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Re: matrix substitution--> Gell-Mann su(3) as a real 6by6 matrix group by substitution

• To: mathgroup at smc.vnet.net
• Subject: [mg67349] Re: matrix substitution--> Gell-Mann su(3) as a real 6by6 matrix group by substitution
• From: Roger Bagula <rlbagula at sbcglobal.net>
• Date: Mon, 19 Jun 2006 00:01:08 -0400 (EDT)
• References: <e665nv$n43$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

This kind of group works is daunting for just about everybody!
It's one of the reasons I bless Mathematica everyday!


I used the this Russian substitution method on a well known matrix group 
Gell-Mann su(3):
a = {{1, 0}, {0, 1}};
b = {{0, 1}, {-1, 0}};
c = {{0, 0}, {0, 0}};
s1 = {{c, a, c}, {a, c, c}, {c, c, c}};
s2 = {{c, -b, c}, {b, c, c}, {c, c, c}};
s3 = {{a, c, c}, {c, -a, c}, {c, c, c}};
s4 = {{c, c, a}, {c, c, c}, {a, c, c}};
s5 = {{c, c, -b}, {c, c, c}, {b, c, c}};
s6 = {{c, c, c}, {c, c, a}, {c, a, c}};
s7 = {{c, c, c}, {c, c, -b}, {c, b, c}};
s8 = {{a, c, c}, {c, a, c}, {c, c, -2*a}}/Sqrt[3];
MatrixForm[s1]
MatrixForm[s2]
MatrixForm[s3]
MatrixForm[s4]
MatrixForm[s5]
MatrixForm[s6]
MatrixForm[s7]
MatrixForm[s8]

I got
s1={{0,0,1,0,0,0},
       {0,0,0,1,0,0},
       {1,0,0,0,0,0},
       {0,1,0,0,0,0},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0}}
s2={{0,0,0,-1,0,0},
       {0,0,1,0,0,0},
       {0,1,0,0,0,0},
       {-1,0,0,0,0,0},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0}}
s3={{1,0,0,0,0,0},
       {0,1,0,0,0,0},
       {0,0,-1,0,0,0},
       {0,0,0,-1,0,0},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0}}
s4={{0,0,1,0,0,0},
       {0,0,0,1,0,0},
       {1,0,0,0,0,0},
       {0,1,0,0,0,0},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0}}
s5={{0,0,0,0,1,0},
       {0,0,0,0,0,1},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0},
       {1,0,0,0,0,0},
       {0,1,0,0,0,0}}
s6={{0,0,0,0,0,-1},
       {0,0,0,0,1,0},
       {0,0,0,0,0,0},
       {0,0,0,0,0,0},
       {0,1,0,0,0,0},
       {-1,0,0,0,0,0}}
s7={{0,0,0,0,0,0},
       {0,0,0,0,0,0},
       {0,0,0,0,1,0},
       {0,0,0,0,0,1},
       {0,0,1,0,0,0},
       {0,0,0,1,0,0}}
s8={{1,0,0,0,0,0},
       {0,1,0,0,0,0},
       {0,0,1,0,0,0},
       {0,0,0,1,0,0},
       {0,0,0,0,-2,0},
       {0,0,0,0,0,-2}}/Sqrt[3]

I welcome someone to check my calculations.
These matrices might  be useful in real number calculations for strong 
field interactions.
It might be possible that double group for this representation can be found.