EXAMPLE: TYPE
1
(1, 7, 5)
12
SEUNG-JO JUNG
Abstract. In this article, as a concrete example, we calculate the
set of G-iraffes using round down functions and the admissible set
1
(1, 7, 5).
of simple roots ∆a for the group G of type 12
Example: type
1
(1, 7, 5)
12
1
Let G be the finite group of type 12
(1, 7, 5) with eigencoordinates
1
3
x, y, z and L the lattice L = Z + Z · 12
(1, 7, 5). Let X denote the
3
quotient variety C /G and Y the economic resolution of X. The toric
fan Σ of Y is shown in Figure 0.1.
To use the recursion process, we need to investigate the cases of type
and of type 51 (1, 2, 3). Let G2 be the group of type 17 (1, 2, 5)
with eigencoordinates ξ2 , η2 , ζ2 and G3 be the group of type 15 (1, 2, 3)
with eigencoordinates ξ3 , η3 , ζ3 . Consider the toric fans Σ2 and Σ3 of
the economic resolutions for the type 71 (1, 2, 5) and the type 15 (1, 2, 3),
respectively.
1
(1, 2, 5)
7
0.1. G-iraffes. We now calculate G-iraffes corresponding to two full
dimensional cones in Σ:
1
1
1
σ4 = Cone 12
(12, 0, 0), 12
(3, 9, 3), 12
(8, 8, 4) ,
1
1
1
τ3 = Cone 12
(1, 7, 5), 12
(3, 9, 3), 12
(8, 8, 4) .
Note that the cones σ4 , τ3 are on the right side of the lowest vector v =
1
(1, 7, 5). Their corresponding cones σ40 , τ30 in Σ3 to σ4 , τ3 , respectively
12
are
(0.1)
σ40 = Cone 51 (5, 0, 0), 15 (1, 2, 3), 15 (1, 1, 4) ,
0
1
1
1
(0.2)
τ3 = Cone 5 (0, 0, 5), 5 (1, 2, 3), 5 (1, 1, 4) .
Observe that the cones σ40 , τ30 are on the left side of Σ3 . To use the
recursion, let G32 be the group of type 21 (1, 1, 1) with eigencoordinates
ξ32 , η32 , ζ32 . Let Σ32 denote the fan of the economic resolution of the
Date: 28th June 2014.
1
2
S.-J. JUNG
↑ e1
v5
σ12
σ11
σ10
σ9
σ8
σ7
σ6
σ5
v10
σ3
σ4
σ2
σ1
v3
v8
τ6
v1
τ2
v6
τ7
τ3
τ9
v4
v11
τ5
τ1
v9
τ8
v2
τ4
τ10
τ0
v7
e3
e2
Figure 0.1. Toric fan of the economic resolution for
↑ e1
↑ e1
σ400
τ30
e2
1
(1, 1, 1)
2
↑ e1
σ40
τ300
e3
1
(1, 7, 5)
12
e3
e2
e3
1
(1, 2, 3)
5
Figure 0.2. Recursion process for
e2
1
(1, 7, 5)
12
1
(1, 7, 5)
12
quotient C2 /G32 . In Σ32 , there exist two cones σ400 , τ300 corresponding to
EXAMPLE: TYPE
1
(1, 7, 5)
12
3
σ40 , τ30 , respectively:
σ400
τ300
= Cone
1
(2, 0, 0), 12 (0, 2, 0), 12 (1, 1, 1)
2
,
= Cone 12 (0, 0, 2), 12 (0, 2, 0), 12 (1, 1, 1) .
As is in Example ??, the G32 -graphs Γ004 and Γ003 corresponding to
σ400 , τ300 are
n
o
00
Γ4 = 1, ζ23 ,
n
o
Γ003 = 1, ξ23 .
Using the left round down function φ32 for 51 (1, 2, 3)
b a+2b+3c
c c
5
ζ32 ,
a
φ32 : ξ3a η3b ζ3c 7→ ξ32
η32
we can see that the corresponding G3 -graphs Γ04 and Γ03 corresponding
to σ40 , τ30 are
n
o
def
ζ3
00
2
Γ04 = φ−1
(Γ
)
=
1,
η
,
η
,
ζ
,
3
3 η3 ,
32
4
3
o
n
def
2
00
,
ξ
,
ξ
η
.
Γ03 = φ−1
)
=
1,
η
,
η
(Γ
3
3
3
3
32
3
2
To get the corresponding G-iraffes Γ4 , Γ3 to σ4 , τ3 , respectively, we use
1
the right round function φ3 for 12
(1, 7, 5):
c
b a+7b+5c
12
φ3 : xa y b z c 7→ ξ3a η3b ζ3
.
We get
Γ4
Γ3
o
n
y y2 y2
2 3 4 z4 z5 z6
=
= 1, y, z , z , z2 , z, z , z , z , y , y , y ,
