nil-Coxeter algebra
In mathematics, the nil-Coxeter algebra, introduced by {{harvtxt|Fomin|Stanley|1994}}, 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{align}
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{align}
These are just the relations for the infinite braid group, together with the relations u{{su|b=i|p=2}} = 0. Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u{{su|b=i|p=2}} = 0 to the relations of the corresponding generalized braid group.
References
- {{Citation | last1=Fomin | first1=Sergey | authorlink1=Sergey Fomin | last2=Stanley | first2=Richard P. | authorlink2=Richard P. Stanley | title=Schubert polynomials and the nil-Coxeter algebra | doi=10.1006/aima.1994.1009 | doi-access=free | mr=1265793 | year=1994 | journal=Advances in Mathematics | issn=0001-8708 | volume=103 | issue=2 | pages=196–207}}