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SUMMARY:K V Subrahmanyam (Chennai Mathematical Institute)
DTSTART:20240320T123000Z
DTEND:20240320T133000Z
DTSTAMP:20260415T110059Z
UID:PHK-cohomology-seminar/99
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/PHK-cohomolo
 gy-seminar/99/">Stabilizer limits and Orbit closures with applications to 
 Geometric Complexity Theory</a>\nby K V Subrahmanyam (Chennai Mathematical
  Institute) as part of Prague-Hradec Kralove seminar Cohomology in algebra
 \, geometry\, physics and statistics\n\n\nAbstract\nLet $G\\subseteq GL(X)
 $ be a reductive group acting on a finite dimensional vector space $V$ ove
 r $\\C$. A central problem in Geometric Complexity Theory is the study poi
 nts $y\,z\\in V$ where (i) $z$ is obtained as the leading term of the acti
 on of a 1-parameter subgroup $\\lambda (t)\\subseteq G$ on $y$\, and (ii) 
 $y$ and $z$ have large distinctive stabilizers $K\,H \\subseteq G$.\n\nWe 
 address the question: under what conditions can (i) and (ii) be simultaneo
 usly satisfied\, i.e\, there exists a 1-PS $\\lambda \\subseteq G$ for whi
 ch $z$ is observed as a limit of $y$.\n\n\nUsing $\\lambda$\, we develop a
  leading term analysis which applies to $V$ as well as to ${\\mathcal G}= 
 Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\\cal K}$ and ${\\cal
  H}$\, the Lie algebras of $K$ and $H$ respectively.\n\nThrough this we co
 nstruct the Lie algebra $\\hat{\\mathcal K} \\subseteq {\\mathcal H}$ whic
 h connects $y$ and $z$ through their Lie algebras. Here $\\hat{\\mathcal K
 }$ is the leading term Lie algebra obtained from ${\\mathcal K}$ by the ad
 joint action of $\\lambda(t)$. We develop the properties of $\\hat{\\mathc
 al K}$ and relate it to the action of ${\\mathcal H}$ on $\\overline{N}=V/
 T_z O(z)$\, the normal slice to the orbit $O(z)$.\n\n\nWe examine the case
  when a semisimple element belongs to both ${\\mathcal H}$ and ${\\mathcal
  K}$. We call this a <em>alignment</em>. We describe some consequences of 
 alignment and relate it to existing work on lower bounds in the case of th
 e determinant and permanent.\n\nWe also connect alignment to <em>intermedi
 ate $G$-varieties</em> $W$ which lie between the orbit closures of $z$ and
  $y$\, i.e. $\\overline{O(z)} \\subsetneq W \\subsetneq O(y)$. These have 
 a direct bearing on representation theoretic as well as geometric properti
 es which connect $z$ and $y$.\n\n\nThis is joint work with Bharat Adsul an
 d Milind Sohoni.\n
LOCATION:https://researchseminars.org/talk/PHK-cohomology-seminar/99/
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