Narrow Results

The Newtonian divided-difference operators generate the nil-Coxeter algebra and semigroup. A bijective correspondence between the nil-Coxeter semigroup and the symmetric group is used to provide braid-like diagrams for the former, and corresponding Reidemeister-type moves for the relations. Conditions are given for similar relations to hold in a skew group ring. Interesting extensions of the nil-Coxeter semigroup are described and given diagrammatic representations.

We give a combinatorial interpretation of a Pieri formula for double Grothendieck polynomials in terms of an interval of the Bruhat order. Another description had been given by Lenart and Postnikov in terms of chain enumerations. We use Lascoux's interpretation of a product of Grothendieck polynomials as a product of two kinds of generators of the 0-Hecke algebra, or sorting operators. In this way, we obtain a direct proof of the result of Lenart and Postnikov and then prove that the set of permutations occurring in the result is actually an interval of the Bruhat order.

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

In the spirit of Bar-Natan’s construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\overrightarrow{x}=(x_1,\dots ,x_n)$, we construct a complex of colored smoothings of the two-strand torus link $T_{2,n}$ in the shape of the Bruhat order on $S_n$, and apply a topological quantum field theory (TQFT) to obtain a chain complex whose Euler characteristic is equal to the Vandermonde determinant evaluated at $\overrightarrow{x}$. A generalization to arbitrary link diagrams is given, yielding categorifications of certain generalized Vandermonde determinants.

We introduce nil-Hecke algebras for Weyl groupoids. We describe a basis and some properties of these algebras which lead to a notion of Bruhat order for Weyl groupoids.