Invariant Polynomials on Tensors Under the Action of a Product of Orthogonal Groups

Author

Lauren Kelly Williams, University of Wisconsin-MilwaukeeFollow

Date of Award

August 2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Jeb F. Willenbring

Committee Members

Jeb F. Willenbring, Fredric Ancel, Allen Bell, Ian Musson, Yi Ming Zou

Abstract

Let K be the product O(n₁) × O(n₂) × … × O(n_r) of orthogonal groups. Let V be the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of degree d homogeneous polynomial functions on V. To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain r-regular graphs on d vertices, as well as a method of associating each invariant to other combinatorial settings such as phylogenetic trees.

Recommended Citation

Williams, Lauren Kelly, "Invariant Polynomials on Tensors Under the Action of a Product of Orthogonal Groups" (2013). Theses and Dissertations. 272.
https://dc.uwm.edu/etd/272