the homogenisation. More specifically if you derived a relation of
the form c*(a_1 + ... + a_k) then I would add the relation (a_1 + ... + a_k) 
whereas you presumably would not.

	If this is the explanation then we can go on to the next case.
This is the first case where I dont know the dimension. In fact I dont
know that this is finite dimensional. A lower bound for the dimension
is 200. Here is the presentation.

		Thankyou for your interest
			Bruce

---------------------------------------------------------------------



(ALGFORMINPUT)
setinvars c, e1, e2, e3, g1, g2, g3;

c*e1 - e1*c,
c*g1 - g1*c,
c*e2 - e2*c,
c*g2 - g2*c,
c*e3 - e3*c,
c*g3 - g3*c,

e1*g1,
g1*e1,
e2*g2,
g2*e2,
e3*g3,
g3*e3,

e1*e1 - c*e1,
g1*g1 - c*g1,
e2*e2 - c*e2,
g2*g2 - c*g2,
e3*e3 - c*e3,
g3*g3 - c*g3,

e1*e3 - e3*e1,
g1*e3 - e3*g1,
e1*g3 - g3*e1,
g1*g3 - g3*g1,



512*g2*g1*g2 + 576*g2*g1*e2 - 432*c*g2*e2 + 
576*g2*e1*g2 + 648*g2*e1*e2 - 96*c*c*g2 - 
512*g1*g2*g1 - 576*g1*g2*e1 - 576*g1*e2*g1 - 
648*g1*e2*e1 + 432*c*g1*e1 + 96*c*c*g1 - 432*c*e2*g2 + 
576*e2*g1*g2 + 648*e2*g1*e2 + 648*e2*e1*g2 + 
729*e2*e1*e2 - 162*c*c*e2 - 576*e1*g2*g1 - 
648*e1*g2*e1 + 432*c*e1*g1 - 648*e1*e2*g1 - 
729*e1*e2*e1 + 162*c*c*e1,
 

 - 768*g2*g1*g2 - 864*g2*g1*e2 + 648*c*g2*e2 - 
864*g2*e1*g2 - 972*g2*e1*e2 + 72*c*g2*e1 + 
144*c*c*g2 + 768*g1*g2*g1 + 768*g1*g2*e1 + 
864*g1*e2*g1 + 864*g1*e2*e1 - 576*c*g1*e1 - 
144*c*c*g1 + 576*c*e2*g2 - 768*e2*g1*g2 -
864*e2*g1*e2 - 
72*c*e2*g1 - 864*e2*e1*g2 - 972*e2*e1*e2 + 
216*c*c*e2 + 864*e1*g2*g1 + 864*e1*g2*e1 - 
648*c*e1*g1 + 972*e1*e2*g1 + 972*e1*e2*e1 - 
216*c*c*e1,
 

256*g2*g1*g2 + 288*g2*g1*e2 - 216*c*g2*e2 + 
288*g2*e1*g2 + 324*g2*e1*e2 - 72*c*g2*e1 - 
48*c*c*g2 - 256*g1*g2*g1 - 192*g1*g2*e1 - 
288*g1*e2*g1 - 216*g1*e2*e1 + 144*c*g1*e1 + 
48*c*c*g1 - 144*c*e2*g2 + 192*e2*g1*g2 +
216*e2*g1*e2 + 
72*c*e2*g1 + 216*e2*e1*g2 + 243*e2*e1*e2 - 
54*c*c*e2 - 288*e1*g2*g1 - 216*e1*g2*e1 + 
216*c*e1*g1 - 324*e1*e2*g1 - 243*e1*e2*e1 + 
54*c*c*e1,
 

 - 768*g2*g1*g2 - 768*g2*g1*e2 + 576*c*g2*e2 - 
864*g2*e1*g2 - 864*g2*e1*e2 + 144*c*c*g2 + 
768*g1*g2*g1 + 864*g1*g2*e1 + 864*g1*e2*g1 + 
972*g1*e2*e1 - 72*c*g1*e2 - 648*c*g1*e1 - 
144*c*c*g1 + 648*c*e2*g2 - 864*e2*g1*g2 -
864*e2*g1*e2 - 
972*e2*e1*g2 - 972*e2*e1*e2 + 216*c*c*e2 + 
768*e1*g2*g1 + 864*e1*g2*e1 + 72*c*e1*g2 - 
576*c*e1*g1 + 864*e1*e2*g1 + 972*e1*e2*e1 - 
216*c*c*e1,
 

