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We study multiplicity space signatures in tensor products of representationsof $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$, and give some applications. Wecompletely classify definite multiplicity spaces for generic tensor products of$\mathfrak{sl}_2$ Verma modules. This provides a classification of a family ofunitary representations of a basic quantized quiver variety, one of the firstsuch classifications for any quantized quiver variety. We use multiplicityspace signatures to provide the first real critical point lower bound forgeneric $\mathfrak{sl}_2$ master functions. As a corollary of this bound, weobtain a simple and asymptotically correct approximation for the number of realcritical points of a generic $\mathfrak{sl}_2$ master function. We obtain aformula for multiplicity space signatures in tensor products of finitedimensional simple $U_q(\mathfrak{sl}_2)$ representations. Our formula alsogives multiplicity space signatures in generic tensor products of$\mathfrak{sl}_2$ Verma modules and generic tensor products of real$U_q(\mathfrak{sl}_2)$ Verma modules. Our results have relations with knottheory, statistical mechanics, quantum physics, and geometric representationtheory.

Signatures of Multiplicity Spaces in tensor products of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$ Representations