New perspectives in combinatorial algebra

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The purpose of this project is to study some classical topics in combinatorics and algebra from the perspective of new modern techniques. The PI is interested in algebraic computations such as the equations that characterize a space, and how they interact with seemingly unrelated computations. The project is broken into three subprojects with a different theme. These include the study of the space of polynomials with repeated roots, super homogeneous spaces, and the use of Lie algebras in representation stability theory. The commonality among them involves viewing specific algebraic computations from a different angle to get a different algebraic computation that surprisingly is much more tractable. This expands on previous work of the PI and his collaborators, and the intention is to take it into new directions. The project will also provide opportunities for training graduate students.The PI will work on three main topics sitting between representation theory and commutative algebra, with applications flowing between these subjects in both directions. The first topic concerns computing the equations and syzygies of multiple root loci in spaces of binary forms. The PI plans to study them all together as the degree of the binary forms grows using certain monad-type constructions. Recently, this was successfully used to give a new proof of the generic Green conjecture on canonically embedded projective curves. The second topic concerns the calculation of syzygies of determinantal-like varieties via their connection to the coherent cohomology of super homogeneous spaces and super analogues of the classical Grothendieck-Springer resolution. This is motivated by the existence of unexpected actions of Lie superalgebras on these syzygies and, in fact, offers a conceptual explanation for their existence. On the other hand, it also offers a new strategy to find super analogues of the Borel-Weil-Bott theorem, which the PI plans to explore. The third topic concerns curried algebras; a concept recently introduced by the PI in collaboration with Andrew Snowden. This provides a deep connection between representation stability theory and constructions in Lie theory such as the Bernstein-Gelfand-Gelfand category O. The PI plans to import these techniques from Lie theory to find and prove new structural results in representation stability and its applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.