Opened 14 years ago

Closed 14 years ago

# [with patch, positive review] G2 fundamental weights were the negative of what they should be.

Reported by: Owned by: bump bump major sage-3.0 combinatorics sage-combinat N/A

In combinat/root_system.py, the fundamental weights for the various root systems are entered by hand. For G2, the fundamental weights were the negatives of what they should be.

```diff -r 80b506b8e07c sage/combinat/root_system.py
--- a/sage/combinat/root_system.py Tue Apr 01 19:18:55 2008 -0700
+++ b/sage/combinat/root_system.py Sat Apr 05 08:40:46 2008 -0700
@@ -788,11 +788,11 @@ class AmbientLattice_g(AmbientLattice_ge
"""
EXAMPLES:
sage: CartanType(['G',2]).root_system().ambient_lattice().fundamental_weights()
-            [(-1, 0, 1), (-2, 1, 1)]
+            [(1, 0, -1), (2, -1, -1)]
"""
return [ c0*self._term(0)+c1*self._term(1)+c2*self._term(2) \
for [c0,c1,c2] in
-                 [[-1,0,1],[-2,1,1]]]
+                 [[1,0,-1],[2,-1,-1]]]

def WeylDim(type, coeffs):

```

### Changed 14 years ago by bump

patch correcting the G2 fundamental weights

### comment:1 Changed 14 years ago by bump

• Owner changed from mhansen to bump
• Status changed from new to assigned

### comment:2 Changed 14 years ago by mabshoff

• Summary changed from G2 fundamental weights were the negative of what they should be. to [with patch, needs review] G2 fundamental weights were the negative of what they should be.

### comment:3 Changed 14 years ago by cremona

• Summary changed from [with patch, needs review] G2 fundamental weights were the negative of what they should be. to [with patch, positive review] G2 fundamental weights were the negative of what they should be.

Quoting from the email to sage-devel:

```I should have added some justification for this conclusion
in the trac report. Instead I'm giving it here. You can
look the weights up in Bourbaki, Lie Groups and Lie
Algebras Ch 4-6 (Appendix) and you can also check
the inner products of the weights with the simple
roots (which are correct). The inner product of
the i-th fundamental weight with the j-th simple
root should be positive if i=j and zero otherwise.
I checked that all the other root systems are right
by examining the output following program on the ambient
lattices. This change had no effect on the output of
the Weyl dimension formula.

def test_lattice(L):
rank = L.ct[1]
roots = L.simple_roots()
weights = L.fundamental_weights()
return [[i,j, roots[i].inner_product(weights[j])] for i in range(rank) for j in range(rank)]
```

I am happy with this (small!) change.

### comment:4 Changed 14 years ago by mabshoff

• Resolution set to fixed
• Status changed from assigned to closed

Merged in Sage 3.0.alpha2.