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

We examine the case when a semisimple element belongs to both ${\mathcal H}$ and ${\mathcal K}$. We call this a 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 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.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( video )

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Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

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