The University of Wisconsin Milwaukee
Thesis Advisor: Jeb F. Willenbring
Thesis Title: Invariant polynomial functions on tensors under a product of orthogonal groups
MA, Mathematics
The University of Wisconsin Milwaukee
BA, Mathematics
The College of New Jersey
Academic Employment
Mercyhurst University
Associate Professor, Mathematics: 2019 - present
Assistant Professor, Mathematics: 2013 - 2019
The adjoint representation of a Lie algebra and the support of Kostant's weight multiplicity formula, with Pamela E Harris and Erik Insko. Journal of Combinatorics, Vol 7, No 1, 2016.
Abstract: Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula is factorial in the rank of the Lie algebra and the value of the partition function is unknown. In this paper we address the difficult question: What are the contributing terms to the multiplicity of the zero weight in the adjoint representation of a finite dimensional Lie algebra? We describe and enumerate the cardinalities of these sets (through linear homogeneous recurrence relations with constant coefficients) for the classical Lie algebras of Type B, C, and D, the Type A case was computed by the first author. In addition, we compute the cardinality of the set of contributing terms for non-zero weight spaces in the adjoint representation. In the Type B case, the cardinality of one such non-zero-weight is enumerated by the Fibonacci numbers. We end with a computational proof of a result of Kostant regarding the exponents of the respective Lie algebra for some low rank examples and provide a section with open problems in this area. Link to online article (paywall) Earlier version available on arXiv
Invariant polynomial functions on tensors under a product of orthogonal groups. Transactions of the American Mathematical Society, Vol 368, No 2, 2016.
Abstract: Let \(K\) be the product \(O_{n_1} \times O_{n_2} \times \cdots \times O_{n_r}\) of orthogonal groups. Let \(V = \bigotimes_{i=1}^r \mathbb{C}^{n_i}\), 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. Link to online article (paywall) Earlier version available on arXiv
The measurement of quantum entanglement and enumeration of graph coverings, with Michael W Hero and Jeb F Willenbring. AMS Contemporary Mathematics Series, Vol 557, 2011.
Abstract: We provide formulas for invariants defined on a tensor product of defining representations of unitary groups, under the action of the product group. This situation has a physical interpretation, as it is related to the quantum mechanical state space of a multi-particle system in which each particle has finitely many outcomes upon observation. Moreover, these invariant functions separate the entangled and unentangled states, and are therefore viewed as measurements of quantum entanglement. When the ranks of the unitary groups are large, we provide a graph theoretic interpretation for the dimension of the invariants of a fixed degree. We also exhibit a bijection between isomorphism classes of finite coverings of connected simple graphs and a basis for the space of invariants. The graph coverings are related to branched coverings of surfaces. Link to online article (paywall) Earlier version available on arXiv