Nil-Coxeter algebra

In mathematics, the nil-Coxeter algebra, introduced by (Fomin Stanley), is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent.

Definition

The nil-Coxeter algebra for the infinite symmetric group is the algebra generated by u₁, u₂, u₃, ... with the relations

$\begin{array}{rl}{u}_i^2& =0,\\ u_i{u}_j& =u_j{u}_i& & \text{ if }|i-j|>1,\\ u_i{u}_j{u}_i& =u_j{u}_i{u}_j& & \text{ if }|i-j|=1.\end{array}$

These are just the relations for the infinite braid group, together with the relations u2i = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u2i = 0 to the relations of the corresponding generalized braid group.

References

• Fomin, Sergey; Stanley, Richard P. (1994), "Schubert polynomials and the nil-Coxeter algebra", Advances in Mathematics 103 (2): 196–207, doi:10.1006/aima.1994.1009, ISSN 0001-8708