2010/07/22 by Zajj Daugherty, Daugherty, Zajj · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.1007.3950
Diagram algebras (e.g. graded braid groups, Hecke algebras, Brauer algebras) arise as tensor power centralizer algebras, algebras of commuting operators for a Lie algebra action on a tensor space. This work explores centralizers of the action of a complex reductive Lie algebra \mathfrakg on tensor space of the form M ⊗ N ⊗ V⊗ k. We define the degenerate two-boundary braid algebra Gk and show that centralizer algebras contain quotients of this algebra in a general setting. As an example, we study in detail the combinatorics of special cases corresponding to Lie algebras \mathfrakgln and \mathfraksln and modules M and N indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra Hkext as a quotient of Gk, and show that a quotient of Hkext is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of Hkext to find that the seminormal representations are indexed by a known family of partitions. The bases for the resulting modules are given by paths in a lattice of partitions, and the action of Hkext is given by combinatorial formulas.