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CS Faculty Talks Date: Wednesday, 9 October 2024 Time: 03:00 - 03:50 PM Venue: Seminar Hall Orbit closures through stabilizer limits & intermediate varieties K V Subrahmanyam Chennai Mathematical Institute. 09-10-2024 Abstract We study how a point z in the projective space of forms with a dis- tinctive stabilizer can be in the orbit closure of another point y with a distinctive stabilizer, under the action of one parameter subgroup λ. These problems arise in the study of the determinant versus permanent problem in algebraic complexity theory. We utilize λ to develop a leading term analysis of Lie algebras and modules. We use it to construct the stabilizer limit Lie algebra ˆK of the same dimension as K and show that ˆK ⊆ H where K and H are the stabilizer Lie algebras of y and z respectively, both being subalge- bras of the the Lie algebra G of the general linear group G acting on polynomials. We introduce the notion of an alignment of y and z through λ and show how this relates to work of Landsberg and Ressayre on the equivariant determinantal complexity of forms and show the consequences of an absence of alignment. We also investigate intermediate closed G-varieties which lie between the orbits of z and y as an aid to obtain a step-wise understanding of the evolution of H from K in the passage of taking the limit under λ. We propose two approaches for constructing intermediate varieties. Based on several examples of both these approaches, we pose the prob- lem of decomposing the limiting process y λ → z as a sequence of limiting processes of the same kind with each step respecting certain geometric and Lie-algebraic conditions. Joint work with Bharat Adsul and Milind Sohoni.
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