Signatures of Multiplicity Spaces in Tensor Products of sl2 and Uq(sl2) Representations, and Applications

We study multiplicity space signatures in tensor products of sl2 and Uq(sl2) representations and theirapplications. We completely classify definite multiplicity spaces for generic tensor products of sl2 Vermamodules. This provides a classification of a family of unitary representations of a basic quantized quivervariety, one of the first such classifications for any quantized quiver variety. We use multiplicity spacesignatures to provide the first real critical point lower bound for generic sl2 master functions. As a corollaryof this bound, we obtain a simple and asymptotically correct approximation for the number of real criticalpoints of a generic sl2 master function. We obtain a formula for multiplicity space signatures in tensorproducts of finite dimensional simple Uq(sl2) representations. Our formula also gives multiplicity spacesignatures in generic tensor products of sl2 Verma modules and generic tensor products of real Uq(sl2)Verma modules. Our results have relations with knot theory, statistical mechanics, quantum physics, andgeometric representation theory.