292
M. A. Walton
operators. We report only preliminary new results, formulas for all rank-2 cases, and
for one of rank 3, related to the Lie algebra A 3 .
But before treating ranks 2 and 3, let us first dispense with the unique rank-1
algebra, A 1 . In this case, the character and weight-polytope lattice sum are identical,
ch λ = B λ = e
λ
+ e
λ−α 1 + e
λ−2α 1 + . . . + e
−λ .
(17)
The Demazure character formulas therefore apply to B λ .
The A 1 Weyl group W has 2 elements, the identity and r 1 = r(α 1 ), where α 1
is the simple root, and only positive root. The longest element of W is therefore
w L = r 1 = r(α 1 ), with a unique reduced decomposition. Applying the Demazure
formulas (12) and (13), we find
B λ = D 1 e
λ
=
d(α 1 ) + 1
e
λ .
(18)
The last expression will turn out to be the most relevant here—see (23) and (25)
below.
The 3 rank-2 algebras can be treated in a unified way. Put the p := dim R +
positive roots of your rank-2 algebra in angular order; label them γ j . So we get, for
A 2 ,
{γ 1 , γ 2 , γ p=3 } = {α 1 , α 1 + α 2 , α 2 } ;
(19)
for B 2 ( ∼ = C 2 ),
{γ 1 , . . . , γ p=4 } = {α 1 , α 1 + α 2 , α 1 + 2α 2 , α 2 } ;
(20)
and for G 2 ,
{γ 1 , . . . , γ p=6 } = {α 1 , α 1 + α 2 , 2α 1 + 3α 2 , α 1 + 2α 2 , α 1 + 3α 2 , α 2 } .
(21)
In a generic weight diagram, these positive roots are parallel to half of the
boundaries, in angular order. They specify a path from the highest weight to the
lowest weight along the polytope edges labeled by γ 1 through γ p , in that order.
Correspondingly, the longest element w L of the Weyl group can be written as a
product of the reflections defined in (15):
w L = r(γ p )r(γ p−1 ) · · · r(γ 2 )r(γ 1 ) .
(22)
The weights of the first p − 1 boundaries can be generated, and then the polytope
can be filled in by γ p -strings of weights. If B λ = B (e λ ), then
B =
d(γ p ) + 1
d(γ p−1 )r(γ p−2 ) · · · r(γ 1 )
+ d(γ p−2 )r(γ p−3 ) · · · r(γ 1 ) + · · · + d(γ 2 )r(γ 1 ) + d(γ 1 ) + 1
.
(23)
M. A. Walton
operators. We report only preliminary new results, formulas for all rank-2 cases, and
for one of rank 3, related to the Lie algebra A 3 .
But before treating ranks 2 and 3, let us first dispense with the unique rank-1
algebra, A 1 . In this case, the character and weight-polytope lattice sum are identical,
ch λ = B λ = e
λ
+ e
λ−α 1 + e
λ−2α 1 + . . . + e
−λ .
(17)
The Demazure character formulas therefore apply to B λ .
The A 1 Weyl group W has 2 elements, the identity and r 1 = r(α 1 ), where α 1
is the simple root, and only positive root. The longest element of W is therefore
w L = r 1 = r(α 1 ), with a unique reduced decomposition. Applying the Demazure
formulas (12) and (13), we find
B λ = D 1 e
λ
=
d(α 1 ) + 1
e
λ .
(18)
The last expression will turn out to be the most relevant here—see (23) and (25)
below.
The 3 rank-2 algebras can be treated in a unified way. Put the p := dim R +
positive roots of your rank-2 algebra in angular order; label them γ j . So we get, for
A 2 ,
{γ 1 , γ 2 , γ p=3 } = {α 1 , α 1 + α 2 , α 2 } ;
(19)
for B 2 ( ∼ = C 2 ),
{γ 1 , . . . , γ p=4 } = {α 1 , α 1 + α 2 , α 1 + 2α 2 , α 2 } ;
(20)
and for G 2 ,
{γ 1 , . . . , γ p=6 } = {α 1 , α 1 + α 2 , 2α 1 + 3α 2 , α 1 + 2α 2 , α 1 + 3α 2 , α 2 } .
(21)
In a generic weight diagram, these positive roots are parallel to half of the
boundaries, in angular order. They specify a path from the highest weight to the
lowest weight along the polytope edges labeled by γ 1 through γ p , in that order.
Correspondingly, the longest element w L of the Weyl group can be written as a
product of the reflections defined in (15):
w L = r(γ p )r(γ p−1 ) · · · r(γ 2 )r(γ 1 ) .
(22)
The weights of the first p − 1 boundaries can be generated, and then the polytope
can be filled in by γ p -strings of weights. If B λ = B (e λ ), then
B =
d(γ p ) + 1
d(γ p−1 )r(γ p−2 ) · · · r(γ 1 )
+ d(γ p−2 )r(γ p−3 ) · · · r(γ 1 ) + · · · + d(γ 2 )r(γ 1 ) + d(γ 1 ) + 1
.
(23)