[PHK_seminar] [Cohomology]Wednesday 20 Mar, K V Subrahmanyam]

[Petr Somberg]

  This is just a gentle remainder of our seminar starting for one hour.

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     Cohomology in algebra, geometry, physics and statistics




         Wednesday, 20 March 2024 - 13:30 to 14:30

         Place: ZOOM meeting




         Speaker: K V Subrahmanyam, Chennai Mathematical Institute

         Title: Stabilizer limits and Orbit closures with applications to
Geometric Complexity Theory



         Abstract


Let $G\subseteq GL(X)$ be a reductive group acting on a finite dimensional
vector space $V$ over $\C$. A central problem in Geometric Complexity
Theory is the study points $y,z\in V$ where (i) $z$ is obtained as the
leading term of the action of a 1-parameter subgroup $\lambda (t)\subseteq
G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H
\subseteq G$.

We address the question: under what conditions can (i) and (ii) be
simultaneously satisfied, i.e, there exists a 1-PS $\lambda \subseteq G$
for which $z$ is observed as a limit of $y$.




Using $\lambda$, 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.

Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal
H}$ which connects $y$ and $z$ through their Lie algebras. Here $\hat{\cal
K} is the leading term Lie algebra obtained from ${\cal K}$ by the adjoint
action of $\lambda(t)$. We develop the properties of $\hat{\cal K}$ and
relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the
normal slice to the orbit $O(z)$.




We examine the case when a semisimple element belongs to both ${\cal H}$
and ${\cal K}$. We call this a {\em alignment}. We describe some
consequences of alignment and relate it to existing work on lower bounds
in the case of the determinant and permanent.

We also connect alignment to {\em intermediate $G$-varieties} $W$ 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 properties which connect $z$
and $y$.




This is joint work with Bharat Adsul and Milind Sohoni.
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We shall open ZOOM at 13.15 for virtual coffee
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          For more information see the seminar web page at

          https://www.math.cas.cz/index.php/events/seminar/16




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