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Geometric Complexity Theory -- Lie Algebraic Methods for Projective Limits of Stable Points | | Bharat Adsul
; Milind Sohoni
; K V Subrahmanyam
; | | Date: |
1 Jan 2022 | | Abstract: | Let $G$ be a connected reductive group acting on a complex vector space $V$
and projective space ${mathbb P}V$. Let $xin V$ and ${cal H}subseteq {cal
G}$ be the Lie algebra of its stabilizer. Our objective is to understand points
$[y]$, and their stabilizers which occur in the vicinity of $[x]$. We construct
an explicit ${cal G}$-action on a suitable neighbourhood of $x$, which we call
the local model at $x$. We show that Lie algebras of stabilizers of points in
the vicinity of $x$ are parameterized by subspaces of ${cal H}$. When ${cal
H}$ is reductive these are Lie subalgebras of ${cal H}$. If the orbit of $x$
is closed this also follows from Luna’s theorem. Our construction involves a
map connected to the local curvature form at $x$. We apply the local model to
forms, when the form $g$ is obtained from the form $f$ as the leading term of a
one parameter family acting on $f$. We show that there is a flattening ${cal
K}_0$ of ${cal K}$, the stabilizer of $f$ which sits as a subalgebra of ${cal
H}$, the stabilizer $g$. We specialize to the case of forms $f$ whose
$SL(X)$-orbits are affine, and the orbit of $g$ is of co-dimension $1$. We show
that (i) either ${cal H}$ has a very simple structure, or (ii) conjugates of
the elements of ${cal K}$ also stabilize $g$ and the tangent of exit. Next, we
apply this to the adjoint action. We show that for a general matrix $X$, the
signatures of nilpotent matrices in its projective orbit closure (under
conjugation) are determined by the multiplicity data of the spectrum of $X$.
Finally, we formulate the path problem of finding paths with specific
properties from $y$ to its limit points $x$ as an optimization problem using
local differential geometry. Our study is motivated by Geometric Complexity
Theory proposed by the second author and Ketan Mulmuley. | | Source: | arXiv, 2201.00135 | | Services: | Forum | Review | PDF | Favorites | |
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