384*g2*g1*g2 + 288*g2*g1*e2 - 504*c*g2*e2 + 
528*g2*e1*g2 + 432*g2*e1*e2 - 132*c*g2*e1 - 
72*c*c*g2 - 384*g1*g2*g1 - 288*g1*g2*e1 - 
528*g1*e2*g1 - 432*g1*e2*e1 + 132*c*g1*e2 + 
504*c*g1*e1 + 72*c*c*g1 - 504*c*e2*g2 +
288*e2*g1*g2 + 
180*e2*g1*e2 + 132*c*e2*g1 + 432*e2*e1*g2 + 
324*e2*e1*e2 - 72*c*c*e2 - 288*e1*g2*g1 - 
180*e1*g2*e1 - 132*c*e1*g2 + 504*c*e1*g1 - 
432*e1*e2*g1 - 324*e1*e2*e1 + 72*c*c*e1,
 


768*g2*g1*g2 + 912*g2*g1*e2 - 252*c*g2*e2 + 
720*g2*e1*g2 + 864*g2*e1*e2 + 60*c*g2*e1 - 
144*c*c*g2 - 768*g1*g2*g1 - 864*g1*g2*e1 - 
720*g1*e2*g1 - 828*g1*e2*e1 - 72*c*g1*e2 + 
264*c*g1*e1 + 144*c*c*g1 - 264*c*e2*g2 +
864*e2*g1*g2 + 
1008*e2*g1*e2 - 60*c*e2*g1 + 828*e2*e1*g2 + 
972*e2*e1*e2 - 216*c*c*e2 - 912*e1*g2*g1 - 
1008*e1*g2*e1 + 72*c*e1*g2 + 252*c*e1*g1 - 
864*e1*e2*g1 - 972*e1*e2*e1 + 216*c*c*e1,
 

256*g2*g1*g2 + 192*g2*g1*e2 - 144*c*g2*e2 + 
288*g2*e1*g2 + 216*g2*e1*e2 - 48*c*c*g2 - 
256*g1*g2*g1 - 288*g1*g2*e1 - 288*g1*e2*g1 - 
324*g1*e2*e1 + 72*c*g1*e2 + 216*c*g1*e1 + 
48*c*c*g1 - 216*c*e2*g2 + 288*e2*g1*g2 +
216*e2*g1*e2 + 
324*e2*e1*g2 + 243*e2*e1*e2 - 54*c*c*e2 - 
192*e1*g2*g1 - 216*e1*g2*e1 - 72*c*e1*g2 + 
144*c*e1*g1 - 216*e1*e2*g1 - 243*e1*e2*e1 + 
54*c*c*e1,
 

768*g2*g1*g2 + 864*g2*g1*e2 - 264*c*g2*e2 + 
720*g2*e1*g2 + 828*g2*e1*e2 + 72*c*g2*e1 - 
144*c*c*g2 - 768*g1*g2*g1 - 912*g1*g2*e1 - 
720*g1*e2*g1 - 864*g1*e2*e1 - 60*c*g1*e2 + 
252*c*g1*e1 + 144*c*c*g1 - 252*c*e2*g2 + 912*e2*g1*g2 + 
1008*e2*g1*e2 - 72*c*e2*g1 + 864*e2*e1*g2 + 
972*e2*e1*e2 - 216*c*c*e2 - 864*e1*g2*g1 - 
1008*e1*g2*e1 + 60*c*e1*g2 + 264*c*e1*g1 - 
828*e1*e2*g1 - 972*e1*e2*e1 + 216*c*c*e1,
 

