Permanents,Doty coalgebras and dominant dimension of Schur algebras

A (hidden) multiplication on _AZ(n,r)$A_{\mathbb{Z}}(n,r)$, the Z$\mathbb{Z}$-dual of the integral Schur algebra _SZ(n,r)$S_{\mathbb{Z}}(n,r)$ 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$p>0$. The image of the induced multiplication on _Ak(n,r)=_AZ(n,r)_⊗Zk$A_k(n,r)=A_{\mathbb{Z}}(n,r){\otimes }_{\mathbb{Z}}k$ turns out to coincide with the Doty coalgebra _Dn,r,p$D_{n,r,p}$ of truncated symmetric powers. Combined with a new straightening formula for bipermanents, it is proved that such a multiplication induces an isomorphism _Ak(n,r)_⊗Sk(n,r)Ak(n,r)≅_Ak(n,r)$A_k(n,r){\otimes }_{S_k(n,r)}{A}_k(n,r)\cong A_k(n,r)$ as _Sk(n,r)$S_k(n,r)$-bimodules if and only if r≤n(p−1)$r\le n(p-1)$, if and only if _Dn,r,p=_Ak(n,r)$D_{n,r,p}=A_k(n,r)$. As a result, _Sk(n,r)$S_k(n,r)$ is a gendo-symmetric algebra, and its dominant dimension is at least two and admits a combinatorial characterization as long as r≤n(p−1)$r\le n(p-1)$.