# coxeterPolynomial -- computes the Coxeter polynomial of a poset

## Synopsis

• Usage:
z = coxeterPolynomial P
z = coxeterPolynomial(P, VariableName => symbol)
• Inputs:
• P, an instance of the type Poset,
• Optional inputs:
• VariableName => , default value t
• Outputs:
• z, , the Coxeter polynomial of $P$

## Description

The Coxeter polynomial of $P$ is the characteristic polynomial of the Coxeter transformation matrix $-M M^{-t}$, where $M$ is the relation matrix. This depends only on the derived category of modules over the incidence algebra.

 i1 : B = booleanLattice 3; i2 : z = coxeterPolynomial B 8 7 6 5 4 3 2 o2 = t + t + t - 2t - 2t - 2t + t + t + 1 o2 : ZZ[t]

## Ways to use coxeterPolynomial :

• "coxeterPolynomial(Poset)"

## For the programmer

The object coxeterPolynomial is .