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

TL;DR

This paper investigates multiplicity space signatures in tensor products of $rak{sl}_2$ and $U_q(rak{sl}_2)$ representations, providing classifications, bounds, and formulas with applications across mathematics and physics.

Contribution

It offers a complete classification of definite multiplicity spaces for generic $rak{sl}_2$ Verma modules and derives formulas for signatures in various tensor products, connecting to multiple fields.

Findings

Classified definite multiplicity spaces for generic $rak{sl}_2$ Verma modules.

Established a real critical point lower bound for $rak{sl}_2$ master functions.

Derived formulas for multiplicity space signatures in quantum and classical tensor products.

Abstract

We study multiplicity space signatures in tensor products of representations of $\mathfrak{sl}_2$ and $U_q(\mathfrak{sl}_2)$, and give some applications. We completely classify definite multiplicity spaces for generic tensor products of $\mathfrak{sl}_2$ Verma modules. This provides a classification of a family of unitary representations of a basic quantized quiver variety, one of the first such classifications for any quantized quiver variety. We use multiplicity space signatures to provide the first real critical point lower bound for generic $\mathfrak{sl}_2$ master functions. As a corollary of this bound, we obtain a simple and asymptotically correct approximation for the number of real critical points of a generic $\mathfrak{sl}_2$ master function. We obtain a formula for multiplicity space signatures in tensor products of finite dimensional simple $U_q(\mathfrak{sl}_2)$ representations. Our formula also gives 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 knot theory, statistical mechanics, quantum physics, and geometric representation theory.