n
o
def −1
y y2 y2
2
= φ3 (Γ02 ) = 1, x, xz, xz 2 , xy, xy
.
,
y,
,
,
,
z,
z
z
z z z2
def
0
φ−1
3 (Γ4 )
Let us consider the following two cones:
1
1
1
σ9 = Cone 12 (12, 0, 0), 12 (9, 3, 9), 12 (4, 4, 8) ,
1
1
1
τ7 = Cone 12
(2, 2, 10), 12
(9, 3, 9), 12
(4, 4, 8) .
Observe that the cones σ9 , τ7 are on the left side of v. The corresponding cones σ90 , τ70 in Σ2 to σ9 , τ7 , respectively are
0
1
1
1
σ9 = Cone 7 (12, 0, 0), 7 (5, 3, 4), 7 (2, 4, 3) ,
τ70 = Cone 17 (1, 2, 5), 71 (5, 3, 4), 71 (2, 4, 3) .
Note that the cones σ90 , τ70 are on the right side of the fan Σ and that
the right side is equal to the fan Σ3 of the economic resolution for
1
(1, 2, 3). Moreover, the cones in Σ3 corresponding to σ90 , τ70 are σ40 , τ20 ,
5
4
S.-J. JUNG
respectively in (0.1). Thus we have the corresponding G23 -graphs Γ009 ,
Γ007 are:
n
o
2
, ζ23 , ηζ23
Γ009 = 1, η23 , η23
,
23
n
o
2
Γ007 = 1, ξ23 , ξ23 η23 , η23 , η23
,
where G23 is the group of type 51 (1, 2, 3) with eigencoordinates ξ23 , η23 , ζ23 .
Using the right round down function φ23 for 17 (1, 2, 5)
c
b a+2b+5c
7
a b
φ23 : ξ2a η2b ζ2c 7→ ξ23
η23 ζ23
,
we can calculate the corresponding G2 -graphs to σ90 , τ70 :
n
o
2
3
def
2 ζ2 ζ2
2
00
,
,
ζ
,
ζ
)
=
1,
η
,
η
Γ09 = φ−1
(Γ
,
2
2
23
2 η2 η2 ,
2
9
n
o
def
00
2
2
Γ07 = φ−1
)
=
1,
ξ
,
ξ
η
,
ξ
ζ
,
η
,
η
,
ζ
,
ζ
(Γ
2
2 2
2 2
2
2
23
7
2
2 .
Lastly, from the left round down function φ2 for
1
(1, 7, 5)
12
b a+7b+5c
c c
12
ζ2 ,
φ2 : xa y b z c 7→ ξ2a η2
it follows that the G-iraffes Γ9 , Γ7 corresponding to σ9 , τ7 are:
o
n
2 3 4 5
2 z2 z2 z2 z3
Γ9 = 1, y, y , y , y , y , z, z , y , y2 , y3 , y3 ,
n
o
Γ7 = 1, x, xy, xy 2 , xy 3 , xz, y, y 2 , y 3 , y 4 , y 5 , z .
1
For each 0 ≤ i ≤ 12, let vi denote the lattice point 12
(7i, i, 12 − i) in
L. For the cones σ in Figure 0.1 in page 2, Table 0.1 in page 6 shows
the corresponding G-iraffe Γσ .
0.2. Admissible set of simple roots. Now we calculate the admis1
sible set of simple roots for 12
(1, 7, 5). Since for the group of type
1
(1,
r
−
1,
1),
the
economic
resolution
is G-Hilb, note that the admissir
1
ble sets of simple roots for 2 (1, 1, 1) and 13 (1, 2, 1) are {ε1 − ε0 }, {ε1 −
ε2 , ε2 −ε0 }, respectively. By the identification (??), that the admissible
sets of simple roots for 15 (1, 2, 3) is
{ε3 − ε4 , ε4 − ε1 , ε1 − ε2 , ε2 − ε0 },
where the underlined root is the added root as in (??). Similarly, the
admissible sets of simple roots for 71 (1, 2, 5) is
{ε5 − ε6 , ε6 − ε3 , ε3 − ε4 , ε4 − ε1 , ε1 − ε2 , ε2 − ε0 }.
1
Lastly, the admissible set of simple roots for 12
(1, 7, 5) is
ε5 − ε6 , ε6 − ε10 , ε10 − ε11 , ε11 − ε8 , ε8 − ε9 , ε9 − ε7 ,
.
ε7 − ε3 , ε3 − ε4 , ε4 − ε1 , ε1 − ε2 , ε2 − ε0
EXAMPLE: TYPE
1
(1, 7, 5)
12
5
Note that corresponding permutation ω is
0 7 2 9 4 11 6 1 8 3 10 5
ω=
.
0 2 1 4 3 7 9 8 11 10 6 5
With the dual basis {θi } with respect to {αi }, the row vectors of the
following matrix is the rays of the admissible Weyl chamber Ca :


