## Abstract

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 _{αi}th 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.

Original language | English |
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Pages (from-to) | 355-392 |

Number of pages | 38 |

Journal | Journal of Algebra |

Volume | 410 |

DOIs | |

Publication status | Published - 15 Jul 2014 |

Externally published | Yes |

## Keywords

- Anti-row iterated Pieri algebras
- Flat deformation
- Gelfand-Tsetlin patterns
- Generalized iterated Pieri rule
- Lowest weight modules
- Sign Hibi cones
- Standard monomial theory