Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2006
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2006

[Date Index] [Thread Index] [Author Index]

Search the Archive

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.


  • Prev by Date: RE: using a previous result
  • Next by Date: Re: using a previous result
  • Previous by thread: Re: matrix substitution-> theta1 minimal Pisot quotient group
  • Next by thread: Sturm-Liouville (eigenvalue/eigenfunction) problems