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

[math]\displaystyle{ \begin{align} u_i^2 & = 0, \\ u_i u_j & = u_j u_i & & \text{ if } |i-j| \gt 1, \\ u_i u_j u_i & = u_j u_i u_j & & \text{ if } |i-j|=1. \end{align} }[/math]

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