poincareSeries(RootSystem,Parabolic,RingElement) -- the generating series of a quotient of the Weyl group by length

Synopsis

• Function: poincareSeries

• Usage:

  poincareSeries(R,P,x)

• Inputs:
  • R, an instance of the type RootSystem,
  • P, an instance of the type Parabolic,
  • x, a ring element,
• Outputs:
  • a ring element, a polynomial in the variable x where the coefficient in front of x^i is the number of cosets in the Weyl group of R modulo the Weyl group of P where the minimal element is of length i.

Description

i1 : R=rootSystemA(3)

o1 = RootSystem{...8...}

o1 : RootSystem

i2 : P=parabolic(R,set{1,2})

o2 = set {1, 2}

o2 : Parabolic

i3 : ZZ[x]

o3 = ZZ[x]

o3 : PolynomialRing

i4 : poincareSeries(R,P,x)

      3    2
o4 = x  + x  + x + 1

o4 : ZZ[x]