Nathaniel Thiem Combinatorial representation theory

My primary research interest is in the interplay between combinatorics and algebraic structures. By employing combinatorial tools such as symmetric functions, partitions, tableaux, graphs, posets, and crystal bases, one can gain significant insight on algebraic and geometric structures such as groups, algebras and rings; and, conversely, the corresponding structure theory can often lead to surprising combinatorial results. Combinatorial representation theory uses combinatorial objects to understand the actions of algebraic structures on vector spaces. A vector space V together with an action by an algebraic structure A is called an A-module. It is natural to ask whether V has any subspace closed under the A-action, because if such a subspace exists, then V in fact decomposes into a direct sum of two A-modules. This type of decomposition suggests that there is some set of “smallest” A-modules, called irreducible modules, that form building blocks from which to construct all other A-modules. Some important problems in combinatorial representation theory therefore include