Hopf Algebras

Papers

• Structure of the Malvenuto-Reutenauer Hopf Algebra of Permutations - Aguiar & Sottile http://www.sciencedirect.com/science/article/pii/S0001870804000787
• Higher Bruhat Orders and Cyclic Hyperplane Arrangements - G. Zieler http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.88.662&rep=rep1&type=pdf
• Noncommutative Symmetric Functions I - VI http://scholar.google.com/scholar?hl=en&q=noncommutative+symmetric+functions&btnG=Search&as_sdt=0%2C22&as_ylo=&as_vis=0

Objectives

• Generalize $QSym$ and $\mathfrak{S}Sym$ to higher Bruhat orders (and Oriented Matroids)
• Find the the generalized descent map $\textbf{Des}: B(n,k) \longrightarrow B(n,k-1)$.

Ideas

• Write the multiplicative laws of $QSym$ and $\mathfrak{S}Sym$ in terms of inversions and try to generalize it to objects of higher Bruhat order.