Permanents, Doty coalgebras and dominant dimension of Schur algebras

A (hidden) multiplication on AZ(n,r)AZ(n,r), the ZZ-dual of the integral Schur algebra SZ(n,r)SZ(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>0p>0. The image of the induced multiplication on Ak(n,r)=AZ(n,r)⊗ZkAk(n,r)=AZ(n,r)⊗Zk turns out to coincide with the Doty coalgebra Dn,r,pDn,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)Ak(n,r)⊗Sk(n,r)Ak(n,r)≅Ak(n,r) as Sk(n,r)Sk(n,r)-bimodules if and only if r≤n(p−1)r≤n(p−1), if and only if Dn,r,p=Ak(n,r)Dn,r,p=Ak(n,r). As a result, Sk(n,r)Sk(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≤n(p−1).