 - 1344*g2*g1*g2 - 1344*g2*g1*e2 + 576*c*g2*e2 - 
1392*g2*e1*g2 - 1404*g2*e1*e2 + 60*c*g2*e1 + 
252*c*c*g2 + 1344*g1*g2*g1 + 1344*g1*g2*e1 + 
1392*g1*e2*g1 + 1404*g1*e2*e1 - 60*c*g1*e2 - 
576*c*g1*e1 - 252*c*c*g1 + 576*c*e2*g2 -
1344*e2*g1*g2 - 
1332*e2*g1*e2 - 60*c*e2*g1 - 1404*e2*e1*g2 - 
1404*e2*e1*e2 + 312*c*c*e2 + 1344*e1*g2*g1 + 
1332*e1*g2*e1 + 60*c*e1*g2 - 576*c*e1*g1 + 
1404*e1*e2*g1 + 1404*e1*e2*e1 - 312*c*c*e1,



512*g3*g2*g3 + 576*g3*g2*e3 - 432*c*g3*e3 + 
576*g3*e2*g3 + 648*g3*e2*e3 - 96*c*c*g3 - 
512*g2*g3*g2 - 576*g2*g3*e2 - 576*g2*e3*g2 - 
648*g2*e3*e2 + 432*c*g2*e2 + 96*c*c*g2 - 432*c*e3*g3 + 
576*e3*g2*g3 + 648*e3*g2*e3 + 648*e3*e2*g3 + 
729*e3*e2*e3 - 162*c*c*e3 - 576*e2*g3*g2 - 
648*e2*g3*e2 + 432*c*e2*g2 - 648*e2*e3*g2 - 
729*e2*e3*e2 + 162*c*c*e2,
 

 - 768*g3*g2*g3 - 864*g3*g2*e3 + 648*c*g3*e3 - 
864*g3*e2*g3 - 972*g3*e2*e3 + 72*c*g3*e2 + 
144*c*c*g3 + 768*g2*g3*g2 + 768*g2*g3*e2 + 
864*g2*e3*g2 + 864*g2*e3*e2 - 576*c*g2*e2 - 
144*c*c*g2 + 576*c*e3*g3 - 768*e3*g2*g3 -
864*e3*g2*e3 - 
72*c*e3*g2 - 864*e3*e2*g3 - 972*e3*e2*e3 + 
216*c*c*e3 + 864*e2*g3*g2 + 864*e2*g3*e2 - 
648*c*e2*g2 + 972*e2*e3*g2 + 972*e2*e3*e2 - 
216*c*c*e2,
 

256*g3*g2*g3 + 288*g3*g2*e3 - 216*c*g3*e3 + 
288*g3*e2*g3 + 324*g3*e2*e3 - 72*c*g3*e2 - 
48*c*c*g3 - 256*g2*g3*g2 - 192*g2*g3*e2 - 
288*g2*e3*g2 - 216*g2*e3*e2 + 144*c*g2*e2 + 
48*c*c*g2 - 144*c*e3*g3 + 192*e3*g2*g3 +
216*e3*g2*e3 + 
72*c*e3*g2 + 216*e3*e2*g3 + 243*e3*e2*e3 - 
54*c*c*e3 - 288*e2*g3*g2 - 216*e2*g3*e2 + 
216*c*e2*g2 - 324*e2*e3*g2 - 243*e2*e3*e2 + 
54*c*c*e2,
 

 - 768*g3*g2*g3 - 768*g3*g2*e3 + 576*c*g3*e3 - 
864*g3*e2*g3 - 864*g3*e2*e3 + 144*c*c*g3 + 
768*g2*g3*g2 + 864*g2*g3*e2 + 864*g2*e3*g2 + 
972*g2*e3*e2 - 72*c*g2*e3 - 648*c*g2*e2 - 
144*c*c*g2 + 648*c*e3*g3 - 864*e3*g2*g3 -
864*e3*g2*e3 - 
972*e3*e2*g3 - 972*e3*e2*e3 + 216*c*c*e3 + 
768*e2*g3*g2 + 864*e2*g3*e2 + 72*c*e2*g3 - 
576*c*e2*g2 + 864*e2*e3*g2 + 972*e2*e3*e2 - 
216*c*c*e2,
 

