Free resolutions of orbit closures
for representations with finitely many orbits
Federico Galetto
Northeastern University
Boston, MA
Combinatorial Algebra meets Algebraic Combinatorics
Université du Québec à Montréal
January 20, 2012
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
1 / 20
The case of determinantal varieties
GLm ˆ GLn ý HompCm , Cn q by row and column operations.
There are finitely many orbits Or “ tmatrices of rank ru for
0 ď r ď minpm, nq.
Or “ tmatrices of rank ď ru is the determinantal variety
defined by the vanishing of minors of order r ` 1.
Or is normal, Cohen-Macaulay and has rational singularities.
The minimal free resolution of CrOr s was calculated by
Lascoux.
We generalize to the setting of a reductive group acting on an
irreducible representation with finitely many orbits.
These representations, classified by Kac, are called of type II and
correspond to Dynkin diagrams with a distinguished node.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
2 / 20
Grading on a Lie algebra
Xn Dynkin diagram
x distinguished node of Xn
Definition (Degree of a root in pXn , xq)
Write the root β as a linear combination of simple roots αi :
β “ c1 α1 ` . . . ` ck´1 αk´1 ` ck αk ` ck`1 αk`1 ` . . . ` cn αn .
If αk is the simple root corresponding to x, deg β “ ck .
If g is the simple Lie algebra corresponding to Xn , then
à
g“
gi
iPZ
where gi is the direct sum of the root spaces with roots of degree i.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
3 / 20
Grading on a Lie algebra
Example (Am`n´1 , m)
1
m´1
m
m`1
m`n´1
εi ´ εj “ pεi ´ εi`1 q ` . . . ` pεm ´ εm`1 q ` . . . ` pεj´1 ´ εj q
$
’
iămăj
&1,
degpεi ´ εj q “ ´1, i ą m ą j
’
%
0,
otherwise
m
slm`n “ g´1 ‘ g0 ‘ g1 “
n
¨
m
n
g0
g1
˚
˚
˝ g´1
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
g0
˛
‹
‹
‚
Northeastern University
4 / 20
Representations with finitely many orbits
Definition (Representation corresponding to pXn , xq)
G 0 ý g1
with G0 the adjoint group of g0 (its Dynkin diagram is Xn zx).
Example (Am`n´1 , m)
1
m´1
m
m`1
m`n´1
G0 “ C˚ ˆ SLm pCq ˆ SLn pCq
g1 “ HompCm , Cn q
The irreducible representations of reductive groups with finitely
many orbits correspond to the pairs pXn , xq.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
5 / 20
Classifying the orbit closures
To list all the orbits of the action we have:
Theorem (Vinberg)
The orbits of G0 ý g1 correspond to some graded subalgebras of g.
A wish list for the orbit closures:
Degeneration partial order (O1 ď O2 ðñ O1 Ď O2 )
Normal? CM? Gorenstein? Rational singularities?
Defining equations
Singular locus
Minimal free resolutions
For the classical types (An , Bn , Cn , Dn ), Lovett showed that the
defining ideals of the orbit closures are given by rank conditions.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
6 / 20
pE7 , 2q: the representation
Example (E7 , 2)
1
3
4
5
6
7
2
g “ g´2 ‘ g´1 ‘ g0 ‘ g1 ‘ g2
G0 “ C˚ ˆ SL7 pCq
Ź
g1 “ 3 C7
The action G0 ý g1 has ten orbits:
O0 , O1 , O2 , O3 , O4 , O5 , O6 , O7 , O8 , O9
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
7 / 20
The geometric technique
AN ˆ G0 {P
Z
q
q1
O
Z desingularization of O
ξ vector bundle on G0 {P
AN
A “ CrAN s
¯
à j´
Ź
Fi :“
H G0 {P, i`j ξ bC Ap´i ´ jq
jě0
Theorem (Weyman)
There exist G0 -equivariant differentials of degree 0
di : Fi Ñ Fi´1
such that F‚ is a complex graded A-modules.
