UC Santa Barbara
A Walk Through Quaternionic Structures
- Author(s): Kelz, Justin M.
- Advisor(s): Jacob, William B
- et al.
In 1980, Murray Marshall proved that the category of
Quaternionic Structures is naturally equivalent to the category of
abstract Witt rings. This paper develops a combinatorial theory
for finite Quaternionic Structures in the case where $1=-1$,
by demonstrating an equivalence between finite quaternionic
structures and Steiner Triple Systems (STSs) with suitable block
colorings. Associated to these STSs are Block Intersection Graphs (BIGs)
with induced vertex colorings. This equivalence allows for
a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associated
to the Witt rings of so-called elementary type, by providing
necessary and sufficient conditions for a BIG associated to
a product or group extension.