Advisor:  Dr. James Wilson

Committee:  Dr. Amit Patel, Dr. Maria Gillespie, Dr. Sanjay Rajopadhye

Title: A Representation Theory for Bilinear Maps

The intent of this paper is to assemble basic concepts and results for a representation theory of biadditive and bilinear maps. Towards this end, we begin with some foundational definitions which are meant to generalize introductory concepts from standard representation and module theory and then build toward a corresponding theory of induction, restriction, and Frobenius Reciprocity. In so doing, we move from the more familiar setting of homogeneous structures to the more general setting of heterogeneous ones and make heavy use of diagrams to express the flow of information and the relation between components of these new heterogeneous object.

That we should wish to observe bilinear maps in the context of representations and modules follows naturally from the observation that bilinear maps, as well as ring multiplications and module actions/scalar multiplications, are distributive products. Induction, restriction, and Frobenius Reciprocity were initially chosen as the primary conclusions of this paper because we expect these concepts or some versions of them to be common to any sufficiently robust representation theory and because these ideas are so called ”gateway” concepts through which more sophisticated results are achieved. This line of inquiry is in service of a broader effort to explicate a structure theory for, and, amongst other things, a category theoretic description of, tensors and tensor products. While the full power and utility of these concepts is still being actively explored, their current uses range from applications in quantum mechanics, machine learning, material sciences, and a host of other areas.

Indeed, when viewed as multidimensional arrays the truly pervasive nature of tensors becomes apparent as we consider the wide scope of data tables used to store and analyze information in fields from accounting to computer science to engineering. It is therefore natural that we would like to explore the characteristics of how these objects interact and we anticipate that these results should hold value beyond purely academic interests.

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