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SOLUTION TO GIT PROBLEM: NONSTABLE LOCI
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Group: A4 RepresentationA4(3,0,0,0,0)
Set of maximal non-stable states:
(1) 1-PS = (1, 1/2, 0, -1/2, -1) yields a state with 20 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 0, 3, 0, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(2) 1-PS = (1, 0, 0, 0, -1) yields a state with 24 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 0, 3, 0, 0), (0, 1, 2, 0, 0), (1, 0, 0, 1, 1), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (0, 0, 0, 3, 0), (1, 1, 1, 0, 0), (0, 0, 1, 2, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (0, 1, 0, 2, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(3) 1-PS = (1, 1, 0, 0, -2) yields a state with 23 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 0, 3, 0, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (0, 0, 0, 3, 0), (1, 1, 1, 0, 0), (0, 0, 1, 2, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (0, 1, 0, 2, 0), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(4) 1-PS = (1, 1/2, -1/2, -1/2, -1/2) yields a state with 19 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 2, 0, 0, 1), (1, 0, 0, 1, 1), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (2, 0, 0, 1, 0), (1, 0, 0, 0, 2), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(5) 1-PS = (1, 0, 0, -1/2, -1/2) yields a state with 19 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 0, 3, 0, 0), (0, 1, 2, 0, 0), (1, 0, 0, 1, 1), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (2, 0, 0, 1, 0), (1, 0, 0, 0, 2), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(6) 1-PS = (1, 1/2, 1/2, -1, -1) yields a state with 22 characters
Maximal nonstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 1, 1, 0), (0, 0, 3, 0, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (3, 0, 0, 0, 0), (2, 1, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (0, 0, 2, 0, 1), (0, 1, 1, 0, 1), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}


**************************************
SOLUTION TO GIT PROBLEM: UNSTABLE LOCI
**************************************
Group: A4 RepresentationA4(3,0,0,0,0)
Set of maximal unstable states:
(1) 1-PS = (1, 7/17, -3/17, -8/17, -13/17) yields a state with 18 characters
Maximal unstable state={(2, 0, 0, 0, 1), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(2) 1-PS = (1, 2/7, 2/7, -3/7, -8/7) yields a state with 20 characters
Maximal unstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 0, 2, 0), (0, 0, 3, 0, 0), (1, 0, 1, 1, 0), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(3) 1-PS = (1, 1, 2/7, -3/7, -13/7) yields a state with 21 characters
Maximal unstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 0, 2, 0), (0, 0, 3, 0, 0), (1, 0, 1, 1, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (0, 1, 0, 2, 0), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(4) 1-PS = (1, 1/6, 1/6, 1/6, -3/2) yields a state with 21 characters
Maximal unstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (1, 0, 0, 2, 0), (0, 0, 3, 0, 0), (1, 0, 1, 1, 0), (0, 1, 2, 0, 0), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (0, 0, 0, 3, 0), (0, 0, 1, 2, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (0, 1, 0, 2, 0), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(5) 1-PS = (1, 2/7, -3/7, -3/7, -3/7) yields a state with 19 characters
Maximal unstable state={(2, 0, 0, 0, 1), (1, 0, 0, 2, 0), (1, 0, 1, 1, 0), (0, 2, 0, 0, 1), (1, 0, 0, 1, 1), (1, 1, 0, 0, 1), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (2, 0, 0, 1, 0), (1, 0, 0, 0, 2), (1, 1, 1, 0, 0), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}
(6) 1-PS = (1, 1, 1, -3/2, -3/2) yields a state with 22 characters
Maximal unstable state={(2, 0, 0, 0, 1), (0, 0, 2, 1, 0), (0, 0, 3, 0, 0), (1, 0, 1, 1, 0), (0, 2, 0, 0, 1), (0, 1, 2, 0, 0), (1, 1, 0, 0, 1), (2, 1, 0, 0, 0), (3, 0, 0, 0, 0), (0, 3, 0, 0, 0), (0, 2, 0, 1, 0), (0, 1, 1, 1, 0), (2, 0, 0, 1, 0), (1, 1, 1, 0, 0), (0, 0, 2, 0, 1), (0, 1, 1, 0, 1), (1, 0, 2, 0, 0), (1, 2, 0, 0, 0), (1, 1, 0, 1, 0), (1, 0, 1, 0, 1), (0, 2, 1, 0, 0), (2, 0, 1, 0, 0)}


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SOLUTION TO GIT PROBLEM: STRICTLY POLYSTABLE LOCI
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Group: A4 RepresentationA4(3,0,0,0,0)
Set of strictly polystable states:
(1) A state with 7 characters
Strictly polystable state={(1, 0, 0, 1, 1), (0, 2, 1, 0, 0), (1, 0, 0, 2, 0), (0, 0, 3, 0, 0), (1, 0, 0, 0, 2), (0, 3, 0, 0, 0), (0, 1, 2, 0, 0)}
(2) A state with 13 characters
Strictly polystable state={(0, 1, 1, 1, 0), (0, 2, 0, 1, 0), (0, 0, 2, 1, 0), (0, 0, 3, 0, 0), (0, 0, 0, 3, 0), (0, 0, 1, 2, 0), (0, 1, 2, 0, 0), (1, 0, 0, 1, 1), (0, 2, 1, 0, 0), (1, 1, 0, 0, 1), (0, 1, 0, 2, 0), (1, 0, 1, 0, 1), (0, 3, 0, 0, 0)}
(3) A state with 5 characters
Strictly polystable state={(1, 0, 0, 2, 0), (0, 1, 1, 1, 0), (0, 0, 3, 0, 0), (1, 0, 1, 0, 1), (0, 2, 0, 0, 1)}
(4) A state with 5 characters
Strictly polystable state={(1, 0, 0, 1, 1), (0, 0, 3, 0, 0), (0, 3, 0, 0, 0), (0, 2, 1, 0, 0), (0, 1, 2, 0, 0)}