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Register a new userThe Horn inequalities from a geometric point of view
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We give an exposition of the Horn inequalities and their triple role characterizing tensor product invariants, eigenvalues of sums of Hermitian matrices, and intersections of Schubert varieties. We follow Belkales geometric method, but assume only basic representation theory and algebraic geometry, aiming for self-contained, concrete proofs. In particular, we do not assume the Littlewood-Richardson rule nor an a priori relation between intersections of Schubert cells and tensor product invariants. Our motivation is largely pedagogical, but the desire for concrete approaches is also motivated by current research in computational complexity theory and effective algorithms.
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This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.
We derive and spell out the structure constants of the $mathbb{Z}_2$-graded algebra $mathfrak{shs}[lambda],$ by using deformed-oscillators techniques in $Aq(2; u),$, the universal enveloping algebra of the Wigner-deformed Heisenberg algebra in 2 dimensions. The use of Weyl ordering of the deformed oscillators is made throughout the paper, via the symbols of the operators and the corresponding associative, non-commutative star product. The deformed oscillator construction was used by Vasiliev in order to construct the higher spin algebras in three spacetime dimensions. We derive an expression for the structure constants of $mathfrak{shs}[lambda],$ and show that they must obey a recurrence relation as a consequence of the associativity of the star product. We solve this condition and show that the $mathfrak{hs}[lambda],$ structure constants are given by those postulated by Pope, Romans and Shen for the Lone Star product.
Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalities of the Mumford cone in the case of the linear representation associated to a quiver and a dimension vector n=(n_x). We give these inequalities in terms of filtered dimension vectors, and we directly adapt Schofields argument to inductively determine the dimension vectors of general subrepresentations in the filtered context. In particular, this gives one further proof of the Horn inequalities for tensor products.
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type [ |A f|_{mathcal{X}}^2 leq C |f|_{mathcal{X}} big|A^2 fbig|_{mathcal{X}}, quad f in dombig(A^2big), ] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.
We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7.
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