−1
0
0
0
0 0 0 1 0 0 0 0
 −1
0 −1
0
0 0 0 1 0 1 0 0 


 −1 −1 −1
0
0 0 0 1 1 1 0 0 


 −1 −1 −1
0 −1 0 0 1 1 1 0 1 


 −1 −1 −1 −1 −1 0 0 1 1 1 1 1 


0 −1 −1 0 0 0 1 1 1 1  .
 −1 −1


0 −1
0 0 0 0 1 0 1 1 
 −1 −1
 −1 −1
0
0
0 0 0 0 0 0 1 1 



 −1 −1
0
0
0
0
1
0
0
0
1
0


 −1 −1
0
0
0 1 1 0 0 0 0 0 
−1
0
0
0
0 1 0 0 0 0 0 0
Mathematics Institute, University of Warwick,
Coventry, CV4 7AL, England
E-mail: [email protected]
6
S.-J. JUNG
Cone Generators
Coordinates
on Uσ
G-iraffes Γσ
σ1
e1 , e2 , v11
1, z, z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 , z 9 , z 10 , z 11
x
, y , z 12
z 5 z 11
σ2
e1 , v10 , v11
1, y, yz , z, z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 , z 9
x y 2 z 11
, ,
z 5 z 10 y
σ3
e1 , v9 , v10
1, y, yz , yz , yz2 , yz3 , yz4 , yz5 , z, z 2 , z 3 , z 4
σ4
e1 , v8 , v9
1, y, yz , yz , yz2 , z, z 2 , z 3 , z 4 , zy , zy , zy
σ5
e1 , v7 , v8
1, y, yz , yz , yz2 , yz2 , yz3 , yz4 , yz4 , yz5 , z, z 2
σ6
e1 , v6 , v7
1, y, z, z 2 , zy , zy , yz 2 , yz 2 , yz 2 , yz 3 , yz 3 , yz 4
σ7
e1 , v5 , v6
1, y, y 2 , y 3 , z, z 2 , zy , zy , yz 2 , yz 2 , yz 2 , yz 3
σ8
e1 , v4 , v5
1, y, y 2 , y 3 , y 4 , y 5 , yz , yz2 , yz2 , z, z 2 , zy
σ9
e1 , v3 , v4
1, y, y 2 , y 3 , y 4 , y 5 , z, z 2 , zy , yz 2 , yz 3 , yz 3
σ10
2
2
2
2
2
2
2
2
5
4
3
3
xz 5 y 3 z 10
, ,
y2 z9 y2
2
3
4
xy y 4 z 9
, ,
z4 z8 y3
6
xz 4 y 5 z 8
, ,
y3 z7 y4
4
3
4
5
5
6
6
xy 2 y 6 z 7
, ,
z3 z6 y5
2
3
3
4
5
5
xy 2 y 7 z 6
, ,
z3 z5 y6
2
xz 2 y 8 z 5
, ,
y5 z4 y7
3
xy 3 y 9 z 4