384*g3*g2*g3 + 288*g3*g2*e3 - 504*c*g3*e3 + 
528*g3*e2*g3 + 432*g3*e2*e3 - 132*c*g3*e2 - 
72*c*c*g3 - 384*g2*g3*g2 - 288*g2*g3*e2 - 
528*g2*e3*g2 - 432*g2*e3*e2 + 132*c*g2*e3 + 
504*c*g2*e2 + 72*c*c*g2 - 504*c*e3*g3 +
288*e3*g2*g3 + 
180*e3*g2*e3 + 132*c*e3*g2 + 432*e3*e2*g3 + 
324*e3*e2*e3 - 72*c*c*e3 - 288*e2*g3*g2 - 
180*e2*g3*e2 - 132*c*e2*g3 + 504*c*e2*g2 - 
432*e2*e3*g2 - 324*e2*e3*e2 + 72*c*c*e2,
 


768*g3*g2*g3 + 912*g3*g2*e3 - 252*c*g3*e3 + 
720*g3*e2*g3 + 864*g3*e2*e3 + 60*c*g3*e2 - 
144*c*c*g3 - 768*g2*g3*g2 - 864*g2*g3*e2 - 
720*g2*e3*g2 - 828*g2*e3*e2 - 72*c*g2*e3 + 
264*c*g2*e2 + 144*c*c*g2 - 264*c*e3*g3 +
864*e3*g2*g3 + 
1008*e3*g2*e3 - 60*c*e3*g2 + 828*e3*e2*g3 + 
972*e3*e2*e3 - 216*c*c*e3 - 912*e2*g3*g2 - 
1008*e2*g3*e2 + 72*c*e2*g3 + 252*c*e2*g2 - 
864*e2*e3*g2 - 972*e2*e3*e2 + 216*c*c*e2,
 

256*g3*g2*g3 + 192*g3*g2*e3 - 144*c*g3*e3 + 
288*g3*e2*g3 + 216*g3*e2*e3 - 48*c*c*g3 - 
256*g2*g3*g2 - 288*g2*g3*e2 - 288*g2*e3*g2 - 
324*g2*e3*e2 + 72*c*g2*e3 + 216*c*g2*e2 + 
48*c*c*g2 - 216*c*e3*g3 + 288*e3*g2*g3 +
216*e3*g2*e3 + 
324*e3*e2*g3 + 243*e3*e2*e3 - 54*c*c*e3 - 
192*e2*g3*g2 - 216*e2*g3*e2 - 72*c*e2*g3 + 
144*c*e2*g2 - 216*e2*e3*g2 - 243*e2*e3*e2 + 
54*c*c*e2,
 

768*g3*g2*g3 + 864*g3*g2*e3 - 264*c*g3*e3 + 
720*g3*e2*g3 + 828*g3*e2*e3 + 72*c*g3*e2 - 
144*c*c*g3 - 768*g2*g3*g2 - 912*g2*g3*e2 - 
720*g2*e3*g2 - 864*g2*e3*e2 - 60*c*g2*e3 + 
252*c*g2*e2 + 144*c*c*g2 - 252*c*e3*g3 + 912*e3*g2*g3 + 
1008*e3*g2*e3 - 72*c*e3*g2 + 864*e3*e2*g3 + 
972*e3*e2*e3 - 216*c*c*e3 - 864*e2*g3*g2 - 
1008*e2*g3*e2 + 60*c*e2*g3 + 264*c*e2*g2 - 
828*e2*e3*g2 - 972*e2*e3*e2 + 216*c*c*e2,
 

