73,404 research outputs found

    Gr\"obner bases of syzygies and Stanley depth

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    Let F. be a any free resolution of a Z^n-graded submodule of a free module over the polynomial ring K[x_1, ..., x_n]. We show that for a suitable term order on F., the initial module of the p'th syzygy module Z_p is generated by terms m_ie_i where the m_i are monomials in K[x_{p+1}, ..., x_n]. Also for a large class of free resolutions F., encompassing Eliahou-Kervaire resolutions, we show that a Gr\"obner basis for Z_p is given by the boundaries of generators of F_p. We apply the above to give lower bounds for the Stanley depth of the syzygy modules Z_p, in particular showing it is at least p+1. We also show that if I is any squarefree ideal in K[x_1, ..., x_n], the Stanley depth of I is at least of order the square root of 2n.Comment: 13 page

    The Maximal C*-Algebra of Quotients as an Operator Bimodule

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    We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra AA as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product AhAA\otimes_h A labelled by the essential closed right ideals of AA into AA. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.Comment: 8 pages; submitte

    The symplectic and algebraic geometry of Horn's problem

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    Horn's problem was the following: given two Hermitian matrices with known spectra, what might be the eigenvalue spectrum of the sum? This linear algebra problem is exactly of the sort to be approached with the methods of modern Hamiltonian geometry (which were unavailable to Horn). The theorem linking symplectic quotients and geometric invariant theory lets one also bring algebraic geometry and representation theory into play. This expository note is intended to elucidate these connections for linear algebraists, in the hope of making it possible to recognize what sort of problems are likely to fall to the same techniques that were used in proving Horn's conjecture.Comment: 16 pages, 1 figure; expository conference paper (second version has inessential cosmetic changes
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