def WeylDim(ct, coeffs):
"""
The Weyl Dimension Formula.
INPUT:
- ``type`` - a Cartan type
- ``coeffs`` - a list of nonnegative integers
The length of the list must equal the rank type[1]. A dominant
weight hwv is constructed by summing the fundamental weights with
coefficients from this list. The dimension of the irreducible
representation of the semisimple complex Lie algebra with highest
weight vector hwv is returned.
EXAMPLES:
For `SO(7)`, the Cartan type is `B_3`, so::
sage: WeylDim(['B',3],[1,0,0]) # standard representation of SO(7)
7
sage: WeylDim(['B',3],[0,1,0]) # exterior square
21
sage: WeylDim(['B',3],[0,0,1]) # spin representation of spin(7)
8
sage: WeylDim(['B',3],[1,0,1]) # sum of the first and third fundamental weights
48
sage: [WeylDim(['F',4],x) for x in [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
[52, 1274, 273, 26]
sage: [WeylDim(['E', 6], x) for x in [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [2, 0, 0, 0, 0, 0]]
[1, 78, 27, 351, 351, 351, 27, 650, 351]
"""
ct = CartanType(ct)
lattice = RootSystem(ct).ambient_space()
rank = ct.rank()
fw = lattice.fundamental_weights()
hwv = lattice.sum(coeffs[i] * fw[i + 1]
for i in range(min(rank, len(coeffs))))
return lattice.weyl_dimension(hwv)
def WeylDim(ct, coeffs):
"""
The Weyl Dimension Formula.
INPUT:
- ``type`` - a Cartan type
- ``coeffs`` - a list of nonnegative integers
The length of the list must equal the rank type[1]. A dominant
weight hwv is constructed by summing the fundamental weights with
coefficients from this list. The dimension of the irreducible
representation of the semisimple complex Lie algebra with highest
weight vector hwv is returned.
EXAMPLES:
For `SO(7)`, the Cartan type is `B_3`, so::
sage: WeylDim(['B',3],[1,0,0]) # standard representation of SO(7)
7
sage: WeylDim(['B',3],[0,1,0]) # exterior square
21
sage: WeylDim(['B',3],[0,0,1]) # spin representation of spin(7)
8
sage: WeylDim(['B',3],[1,0,1]) # sum of the first and third fundamental weights
48
sage: [WeylDim(['F',4],x) for x in [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
[52, 1274, 273, 26]
sage: [WeylDim(['E', 6], x) for x in [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [2, 0, 0, 0, 0, 0]]
[1, 78, 27, 351, 351, 351, 27, 650, 351]
"""
ct = CartanType(ct)
lattice = RootSystem(ct).ambient_space()
rank = ct.rank()
fw = lattice.fundamental_weights()
hwv = lattice.sum(coeffs[i]*fw[i+1] for i in range(min(rank, len(coeffs))))
return lattice.weyl_dimension(hwv)
