Dominant weight multiplicities in hybrid characters of Bn, Cn, F4, G2

TL;DR

This paper introduces hybrid characters for certain Lie groups, analyzing their dominant weight multiplicities and deriving formulas for hybrid dimensions, which differ from traditional characters due to the distinct action on long and short roots.

Contribution

It defines hybrid characters for Lie groups with two root lengths and determines their dominant weight multiplicities and hybrid dimension formulas.

Findings

Dominant weight multiplicities for hybrid characters are explicitly determined.

Formulas for hybrid dimensions are derived as zero-degree terms in power expansions.

Hybrid characters exhibit different Weyl group actions on long and short roots.

Abstract

The characters of irreducible finite dimensional representations of compact simple Lie group G are invariant with respect to the action of the Weyl group W(G) of G. The defining property of the new character-like functions ("hybrid characters") is the fact that W(G) acts differently on the character term corresponding to the long roots than on those corresponding to the short roots. Therefore the hybrid characters are defined for the simple Lie groups with two different lengths of their roots. Dominant weight multiplicities for the hybrid characters are determined. The formulas for "hybrid dimensions" are also found for all cases as the zero degree term in power expansion of the "hybrid characters".