In mathematics, the nil-Coxeter algebra, introduced by , is an algebra similar to the group algebra of a

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

except that the generators are

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the cla ...

Definition

The nil-Coxeter algebra for the infinite

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

is the algebra generated by ''u''₁, ''u''₂, ''u''₃, ... with the relations :

\begin
u_i^2 & = 0, \\
u_i u_j & = u_j u_i & & \text , i-j,  > 1, \\
u_i u_j u_i & = u_j u_i u_j & & \text , i-j, =1.
\end

These are just the relations for the infinite

braid group A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-strande ...

, together with the relations ''u'' = 0. Similarly one can define a nil-Coxeter algebra for any

Coxeter system In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

, by adding the relations ''u'' = 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 Representation theory