2,015 research outputs found

    Vertex operators arising from Jacobi-Trudi identities

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    We give an interpretation of the boson-fermion correspondence as a direct consequence of Jacobi-Trudi identity. This viewpoint enables us to construct from a generalized version of the Jacobi-Trudi identity the action of Clifford algebra on polynomial algebras that arrives as analogues of the algebra of symmetric functions. A generalized Giambelli identity is also proved to follow from that identity. As applications, we obtain explicit formulas for vertex operators corresponding to characters of the classical Lie algebras, shifted Schur functions, and generalized Schur symmetric functions associated to linear recurrence relations.Comment: 23 page

    False theta functions and companions to Capparelli's identities

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    Capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions, where were motivated by the calculation of characters for the standard modules of certain affine Lie algebras and by vertex operator theory. These identities were subsequently proved and refined by Andrews, who related them to Jacobi theta functions, and also by Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two new companions to Capparelli's identities, where the evaluations are expressed in terms of Jacobi theta functions and false theta functions.Comment: 17 pages; references update

    Acyclic Jacobi Diagrams

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    We propose a simple new combinatorial model to study spaces of acyclic Jacobi diagrams, in which they are identified with algebras of words modulo operations. This provides a starting point for a word-problem type combinatorial investigation of such spaces, and provides fresh insights on known results.Comment: 18 pages, 7 figures. Refernces added. Section 2 rewritten. Proof of Theorem 1.1 rewritten. To appear in Kobe J. Mat

    On the Lie enveloping algebra of a post-Lie algebra

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    We consider pairs of Lie algebras gg and gˉ\bar{g}, defined over a common vector space, where the Lie brackets of gg and gˉ\bar{g} are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra U(g)U(g). This permits us to define another associative product on U(g)U(g), which gives rise to a Hopf algebra isomorphism between U(gˉ)U(\bar{g}) and a new Hopf algebra assembled from U(g)U(g) with the new product. For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
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