 - 1344*g3*g2*g3 - 1344*g3*g2*e3 + 576*c*g3*e3 - 
1392*g3*e2*g3 - 1404*g3*e2*e3 + 60*c*g3*e2 + 
252*c*c*g3 + 1344*g2*g3*g2 + 1344*g2*g3*e2 + 
1392*g2*e3*g2 + 1404*g2*e3*e2 - 60*c*g2*e3 - 
576*c*g2*e2 - 252*c*c*g2 + 576*c*e3*g3 -
1344*e3*g2*g3 - 
1332*e3*g2*e3 - 60*c*e3*g2 - 1404*e3*e2*g3 - 
1404*e3*e2*e3 + 312*c*c*e3 + 1344*e2*g3*g2 + 
332*e2*g3*e2 + 60*c*e2*g3 - 576*c*e2*g2 + 
1404*e2*e3*g2 + 1404*e2*e3*e2 - 312*c*c*e2,

,

From joeb@matematik.su.se Thu Jun  5 13:57:19 1997
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From: Jorgen Backelin <joeb@matematik.su.se>
Received: (joeb@localhost) by pavidus.matematik.su.se (8.8.5/8.6.9) id NAA05511; Thu, 5 Jun 1997 13:57:16 +0200 (MET DST)
Date: Thu, 5 Jun 1997 13:57:16 +0200 (MET DST)
Message-Id: <199706051157.NAA05511@pavidus.matematik.su.se>
To: bww@maths.nott.ac.uk, joeb@matematik.su.se
Subject: Re: Not expected results.
Status: R

Dear Bruce,

> I found four corrections needed to make the relations homogeneous.
> You only pointed out two. I hope you spotted the other two.

I don't quite understand. I found in total five errors in two
polynomials; the first of these polynomials contained three
errors, and the second one contained two errors. I pointed out
all these four errors in my letter to you.

I found no other errors; in fact, the bergman procedure TESTINDATA
declared the other polynomials to be homogeneous. Would you please
check once more whether or not there were some errors beside the
five ones marked between >> and << in my preceeding letters?

> I dont understand how you got
>
> 1 + 4*z + 8*z^2 + 9*z^3 + 0*z^4 - 2*z^5
>
> The only explanation I can think of is that this has something to do
> with the homogenisation. More specifically if you derived a relation of
> the form c*(a_1 + ... + a_k) then I would add the relation (a_1 + ... + a_k) 
> whereas you presumably would not.

Bergman adds the relation  c*(a_1 + ... + a_k) (in the distributed form
c*a_1 + ... + c*a_k). Since you put the c first in your setinvars
( = "set input variables") command, and the ordering is a bit funny in
the non-commutative case, actually any Groebner basis element of degree
3 or more is divisible by a power of c iff its leading monomial is.
Indeed, the output Groebner basis contained two such polynomials of
degree 5:


% 5
-16*c*c*e1*g2*g1-9*c*c*e1*e2*g1,
   16*c*c*e2*g1*g2+9*c*c*e2*e1*g2,

and factoring out the c^2 and adding the corresponding degree three
relations to the input indeed yielded the output

+13*z^2
+20*z^3
+20*z^4
+20*z^5
+20*z^6

corresponding to the series you expected.

However, there could not be any similar explanation in the case
of the added extra relations calculations. Recall that you wrote

> These relations are intended to be consequences of the above relations.
> I guess one way to check this would be to include them and see if it
> makes any difference.
>
> 16*e1*g2*g1 - 9*e1*e2*g1,
> 16*g1*g2*e1 - 9*g1*e2*e1,
> 16*e2*g1*g2 - 9*e2*e1*g2,
> 16*g2*g1*e2 - 9*g2*e1*e2,
>
> 24*e2*g1*e2 - 24*e1*g2*e1 + 9*e2*e1*e2 - 9*e1*e2*e1 + 2*c*c*e1 - 2*c*c*e2,
>
> 72*g2*e1*g2 - 72*g1*e2*g1 + 112*g2*g1*g2,

and that when I added them I got the Hilbert series output

+13*z^2
+17*z^3
+9*z^4
+6*z^5
+6*z^6

showing that indeed there are relations among these which are not
consequences of the old ones. In fact, then among others c^4*g1 and
c^4*g2 turn generators, whence clearly the quotient algebra is much
smaller that before. Here comes the full Groebner basis output from 
that run:

