Tuesday, December 12, 14:00 and 14:50, 7.527

Richard Dipper, q-partition algebras 1 and 2.

Abstract:
Let F be a field and V an n-dimensional F-vector space. The action of the symmetric group Sym(r) on the r-fold tensor power V^{⊗ r} of V by place permutations can be q-deformed to the action of the Iwahori-Hecke H(q,r) algebra associated with Sym(r). This action and the natural quantum-GL(n)-action on tensor space are in Schur-Weyl duality. On the other hand, the symmetric group Sym(n) on a fixed basis of V acts on tensor space as well, restricting the natural action of GL(n) to its subgroup Sym(n). The endomorphism algebra is known to be the partition algebra P_r(n). Since H(q,n) does not have a Hopf coproduct there is no obvious q-deformation of the letter action of Sym(n) on tensor space to H(q,n).

The first part of my talk concerns the construction of such a q-deformation. Thus the endomorphism algebra of this action is a q-deformation of P_r(n). In the second part of my talk I shall outline a proof, that in the special case of prime powers q this q-deformation is isomorphic to the q-partition algebra defined by Halverson and Thiem by different means a few years ago.

(joint work with Geetha Thangavelu)