DEGENERATE TWO-BOUNDARY CENTRALIZER ALGEBRAS

作者：Daugherty Zajj^*

来源：Pacific Journal of Mathematics, 2012, 258(1): 91-142.

• representation theory
• combinatorics
• Lie theory
• Hecke algebras
• braid groups
• diagram algebras
• tantalizers
• Schur-Weyl duality

Diagram algebras (for example, graded braid groups, Hecke algebras and 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 g on tensor space of the form M circle times N circle times V-circle times k. We define the degenerate two-boundary braid algebra g(k) 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 gl(n) and sl(n) and modules M and N indexed by rectangular partitions. For this setting, we define the degenerate extended two-boundary Hecke algebra H-k(ext) as a quotient of g(k), and show that a quotient of H-k(ext) is isomorphic to a large subalgebra of the centralizer. We further study the representation theory of H-k(ext) 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 H-k(ext) is given by combinatorial formulas.

• 出版日期2012-7