, ,
z2 z3 y8
e1 , v2 , v3
1, y, y 2 , y 3 , y 4 , y 5 , y 6 , yz , yz , yz , yz , z
xz y 10 z 3
, ,
y6 z2 y9
σ11
e1 , v1 , v2
1, y, y 2 , y 3 , y 4 , y 5 , y 6 , z, yz , yz2 , yz3 , yz4
xy 4 y 11 z 2
, z1 , y10
z
σ12
e1 , e3 , v1
1, y, y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 , y 10 , y 11
x
, y 12 , yz11
y7
τ1
e2 , v9 , v11
1, x, xz, xz 2 , xz 3 , xz 4 , x2 , x2 z, z, z 2 , z 3 , z 4
5
x3
, y ,z
z 3 x2 z x
τ2
v9 , v10 , v11
1, x, z, xz, z 2 , xz 2 , z 3 , xz 3 , z 4 , xz 4 , y, yz
τ3
v7 , v8 , v9
1, x, xy, xy
, xz, xz 2 , y, yz , yz , yz2 , z, z 2
z
τ4
e2 , v7 , v9
1, x, x2 , x3 , x4 , xz, xz 2 , x2 z, x3 z, x4 z, z, z 2
τ5
v4 , v6 , v7
1, x, xy, xz, xz 2 , xzy , x2 , x2 y, y, z, z 2 , zy
τ6
v4 , v5 , v6
1, x, xy, xz, xz 2 , xzy , y, y 2 , y 3 , z, z 2 , zy
3
x2 y 5
, , z
y 2 xz 2 xy 2
τ7
v2 , v3 , v4
1, x, xy, xy 2 , xy 3 , xz, y, y 2 , y 3 , y 4 , y 5 , z
x2 y 6 z 2
, ,
y 2 xz xy 3
τ8
v2 , v4 , v7
1, x, xy, xz, x2 , x2 y, x3 , x3 y, x4 , x4 y, y, z
x5 y 2 z 2
, ,
z x2 x3 y
τ9
e3 , v1 , v2
1, x, xy, xy 2 , xy 3 , xy 4 , y, y 2 , y 3 , y 4 , y 5 , y 6
x2 y 7
, , z
y 2 x xy 4
τ10
e3 , v2 , v7
1, x, xy, x2 , x2 y, x3 , x3 y, x4 , x4 y, x5 , x6 , y
x7 y 2 z
, ,
y x2 x5
τ0
e2 , e3 , v7
1, x, x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 , x10 , x11
x12 , xy7 , xz5
2
3
5
5
6
2
6
2
7
2
2
8
9
2
2
Table 0.1. G-iraffes for G =
x2 z y 3 z 4
, xz4 , xy
y
2
2
2
1
(1, 7, 5)
12
x2 z y 2 z 5
, xz5 , x
y
3
x5
, y ,z
z x2 z x3
x3 y y 2 z 3
, ,
z 2 x2 xy