multiplication table for conformal geometric algebra?

Question

I'm trying to find the full multiplication table for conformal geometric algebra (should be a 32 by 32 matrix). It does not seem to be available in explicit form anywhere on the web. Can anybody help with a link or a paste? Thank you.

Answer 1

I hacked together a Python program to generate the multiplication table for standard basis elements. From the first few entries you can see I'm using $a,b,c,d$ to be the basis elements squaring to $1$, and $e$ to be the basis element squaring to $-1$.

(PS: Oh yeah, and I left off the identity row and column because they're obvious computations. So this is a slightly less than complete multiplication table for the basis.)

     | a     b     c     d     e     ab    ac    ad    ae    bc    bd    be    cd    ce    de    cde   bde   bce   bcd   ade   ace   acd   abe   abd   abc   abcd  abce  abde  acde  bcde  abcde
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
a    | 1     ab    ac    ad    ae    b     c     d     e     abc   abd   abe   acd   ace   ade   acde  abde  abce  abcd  de    ce    cd    be    bd    bc    bcd   bce   bde   cde   abcde bcde
b    |-ab    1     bc    bd    be   -a    -abc  -abd  -abe   c     d     e     bcd   bce   bde   bcde  de    ce    cd   -abde -abce -abcd -ae   -ad   -ac   -acd  -ace  -ade  -abcde cde  -acde
c    |-ac   -bc    1     cd    ce    abc  -a    -acd  -ace  -b    -bcd  -bce   d     e     cde   de   -bcde -be   -bd   -acde -ae   -ad    abce  abcd  ab    abd   abe   abcde-ade  -bde   abde
d    |-ad   -bd   -cd    1     de    abd   acd  -a    -ade   bcd  -b    -bde  -c    -cde   e    -ce   -be    bcde  bc   -ae    acde  ac    abde  ab   -abcd -abc  -abcde abe   ace   bce  -abce
e    |-ae   -be   -ce   -de   -1     abe   ace   ade   a     bce   bde   b     cde   c     d    -cd   -bd   -bc   -bcde -ad   -ac   -acde -ab   -abde -abce  abcde abc   abd   acd   bcd  -abcd
ab   |-b     a     abc   abd   abe  -1    -bc   -bd   -be    ac    ad    ae    abcd  abce  abde  abcde ade   ace   acd  -bde  -bce  -bcd  -e    -d    -c    -cd   -ce   -de   -bcde  acde -cde
ac   |-c    -abc   a     acd   ace   bc   -1    -cd   -ce   -ab   -abcd -abce  ad    ae    acde  ade  -abcde-abe  -abd  -cde  -e    -d     bce   bcd   b     bd    be    bcde -de   -abde  bde
ad   |-d    -abd  -acd   a     ade   bd    cd   -1    -de    abcd -ab   -abde -ac   -acde  ae   -ace  -abe   abcde abc  -e     cde   c     bde   b    -bcd  -bc   -bcde  be    ce    abce -bce
ae   |-e    -abe  -ace  -ade  -a     be    ce    de    1     abce  abde  ab    acde  ac    ad   -acd  -abd  -abc  -abcde-d    -c    -cde  -b    -bde  -bce   bcde  bc    bd    cd    abcd -bcd
bc   | abc  -c     b     bcd   bce  -ac    ab    abcd  abce -1    -cd   -ce    bd    be    bcde  bde  -cde  -e    -d     abcde abe   abd  -ace  -acd  -a    -ad   -ae   -acde  abde -de   -ade
bd   | abd  -d    -bcd   b     bde  -ad   -abcd  ab    abde  cd   -1    -de   -bc   -bcde  be   -bce  -e     cde   c     abe  -abcde-abc  -ade  -a     acd   ac    acde -ae   -abce  ce    ace
be   | abe  -e    -bce  -bde  -b    -ae   -abce -abde -ab    ce    de    1     bcde  bc    bd   -bcd  -d    -c    -cde   abd   abc   abcde a     ade   ace  -acde -ac   -ad   -abcd  cd    acd
cd   | acd   bcd  -d     c     cde   abcd -ad    ac    acde -bd    bc    bcde -1    -de    ce   -e     bce  -bde  -b     ace  -ade  -a     abcde abc  -abd  -ab   -abde  abce -ae   -be   -abe
