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    Characters Bar & Grill

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    Bar & grill. Theater themed menu. Artistic charactures of celebrities. Geographical location: Unknown

    The Conformal Characters

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    We revisit the study of the multiplets of the conformal algebra in any dimension. The theory of highest weight representations is reviewed in the context of the Bernstein-Gelfand-Gelfand category of modules. The Kazhdan-Lusztig polynomials code the relation between the Verma modules and the irreducible modules in the category and are the key to the characters of the conformal multiplets (whether finite dimensional, infinite dimensional, unitary or non-unitary). We discuss the representation theory and review in full generality which representations are unitarizable. The mathematical theory that allows for both the general treatment of characters and the full analysis of unitarity is made accessible. A good understanding of the mathematics of conformal multiplets renders the treatment of all highest weight representations in any dimension uniform, and provides an overarching comprehension of case-by-case results. Unitary highest weight representations and their characters are classified and computed in terms of data associated to cosets of the Weyl group of the conformal algebra. An executive summary is provided, as well as look-up tables up to and including rank four.Comment: 41 pages, many figure

    Quantum Cluster Characters

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    Let \FF be a finite field and (Q,\bfd) an acyclic valued quiver with associated exchange matrix B~\tilde{B}. We follow Hubery's approach \cite{hub1} to prove our main conjecture of \cite{rupel}: the quantum cluster character gives a bijection from the isoclasses of indecomposable rigid valued representations of QQ to the set of non-initial quantum cluster variables for the quantum cluster algebra \cA_{|\FF|}(\tilde{B},\Lambda). As a corollary we find that, for any rigid valued representation VV of QQ, all Grassmannians of subrepresentations Gr_\bfe^V have counting polynomials.Comment: material reorganized, some proofs rewritte

    Generic cluster characters

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    Let \CC be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object TT. Under a constructibility condition we prove the existence of a set \mathcal G^T(\CC) of generic values of the cluster character associated to TT. If \CC has a cluster structure in the sense of Buan-Iyama-Reiten-Scott, \mathcal G^T(\CC) contains the set of cluster monomials of the corresponding cluster algebra. Moreover, these sets coincide if C\mathcal C has finitely many indecomposable objects. When \CC is the cluster category of an acyclic quiver and TT is the canonical cluster-tilting object, this set coincides with the set of generic variables previously introduced by the author in the context of acyclic cluster algebras. In particular, it allows to construct Z\Z-linear bases in acyclic cluster algebras.Comment: 24 pages. Final Version. In particular, a new section studying an explicit example was adde
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