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

Let (Γ,⪰)(Γ,⪰) be a finite poset. The set Z⩾0Γ,⪰ of all order preserving functions f:Γ→Z⩾0f: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⊆ΓB⊆Γ, we consider the setΩA,B(Γ):={f∈ZΓ,⪰: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,lAn,k,l whose structure encodes the decomposition of tensor productρ⊗Sα1(Cn⁎)⊗⋯⊗Sαl(Cn⁎)ρ⊗Sα1(Cn⁎)⊗⋯⊗Sαl(Cn⁎) where ρ   is a polynomial representation of GLnGLn and Sαi(Cn⁎)Sαi(Cn⁎) is the αiαith symmetric power of Cn⁎Cn⁎, the dual of the standard representation of GLnGLn on CnCn. We call An,k,lAn,k,l an anti-row iterated Pieri algebra for GLnGLn. We show that a certain sign Hibi cone Ωn,k,lΩn,k,l is naturally associated with An,k,lAn,k,l and we construct a basis for An,k,lAn,k,l indexed by the elements of Ωn,k,lΩn,k,l. We further show that this basis contains all the standard monomials on a set of algebra generators of An,k,lAn,k,l.