ce   | ace   bce  -e    -cde  -c     abce -ae   -acde -ac   -be   -bcde -bc    de    1     cd   -d     bcd   b     bde   acd   a     ade  -abc  -abcde-abe   abde  ab    abcd -ad   -bd   -abd
de   | ade   bde   cde  -e    -d     abde  acde -ae   -ad    bcde -be   -bd   -ce   -cd    1     c     b    -bcd  -bce   a    -acd  -ace  -abd  -abe   abcde-abce -abcd  ab    ac    bc    abc
cde  |-acde -bcde  de   -ce   -cd    abcde-ade   ace   acd  -bde   bce   bcd  -e    -d     c     1    -bc    bd    be   -ac    ad    ae   -abcd -abce  abde -abe  -abd   abc  -a    -b     ab
bde  |-abde  de    bcde -be   -bd   -ade  -abcde abe   abd   cde  -e    -d    -bce  -bcd   b     bc    1    -cd   -ce   -ab    abcd  abce  ad    ae   -acde  ace   acd  -a    -abc   c    -ac
bce  |-abce  ce   -be   -bcde -bc   -ace   abe   abcde abc  -e    -cde  -c     bde   b     bcd  -bd    cd    1     de   -abcd -ab   -abde  ac    acde  ae   -ade  -a    -acd   abd  -d     ad
bcd  |-abcd  cd   -bd    bc    bcde -acd   abd  -abc  -abcde-d     c     cde  -b    -bde   bce  -be    ce   -de   -1    -abce  abde  ab   -acde -ac    ad    a     ade  -ace   abe  -e     ae
ade  | de    abde  acde -ae   -ad    bde   cde  -e    -d     abcde-abe  -abd  -ace  -acd   a     ac    ab   -abcd -abce  1    -cd   -ce   -bd   -be    bcde -bce  -bcd   b     c     abc   bc
ace  | ce    abce -ae   -acde -ac    bce  -e    -cde  -c    -abe  -abcde-abc   ade   a     acd  -ad    abcd  ab    abde  cd    1     de   -bc   -bcde -be    bde   b     bcd  -d    -abd  -bd
acd  | cd    abcd -ad    ac    acde  bcd  -d     c     cde  -abd   abc   abcde-a    -ade   ace  -ae    abce -abde -ab    ce   -de   -1     bcde  bc   -bd   -b    -bde   bce  -e    -abe  -be
abe  | be   -ae   -abce -abde -ab   -e    -bce  -bde  -b     ace   ade   a     abcde abc   abd  -abcd -ad   -ac   -acde  bd    bc    bcde  1     de    ce   -cde  -c    -d    -bcd   acd   cd
abd  | bd   -ad   -abcd  ab    abde -d    -bcd   b     bde   acd  -a    -ade  -abc  -abcde abe  -abce -ae    acde  ac    be   -bcde -bc   -de   -1     cd    c     cde  -e    -bce   ace   ce
abc  | bc   -ac    ab    abcd  abce -c     b     bcd   bce  -a    -acd  -ace   abd   abe   abcde abde -acde -ae   -ad    bcde  be    bd   -ce   -cd   -1    -d    -e    -cde   bde  -ade  -de
abcd |-bcd   acd  -abd   abc   abcde-cd    bd   -bc   -bcde -ad    ac    acde -ab   -abde  abce -abe   ace  -ade  -a    -bce   bde   b    -cde  -c     d     1     de   -ce    be   -ae    e
abce |-bce   ace  -abe  -abcde-abc  -ce    be    bcde  bc   -ae   -acde -ac    abde  ab    abcd -abd   acd   a     ade  -bcd  -b    -bde   c     cde   e    -de   -1    -cd    bd   -ad    d
abde |-bde   ade   abcde-abe  -abd  -de   -bcde  be    bd    acde -ae   -ad   -abce -abcd  ab    abc   a    -acd  -ace  -b     bcd   bce   d     e    -cde   ce    cd   -1    -bc    ac   -c
acde |-cde  -abcde ade  -ace  -acd   bcde -de    ce    cd   -abde  abce  abcd -ae   -ad    ac    a    -abc   abd   abe  -c     d     e    -bcd  -bce   bde  -be   -bd    bc   -1    -ab    b
bcde | abcde-cde   bde  -bce  -bcd  -acde  abde -abce -abcd -de    ce    cd   -be   -bd    bc    b    -c     d     e     abc  -abd  -abe   acd   ace  -ade   ae    ad   -ac    ab   -1    -a
abcde| bcde -acde  abde -abce -abcd -cde   bde  -bce  -bcd  -ade   ace   acd  -abe  -abd   abc   ab   -ac    ad    ae    bc   -bd   -be    cd    ce   -de    e     d    -c     b    -a    -1

