#### Date of Award

Spring 1-1-2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Richard M. Green

#### Second Advisor

Nathaniel Thiem

#### Third Advisor

Eric Stade

#### Fourth Advisor

Martin Walter

#### Fifth Advisor

Amanda Schaeffer Fry

#### Abstract

In this thesis, I present a combinatorial formula for a symmetric invariant quartic form on a spin module for the simple Lie algebra d6. This formula relies on a description of this spin module as a vector space with weights, and weight vectors, indexed by ideals of a particular heap. I describe a new statistic, the profile, on pairs of heap ideals. The profile efficiently encodes the shape of the symmetric difference between the two ideals and demonstrates the available actions of the Weyl group and Lie algebra on any given pair. From the profile, I identify a property called a crossing. The actions of the Weyl group and Lie algebra on pairs of weights may be interpreted as adding or removing crossings between the corresponding ideals. Using the crossings, I present a formula for the symmetric invariant quartic form on a spin module for d6, and discuss potential applications to other closely related minuscule representations.

#### Recommended Citation

Shannon, Erica Hilary, "Computing Invariant Forms for Lie Algebras Using Heaps" (2016). *Mathematics Graduate Theses & Dissertations*. 46.

https://scholar.colorado.edu/math_gradetds/46