268 research outputs found

    Grossberg-Karshon twisted cubes and hesitant jumping walk avoidance

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    Let GG be a complex simply-laced semisimple algebraic group of rank rr and BB a Borel subgroup. Let i∈[r]n\mathbf i \in [r]^n be a word and let β„“=(β„“1,…,β„“n)\mathbf \ell = (\ell_1,\dots,\ell_n) be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to i\mathbf i and β„“\mathbf \ell called a twisted cube, whose lattice points encode the character of a BB-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by i\mathbf i and β„“\mathbf \ell. In recent work, the author and Harada described precisely when the Grossberg-Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence β„“\mathbf \ell comes from a weight Ξ»\lambda of GG. However, not every integer sequence β„“\mathbf \ell comes from a weight of GG. In the present paper, we interpret untwistedness of Grossberg-Karshon twisted cubes associated to any word i\mathbf i and any integer sequence β„“\mathbf \ell using the combinatorics of i\mathbf i and β„“\mathbf \ell. Indeed, we prove that the Grossberg-Karshon twisted cube is untwisted precisely when i\mathbf i is hesitant-jumping-β„“\mathbf \ell-walk-avoiding.Comment: Keywords: Grossberg-Karshon twisted cubes, pattern avoidance, character formula, generalized Demazure modules. arXiv admin note: text overlap with arXiv:1407.854

    Algebraic and geometric properties of flag Bott-Samelson varieties and applications to representations

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    We introduce the notion of flag Bott-Samelson variety as a generalization of Bott-Samelson variety and flag variety. Using a birational morphism from an appropriate Bott-Samelson variety to a flag Bott-Samelson variety, we compute Newton-Okounkov bodies of flag Bott-Samelson varieties as generalized string polytopes, which are applied to give polyhedral expressions for irreducible decompositions of tensor products of GG-modules. Furthermore, we show that flag Bott-Samelson varieties are degenerated into flag Bott manifolds with higher rank torus actions, and find the Duistermaat-Heckman measures of the moment map images of flag Bott-Samelson varieties with the torus action together with invariant closed 22-forms

    Maximum Gap in (Inverse) Cyclotomic Polynomial

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    Let g(f)g(f) denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial ff. Let Ξ¦n\Phi_n denote the nn-th cyclotomic polynomial and let Ξ¨n\Psi_n denote the nn-th inverse cyclotomic polynomial. In this note, we study g(Ξ¦n)g(\Phi_n) and g(Ξ¨n)g(\Psi_n) where nn is a product of odd primes, say p1<p2<p3p_1 < p_2 < p_3, etc. It is trivial to determine g(Ξ¦p1)g(\Phi_{p_1}), g(Ξ¨p1)g(\Psi_{p_1}) and g(Ξ¨p1p2)g(\Psi_{p_1p_2}). Hence the simplest non-trivial cases are g(Ξ¦p1p2)g(\Phi_{p_1p_2}) and g(Ξ¨p1p2p3)g(\Psi_{p_1p_2p_3}). We provide an exact expression for g(Ξ¦p1p2).g(\Phi_{p_1p_2}). We also provide an exact expression for g(Ξ¨p1p2p3)g(\Psi_{p_1p_2p_3}) under a mild condition. The condition is almost always satisfied (only finite exceptions for each p1p_1). We also provide a lower bound and an upper bound for g(Ξ¨p1p2p3)g(\Psi_{p_1p_2p_3})
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