On the Representation Theory of Negative Spin

TL;DR

This paper introduces a new class of infinite-dimensional, indefinite inner product representations of su(2) with negative spin, analyzing their structure and properties using generalized characters.

Contribution

It constructs negative spin irreducible representations of su(2), analyzes their decomposition, and defines effective fractional dimensions, expanding the representation theory of su(2).

Findings

Negative spin representations are infinite-dimensional with indefinite inner products.

Explicit reduction formulas for products of positive and negative representations are provided.

Effective dimensions of negative spin representations are fractional and behave consistently under operations.

Abstract

We construct a class of negative spin irreducible representations of the su(2) Lie algebra. These representations are infinite-dimensional and have an indefinite inner product. We analyze the decomposition of arbitrary products of positive and negative representations with the help of generalized characters and write down explicit reduction formulae for the products. From the characters, we define effective dimensions for the negative spin representations, find that they are fractional, and point out that the dimensions behave consistently under multiplication and decomposition of representations.