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On duality and negative dimensions in the theory of Lie groups and symmetric spaces
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abstract
We give one more interpretation of the symbolic formulae $U(-N)=U(N)$ and $Sp(-2N)=SO(2N)$ by comparing the values of certain Casimir operators in the corresponding tensor representations. We show also that such relations can be extended to the classical symmetric spaces using Macdonald duality for Jack and Jacobi symmetric functions.
Forward citations
Cited by 3 Pith papers
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$N \leftrightarrow -N$ duality of SU(N) for stable sequences of representations
For stable sequences D(λ,τ) of SU(N) representations, dimensions satisfy dim(D(λ,τ),N) = (-1)^{Area(λ)+Area(τ)} dim(D(τ,λ),-N) and second-order Casimir eigenvalues satisfy C(D(λ,τ),N) = -C(D(λ^T,τ^T),-N).
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Torus knots in adjoint representation and Vogel's universality
Universal adjoint invariants for T[4,n] torus knots are constructed via Vogel's universality, completing the T[4,n] case after previous T[2,n] and T[3,n] results.
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Vogel's universality and Macdonald dimensions
The paper gives a single rational formula for adjoint Macdonald dimensions that unifies the simply laced Lie algebras A_n, D_n, E6, E7, E8, plus explicit mixed-root-system dimension formulas.
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