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SOLUTION TO GIT PROBLEM: NONSTABLE LOCI
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Group: D4 RepresentationD4(1/2,1/2,1/2,-1/2)
Set of maximal non-stable states:
(1) 1-PS = (1, 1, 1, -1) yields a state with 7 characters
Maximal nonstable state={(1/2, 1/2, -1/2, 1/2), (1/2, -1/2, 1/2, 1/2), (-1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2), (-1/2, -1/2, 1/2, -1/2), (1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2)}


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SOLUTION TO GIT PROBLEM: UNSTABLE LOCI
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Group: D4 RepresentationD4(1/2,1/2,1/2,-1/2)
Set of maximal unstable states:
(1) 1-PS = (1, 0, 0, 0) yields a state with 4 characters
Maximal unstable state={(1/2, -1/2, 1/2, 1/2), (1/2, 1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2), (1/2, 1/2, -1/2, 1/2)}
(2) 1-PS = (1, 1, 1, 1) yields a state with 4 characters
Maximal unstable state={(1/2, -1/2, 1/2, 1/2), (1/2, 1/2, 1/2, -1/2), (-1/2, 1/2, 1/2, 1/2), (1/2, 1/2, -1/2, 1/2)}


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SOLUTION TO GIT PROBLEM: STRICTLY POLYSTABLE LOCI
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Group: D4 RepresentationD4(1/2,1/2,1/2,-1/2)
Set of strictly polystable states:
(1) A state with 2 characters
Strictly polystable state={(1/2, -1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2)}
(2) A state with 4 characters
Strictly polystable state={(1/2, -1/2, 1/2, 1/2), (-1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2), (1/2, -1/2, -1/2, -1/2)}
(3) A state with 6 characters
Strictly polystable state={(1/2, 1/2, -1/2, 1/2), (1/2, -1/2, 1/2, 1/2), (-1/2, 1/2, -1/2, -1/2), (-1/2, 1/2, 1/2, 1/2), (-1/2, -1/2, 1/2, -1/2), (1/2, -1/2, -1/2, -1/2)}