qdim -- Compute principal specialization of character or quantum dimension

Synopsis

Usage:

qdim M

qdim(M,l)
Inputs:
- M, an instance of the type LieAlgebraModule,
- l, an integer,
Outputs:
- P, a ring element,

Description

qdim M computes the principal specialization of the character of M. qdim (M,l) evaluates it modulo the appropriate cyclotomic polynomial, so that upon specialization of the variable $q$ to be the corresponding root of unity of smallest positive argument, it provides the quantum dimension of M.

i1 : g=simpleLieAlgebra("A",2)

o1 = g

o1 : simple LieAlgebra

i2 : W=weylAlcove(g,3)

o2 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1},
     ------------------------------------------------------------------------
     {3, 0}}

o2 : List

i3 : L=LL_(1,1) (g)

o3 = L

o3 : irreducible LieAlgebraModule over g

i4 : M=matrix table(W,W,(v,w)->fusionCoefficient(L,LL_v g,LL_w g,3))

o4 = | 0 0 0 0 0 1 0 0 0 0 |
     | 0 1 0 0 0 0 1 1 0 0 |
     | 0 0 1 0 1 0 0 0 1 0 |
     | 0 0 0 0 0 1 0 0 0 0 |
     | 0 0 1 0 1 0 0 0 1 0 |
     | 1 0 0 1 0 2 0 0 0 1 |
     | 0 1 0 0 0 0 1 1 0 0 |
     | 0 1 0 0 0 0 1 1 0 0 |
     | 0 0 1 0 1 0 0 0 1 0 |
     | 0 0 0 0 0 1 0 0 0 0 |

              10       10
o4 : Matrix ZZ   <-- ZZ

i5 : first eigenvalues M

o5 = 3

o5 : CC (of precision 53)

i6 : qdim L

      4     2         -2    -4
o6 = q  + 2q  + 2 + 2q   + q

o6 : ZZ[q]

i7 : qdim (L,3)

o7 = 3

        ZZ[q]
o7 : -----------
      4    2
     q  - q  + 1

Ways to use qdim :

qdim(LieAlgebraModule)
qdim(LieAlgebraModule,ZZ)

For the programmer

The object qdim is a method function.