Answer 2

\begin{pmatrix}1 & {{e}_{1}} & {{e}_{2}} & {{e}_{3}} & {{e}_{4}} & {{e}_{1,2}} & {{e}_{1,3}} & {{e}_{1,4}} & {{e}_{2,3}} & {{e}_{2,4}} & {{e}_{3,4}} & {{e}_{1,2,3}} & {{e}_{1,2,4}} & {{e}_{1,3,4}} & {{e}_{2,3,4}} & {{e}_{1,2,3,4}}\\ {{e}_{1}} & 1 & {{e}_{1,2}} & {{e}_{1,3}} & {{e}_{1,4}} & {{e}_{2}} & {{e}_{3}} & {{e}_{4}} & {{e}_{1,2,3}} & {{e}_{1,2,4}} & {{e}_{1,3,4}} & {{e}_{2,3}} & {{e}_{2,4}} & {{e}_{3,4}} & {{e}_{1,2,3,4}} & {{e}_{2,3,4}}\\ {{e}_{2}} & -{{e}_{1,2}} & 1 & {{e}_{2,3}} & {{e}_{2,4}} & -{{e}_{1}} & -{{e}_{1,2,3}} & -{{e}_{1,2,4}} & {{e}_{3}} & {{e}_{4}} & {{e}_{2,3,4}} & -{{e}_{1,3}} & -{{e}_{1,4}} & -{{e}_{1,2,3,4}} & {{e}_{3,4}} & -{{e}_{1,3,4}}\\ {{e}_{3}} & -{{e}_{1,3}} & -{{e}_{2,3}} & 1 & {{e}_{3,4}} & {{e}_{1,2,3}} & -{{e}_{1}} & -{{e}_{1,3,4}} & -{{e}_{2}} & -{{e}_{2,3,4}} & {{e}_{4}} & {{e}_{1,2}} & {{e}_{1,2,3,4}} & -{{e}_{1,4}} & -{{e}_{2,4}} & {{e}_{1,2,4}}\\ {{e}_{4}} & -{{e}_{1,4}} & -{{e}_{2,4}} & -{{e}_{3,4}} & -1 & {{e}_{1,2,4}} & {{e}_{1,3,4}} & {{e}_{1}} & {{e}_{2,3,4}} & {{e}_{2}} & {{e}_{3}} & -{{e}_{1,2,3,4}} & -{{e}_{1,2}} & -{{e}_{1,3}} & -{{e}_{2,3}} & {{e}_{1,2,3}}\\ {{e}_{1,2}} & -{{e}_{2}} & {{e}_{1}} & {{e}_{1,2,3}} & {{e}_{1,2,4}} & -1 & -{{e}_{2,3}} & -{{e}_{2,4}} & {{e}_{1,3}} & {{e}_{1,4}} & {{e}_{1,2,3,4}} & -{{e}_{3}} & -{{e}_{4}} & -{{e}_{2,3,4}} & {{e}_{1,3,4}} & -{{e}_{3,4}}\\ {{e}_{1,3}} & -{{e}_{3}} & -{{e}_{1,2,3}} & {{e}_{1}} & {{e}_{1,3,4}} & {{e}_{2,3}} & -1 & -{{e}_{3,4}} & -{{e}_{1,2}} & -{{e}_{1,2,3,4}} & {{e}_{1,4}} & {{e}_{2}} & {{e}_{2,3,4}} & -{{e}_{4}} & -{{e}_{1,2,4}} & {{e}_{2,4}}\\ {{e}_{1,4}} & -{{e}_{4}} & -{{e}_{1,2,4}} & -{{e}_{1,3,4}} & -{{e}_{1}} & {{e}_{2,4}} & {{e}_{3,4}} & 1 & {{e}_{1,2,3,4}} & {{e}_{1,2}} & {{e}_{1,3}} & -{{e}_{2,3,4}} & -{{e}_{2}} & -{{e}_{3}} & -{{e}_{1,2,3}} & {{e}_{2,3}}\\ {{e}_{2,3}} & {{e}_{1,2,3}} & -{{e}_{3}} & {{e}_{2}} & {{e}_{2,3,4}} & -{{e}_{1,3}} & {{e}_{1,2}} & {{e}_{1,2,3,4}} & -1 & -{{e}_{3,4}} & {{e}_{2,4}} & -{{e}_{1}} & -{{e}_{1,3,4}} & {{e}_{1,2,4}} & -{{e}_{4}} & -{{e}_{1,4}}\\ {{e}_{2,4}} & {{e}_{1,2,4}} & -{{e}_{4}} & -{{e}_{2,3,4}} & -{{e}_{2}} & -{{e}_{1,4}} & -{{e}_{1,2,3,4}} & -{{e}_{1,2}} & {{e}_{3,4}} & 1 & {{e}_{2,3}} & {{e}_{1,3,4}} & {{e}_{1}} & {{e}_{1,2,3}} & -{{e}_{3}} & -{{e}_{1,3}}\\ {{e}_{3,4}} & {{e}_{1,3,4}} & {{e}_{2,3,4}} & -{{e}_{4}} & -{{e}_{3}} & {{e}_{1,2,3,4}} & -{{e}_{1,4}} & -{{e}_{1,3}} & -{{e}_{2,4}} & -{{e}_{2,3}} & 1 & -{{e}_{1,2,4}} & -{{e}_{1,2,3}} & {{e}_{1}} & {{e}_{2}} & {{e}_{1,2}}\\ {{e}_{1,2,3}} & {{e}_{2,3}} & -{{e}_{1,3}} & {{e}_{1,2}} & {{e}_{1,2,3,4}} & -{{e}_{3}} & {{e}_{2}} & {{e}_{2,3,4}} & -{{e}_{1}} & -{{e}_{1,3,4}} & {{e}_{1,2,4}} & -1 & -{{e}_{3,4}} & {{e}_{2,4}} & -{{e}_{1,4}} & -{{e}_{4}}\\ {{e}_{1,2,4}} & {{e}_{2,4}} & -{{e}_{1,4}} & -{{e}_{1,2,3,4}} & -{{e}_{1,2}} & -{{e}_{4}} & -{{e}_{2,3,4}} & -{{e}_{2}} & {{e}_{1,3,4}} & {{e}_{1}} & {{e}_{1,2,3}} & {{e}_{3,4}} & 1 & {{e}_{2,3}} & -{{e}_{1,3}} & -{{e}_{3}}\\ {{e}_{1,3,4}} & {{e}_{3,4}} & {{e}_{1,2,3,4}} & -{{e}_{1,4}} & -{{e}_{1,3}} & {{e}_{2,3,4}} & -{{e}_{4}} & -{{e}_{3}} & -{{e}_{1,2,4}} & -{{e}_{1,2,3}} & {{e}_{1}} & -{{e}_{2,4}} & -{{e}_{2,3}} & 1 & {{e}_{1,2}} & {{e}_{2}}\\ {{e}_{2,3,4}} & -{{e}_{1,2,3,4}} & {{e}_{3,4}} & -{{e}_{2,4}} & -{{e}_{2,3}} & -{{e}_{1,3,4}} & {{e}_{1,2,4}} & {{e}_{1,2,3}} & -{{e}_{4}} & -{{e}_{3}} & {{e}_{2}} & {{e}_{1,4}} & {{e}_{1,3}} & -{{e}_{1,2}} & 1 & -{{e}_{1}}\\ {{e}_{1,2,3,4}} & -{{e}_{2,3,4}} & {{e}_{1,3,4}} & -{{e}_{1,2,4}} & -{{e}_{1,2,3}} & -{{e}_{3,4}} & {{e}_{2,4}} & {{e}_{2,3}} & -{{e}_{1,4}} & -{{e}_{1,3}} & {{e}_{1,2}} & {{e}_{4}} & {{e}_{3}} & -{{e}_{2}} & {{e}_{1}} & -1\end{pmatrix}