Date:  01/13/2016

File: reflection.gp

This is a little code which permits the computation of the spinor norm of an orthogonal map.  This is needed for our paper Weyl groups of ... with A. Feingold.  The functions included in the file reflection.gp are:

base_standard

pairing

projection

orthogonal_basis

orthogonal_basis1

apply_aut

reflection

composition

product_of_reflection

spinor_norm


In gp, one can type "?name_of_function" in order to get a short description of what the function does.

Below, one finds an example of the calculation of the spinor norm of the unique non-trivial outer automorphism for A_{3}^{++}.




EXAMPLE:



daniel@bigbang:~/Desktop/paper_update/computation$ gp
Reading GPRC: /Users/daniel/.gprc ...Done.

                                                                                   GP/PARI CALCULATOR Version 2.7.1 (released)
                                                                          i386 running darwin (x86-64/GMP-6.0.0 kernel) 64-bit version
                                                              compiled: Jul 14 2014, Apple LLVM version 5.1 (clang-503.0.40) (based on LLVM 3.4svn)
                                                                                            threading engine: single
                                                                                 (readline v6.3 enabled, extended help enabled)

                                                                                     Copyright (C) 2000-2014 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?12 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000
? 
? 
? A = matrix(5,5)
%1 = 
[0 0 0 0 0]

[0 0 0 0 0]

[0 0 0 0 0]

[0 0 0 0 0]

[0 0 0 0 0]

? for(i=1,5,A[i,i] = 2)
? A
%3 = 
[2 0 0 0 0]

[0 2 0 0 0]

[0 0 2 0 0]

[0 0 0 2 0]

[0 0 0 0 2]

? for(i=1,4,A[i,i+1] = -1)
? A
%5 = 
[2 -1  0  0  0]

[0  2 -1  0  0]

[0  0  2 -1  0]

[0  0  0  2 -1]

[0  0  0  0  2]

? for(i=2,5,A[i,i-1] = -1)
? A
%7 = 
[ 2 -1  0  0  0]

[-1  2 -1  0  0]

[ 0 -1  2 -1  0]

[ 0  0 -1  2 -1]

[ 0  0  0 -1  2]

? A[2,5] = A[5,2] = -1
%8 = -1
? 
? 
? A
%9 = 
[ 2 -1  0  0  0]

[-1  2 -1  0 -1]

[ 0 -1  2 -1  0]

[ 0  0 -1  2 -1]

[ 0 -1  0 -1  2]

? A = A*(1/2);
? A
%11 = 
[   1 -1/2    0    0    0]

[-1/2    1 -1/2    0 -1/2]

[   0 -1/2    1 -1/2    0]

[   0    0 -1/2    1 -1/2]

[   0 -1/2    0 -1/2    1]

? 
? 
? read("reflection.gp")
time = 1 ms.
? base_standard(5)
%13 = [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 1, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]]
? 
? automo = %13;
? 
? 
? automo[3] = %13[5];
? automo[5] = %13[3];
? automo
%17 = [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0]]
? 
? 
? product_of_reflection(A,automo)
time = 2 ms.
%18 = [[1, 0, 0, 0, 0], [2, 0, 0, 0, 0], [1/2, 1, 0, 0, 0], [1, 2, 0, 0, 0], [0, 0, -1, 0, 1], [1/4, 1/2, 3/4, 1, 0], [1/2, 1, 3/2, 2, 0], [4/5, 8/5, 7/5, 6/5, 1], [8/5, 16/5, 14/5, 12/5, 2]]
? spinor_norm(A,%18)
%19 = 18
? factor(%)
%20 = 
[2 1]

[3 2]