   --------------      ---------------    --------------

% 2
-e1*c+c*e1,
   e1*e1-c*e1,
   e1*g1,
   -e2*c+c*e2,
   e2*e2-c*e2,
   e2*g2,
   -g1*c+c*g1,
   g1*e1,
   g1*g1-c*g1,
   -g2*c+c*g2,
   g2*e2,
   g2*g2-c*g2,
   
% 3
-c*g2*e1-c*g1*e2+c*e2*g1+c*e1*g2,
   16*e1*g2*g1-9*e1*e2*g1,
   9*e2*e1*e2-9*e1*e2*e1-8*c*g1*e2+8*c*e1*g2-2*c*c*e2+2*c*c*e1,
   3*e2*g1*e2-3*e1*g2*e1+c*g1*e2-c*e1*g2,
   16*e2*g1*g2-9*e2*e1*g2,
   -g1*e2*e1+e2*e1*g2,
   16*g1*g2*e1-9*e2*e1*g2,
   16*g1*g2*g1+3*c*c*g2-3*c*c*g1,
   g2*e1*e2-e1*e2*g1,
   4*g2*e1*g2-4*g1*e2*g1+7*c*g1*e2-7*c*e1*g2,
   16*g2*g1*e2-9*e1*e2*g1,
   8*g2*g1*g2-9*c*g1*e2+9*c*e1*g2,
   
% 4
c*c*e1*g2,
   c*c*e2*g1,
   c*c*g1*e2,
   -c*c*g1*g2+c*c*c*g2,
   c*c*g2*g1-c*c*c*g2,
   c*e1*g2*e1-c*e1*e2*g1,
   -c*e2*e1*g2+c*e1*e2*g1,
   -16*c*g1*e2*g1+9*c*e1*e2*g1,
   e1*e2*e1*g2,
   e2*e1*g2*e1-c*e1*e2*g1,
   
% 5
c*c*c*c*g1,
   c*c*c*c*g2,
   c*c*e1*e2*g1,
   
Done

   --------------      ---------------    --------------

So, I would still like to know a bit more about the interpretation
of the output before we start on much higher levels. I'll run your
next level problem input anyhow.

                Joergen Backelin

From bww@maths.nott.ac.uk Thu Jun  5 15:12:21 1997
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From: Bruce W Westbury <bww@maths.nott.ac.uk>
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Message-Id: <9706051312.AA15122@claret>
Subject: Re: Not expected results.
To: joeb@matematik.su.se (Jorgen Backelin)
Date: Thu, 5 Jun 1997 14:12:17 +0100 (BST)
In-Reply-To: <199706051157.NAA05511@pavidus.matematik.su.se> from "Jorgen Backelin" at Jun 5, 97 01:57:16 pm
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Status: R

Dear Joergen
	We seem to have come to some agreement on this first example.
At least in the sense that your output is consistent with my previous
understanding. I would prefer it if there were no relations divisible
by c. Is it possible to avoid this by playing with the ordering? for 
example putting c at the end of the list of generators? If not, 
is it possible to tell Bergman this is what I want?
	There seems to be a problem with the extra relations. I will
check this as the most likely explanation is that I have made a mistake.
	The output I want is (1-z)*H(z) where H(z) is the Hilbert series
your program produces. If there are no relations divisible by c and the
algebra is finite dimensional then this is a polynomial with positive
coefficients and the dimension of the algebra is the sum of the coefficients.
I can construct finite dimensional irreducible representations of this
algebra. The sum of the squares of the dimensions is 200. If your program
says that the dimension is 200 then the algebra is semi-simple and I have
got all the ireducible representations. If the dimension is more then
there are further irreducible representations or the algebra is not
semi-simple.

	These algebras are quotients of the braid group algebras in which
the braid group generators satisfy a cubic relation. These conditions do
not define finite dimensional algebras. The problem is to impose additional
relations so that the algebras are finite dimensional. I then want to
study the representation theory of these algebras.

	I would also be interested in knowing wether your program can
help with the representation theory. For example is it possible to find
a basis of the left ideal generated by an idempotent? A reference that may
be relevant is Charles C. Sims "Computation with finitely presented groups"
Chapter 10.

		Bruce

