Projective geometry, invariants and momentum
Final Report Abstract
The project is concerned, in broad terms, with the study of symmetries and symmetry-invariant functions on geometric objects. More concretely, a key role is played by a class of geometric objects, so-called flag varieties, whose importance is arguably due to their many incarnations and prominence in various contexts - algebraic geometry, differential geometry, quantum mechanics, synthetic geometry etc. The results of the project are published in three articles, as explained below. The first article, entitled “Secant varieties and degrees of invariants”, addresses the generators of the ring of invariant polynomials on a representation space of a complex reductive group. A classical theorem of Hilbert asserts that this ring is finitely generated, but the general proof is famously nonconstructive. Closed formulae, or effective algorithms are known only in very special cases. I have studied properties of the generators related to the geometry of closed projective orbits, which are flag varieties, and their secant varieties. Remarkably, secant varieties have not been used systematically in invariant theory, although they fit naturally in the context. I was able to derive new properties of the generating invariants, including lower bounds for their degrees, as well as an existence theorem for generators of certain degrees having certain explicit monomials. The second article, entitled “Quantum marginals from doubly excited states”, coauthored with T. Maciazek, focuses on representations of compact groups and the resulting momentum maps. These notions have direct interpretations in the contexts of quantum mechanics. The so-called orbit of coherent states is a flag variety. Remarkably, the concrete momentum images of certain representations, e.g. the fundamental representations of the unitary groups, known as fermions in physics, remain a mystery despite the long standing demand for application purposes. One of our goals is to extend the dictionary between quantum mechanics and projective geometry. We exhibit relations between momentum images and the classical geometric notions of osculating spaces and Gaussian fundamental forms. We show that the convexity of the full momentum image is equivalent to the nondegeneracy of the second fundamental form of the coherent orbit. We determine a class of representations, whose momentum image is entirely determined by the second fundamental form. The third article is entitled “Unstable loci in flag varieties and variations of quotients”, coauthored with H. Seppanen, and contains, in my opinion, the most substantial part of my work within this project. It focuses on actions on flag varieties by reductive subgroups of the symmetry group. We study relations between the geometry of the subgroup-orbits and subgroup-invariants in irreducible representations of the symmetry group, in the context of Geometric Invariant Theory for line bundles on the variety. We give a closed formula for the unstable locus - vanishing locus of the nonconstant homogeneous invariants in the section ring of a given bundle. We describe the GIT equivalence classes of line bundles. We make the remarkable observation that the dimension of the unstable locus jumps by at most 1 along “continuous” variations of the bundle. We derive the existence and a description of a sequence of nested cones in the Picard group of the flag variety, defined by bounds on the codimension of the unstable locus. We show that a suitable GIT-quotient contains the global information on all invariants for the given subgroup, and prove a number of interesting properties of such quotients. We devise several tests for the presence of the desired quotients and show that generically they do exist.
Publications
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Quantum marginals from pure doubly excited states, J. Phys. A: Math. Theor. 50 (2017), p. 365-404
T. Maciazek, V. Tsanov
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Secant varieties and degrees of invariants
V. Tsanov
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Unstable loci in flag varieties and variation of quotients
H. Seppanen, V. Tsanov
