Symmetry Adapted Gram Spectrahedra
summary
Constructs a symmetry-adapted Gram spectrahedron by restricting Gram matrices to the fixed-point subspace for the symmetric group, uses a representation-theory change of basis to block-diagonalize Q and reduce PSD checks to smaller blocks, and gives exact counts: general binary degree 2d has 2^{d-1} rank-two representations, while a symmetric binary 2d keeps 2^{d/2} if d is even or 2^{(d-1)/2} if d is odd. Examples: symmetric ternary quartics yield a 2D spectrahedron given by a parabola and a hyperbola; simulations show rank six appears most often on symmetric sextic boundaries.
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chapters (8)
- 0:00Start
- 1:03Joint Work
- 1:55Polynomial Optimization and SOS relaxation
- 6:04Geometry of Gram Spectrahedra
- 7:00Example: Binary Sextics
- 12:47Binary Octics
- 25:30Symmetric Binary Sextics
- 27:49Symmetric Ternary Sextics
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