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Register a new userNewell-Littlewood numbers II: extended Horn inequalities
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The Newell-Littlewood numbers $N_{mu, u,lambda}$ are tensor product multiplicities of Weyl modules for classical Lie groups, in the stable limit. For which triples of partitions $(mu, u,lambda)$ does $N_{mu, u,lambda}>0$ hold? The Littlewood-Richardson coefficient case is solved by the Horn inequalities (in work of A. Klyachko and A. Knutson-T. Tao). We extend these celebrated linear inequalities to a much larger family, suggesting a general solution.
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We provide two shifted analogues of the tableau switching process due to Benkart, Sottile, and Stroomer, the shifted tableau switching process and the modified shifted tableau switching process. They are performed by applying a sequence of specially contrived elementary transformations called {em switches} and turn out to have some spectacular properties. For instance, the maps induced from these algorithms are involutive and behave very nicely with respect to shifted Young tableaux whose reading words satisfy the lattice property. As an application, we give combinatorial interpretations of Schur $P$- and $Q$-function identities. We also demonstrate the relationship between the shifted tableau switching process and the shifted $J$-operation due to Worley.
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.
In 1943, Littlewood and Offord proved the first anti-concentration result for sums of independent random variables. Their result has since then been strengthened and generalized by generations of researchers, with applications in several areas of mathematics. In this paper, we present the first non-abelian analogue of Littlewood-Offord result, a sharp anti-concentration inequality for products of independent random variables.
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This work lies across three areas (in the title) of investigation that are by themselves of independent interest. A problem that arose in quantum computing led us to a link that tied these areas together. This link consists of a single formal power series with a multifaced interpretation. The deeper exploration of this link yielded results as well as methods for solving some numerical problems in each of these separate areas.
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