Sign Hibi cones and the anti-row iterated Pieri algebras for the general linear groups

Let (Γ, {precedes above singleline equals sign}) be a finite poset. The set Z≥0Γ,{precedes above singleline equals sign} of all order preserving functions f:Γ→Z≥0 forms a semigroup, and is called a Hibi cone. It has a simple structure and has been used to describe the structure of some algebras of interest in representation theory. Now for A, B ⊆ Γ, we consider the set. ΩA,B(Γ):={f∈ZΓ,{precedes above singleline equals sign}:f(A)≥0,f(B)≤0}. It is also a semigroup. We call it a sign Hibi cone. We will develop the structure theory for the sign Hibi cones. Next, we construct an algebra An,k,l whose structure encodes the decomposition of tensor productρ⊗Sα1(Cn*)⊗⋯⊗Sαl(Cn*) where ρ is a polynomial representation of _GLn and Sαi(Cn*) is the _αith symmetric power of Cn*, the dual of the standard representation of _GLn on Cn. We call An,k,l an anti-row iterated Pieri algebra for _GLn. We show that a certain sign Hibi cone _Ωn,k,l is naturally associated with An,k,l and we construct a basis for An,k,l indexed by the elements of _Ωn,k,l. We further show that this basis contains all the standard monomials on a set of algebra generators of An,k,l.