Ming Fang (Chinese Academy of Sciences, Beijing), Comultiplication, Hochschild cocomplex and dominant dimension.

Abstract:

Every symmetric algebra (or even Frobenius algebra) has a canonical coalgebra structure, though not compatible with the algebra structure. With this comultiplication, particularly the counit, we will have an exact Hochschild cocomplex.

In this talk, based on the joint work with S. Koenig, I will first introduce the gendo-symmetric algebras, a generalization of symmetric algebras and the associated canonical comultiplications. We will show that the exactness of the corresponding Hochschild cocomplex at the first few terms and the dominant dimension of the algebra determine each other so that the dominant dimension measures how far a gendo-symmetric algebra could have a compatible counit.

Ming Fang (Chinese Academy of Sciences, Beijing), Permanents, Doty coalgebras and dominant dimension of Schur algebras.

Abstract:

A (hidden) multiplication on A_Z(n,r), the Z-dual of the integral Schur algebra S_Z(n,r) is explicitly constructed, possibly without a unit. The image of the multiplication map is shown to be spanned by bipermanents. Let k be any field of characteristic p>0. The image of the induced multiplication on A_k(n,r)=A_Z(n,r)\otimes_Z k turns out to coincide with the Doty coalgebra D_{n,r,p} of truncated symmetric powers. Combined with a new straightening formula for bipermanents, it is proved that such a multiplication induces A_k(n,r)\otimes_{S_(n,r)}A_k(n,r)\cong A_k(n,r) as S_k(n,r)-bimodules if and only if r\leq n(p-1), if and only if D_{n,r,p}=A_{k}(n,r). As a result, S_k(n,r) is a gendo-symmetric algebra, and its dominant dimension is at least two and admits a combinatorial characterization as long as r\leq n(p-1).