Under some additional hypotheses, F‚ is a free resolution of
CrN pOqs as a graded A-module.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
8 / 20
The expected resolutions (Kraśkiewicz & Weyman)
Fi :“
à
´
¯
Ź
H j G0 {P, i`j ξ bC Ap´i ´ jq
jě0
If ξ is semisimple:
§
§
compute the cohomology groups directly
get the minimal free resolution of CrN pOqs
Otherwise (most cases):
§
§
§
§
compute the equivariant Euler characteristic of F‚
get the Hilbert polynomial of CrN pOqs
get the expected minimal free resolution of CrN pOqs as a
complex of G0 -modules and maps
construct the resolution explicitly using M2
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
9 / 20
pE7 , 2q: the expected resolution of O7
The Hilbert polynomial:
1 ` 4t ` 10t2 ` 20t3 ` 35t4 ` 56t5 ` 49t6 ` 28t7 ` 7t8
The expected resolution:
Sp07 q C7 b A
Sp34 ,23 q C7 b Ap´6q
Sp4,35 ,2q C7 b Ap´7q
Sp52 ,45 q C7 b Ap´10q
Sp6,56 q C7 b Ap´12q
Ź
where A “ Symp 3 C7 q “ Crxijk | 1 ď i ă j ă k ď 7s.
0
Forgetting the G0 structure:
A
Ap´6q35
Ap´7q48
Ap´10q21
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Ap´12q7
0
Northeastern University
10 / 20
Constructing the complex
Write a differential di explicitly
Resolve coker di and coker d|i
Splice the resulting complexes
coker di
0
Fi´1
H‚
˚
Fi´1
H‚˚
Fi´1
di
d|i
di
Fi
T‚
Fi˚
coker d|i
Fi
T‚
0
0
Resolve using the commands
res
syz (step by step)
res/syz with DegreeLimit option
res with SyzygyLimit option
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
11 / 20
pE7 , 2q: the differential for O7
d2 : Sp4,35 ,2q C7 b Ap´7q ÝÑ Sp34 ,23 q C7 b Ap´6q
¨
0
˚ 0
˚
˚ 0
˚
˚ x167
˚
˚ 0
˚ 0
˚
˚ 0
˚
˚´x
˚
157
˚ 0
˚
˚ 0
˚
˚ x
˚ 147
˚ 0
˚
˚´x137
˚
˚ 0
˚
˚ 0
˚
˚ 0
˚
˚ 0
˚
˚ x156
˚
˚ 0
˚
˚ 0
˚
˚´x146
˚
˚ 0
˚
˚ x136
˚ 0
˚
˝ 0
...
0
0
0
0
x267
0
0
0
´x257
0
0
x247
0
´x237
0
0
0
0
x256
0
0
´x246
0
x236
0
...
0
0
x167
0
0
0
´x157
0
0
x147
0
0
´x127
0
0
0
x156
0
0
´x146
0
0
x126
0
0
...
0
0
0
0
x367
0
0
0
´x357
0
0
x347
0
0
´x237
0
0
0
x356
0
0
´x346
0
0
x236
...
0
0
x267
0
0
0
´x257
0
0
x247
0
0
0
´x127
0
0
x256
0
0
´x246
0
0
0
x126
0
...
0
0
0
x367
0
0
0
´x357
0
0
x347
0
0
0
´x137
0
0
x356
0
0
´x346
0
0
0
x136
...
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
0
x167
0
0
0
´x157
0
0
0
x137
´x127
0
0
0
0
x156
0
0
0
´x136
x126
0
0
0
0
...
0
0
0
0
x467
0
0
0
´x457
0
0
0
0
x347
´x247
0
0
0
x456
0
0
0
0
´x346
x246
...
0
x267
0
0
0
´x257
0
0
0
x237
0
´x127
0
0
0
x256
0
0
0
´x236
0
x126
0
0
0
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
Northeastern University
12 / 20
Proving exactness
Theorem (Buchsbaum–Eisenbud)
Let A be a ring and
F‚ : F0
d1
F1
...
Fn´1
dn
Fn
0
a complex of free A-modules. F‚ is exact ðñ @k “ 1, . . . , n
1
rk Fk “ rk dk ` rk dk`1 ;
2
depth Ipdk q ě k where Ipdk q is the ideal of A generated by
maximal non vanishing minors of dk .
Since our complex is G0 -equivariant:
1
need only be checked at the representative of the generic orbit;
2
can be verified by comparing k with the codimension of the
orbits where the rank of dk drops.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
13 / 20
pE7 , 2q: the resolution of CrO7 s
A
d1
Ap´6q35
#
0
1
2
3
4
5
6
7
8
9
dim
0
13
20
21
25
26
28
31
34
35
d2
Ap´7q48
rkpd1 q
0
0
0
0
0
0
0
0
1
1
d3
Ap´10q21
rkpd2 q
0
13
20
21
25
26
28
31
34
34
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
d4
rkpd3 q
0
0
0
6
3
6
6
11
14
14
Ap´12q7
0
rkpd4 q
0
0
1
1
3
6
4
6
7
7
Northeastern University
14 / 20
pE7 , 2q: the defining equations of O7
x2345 x236 x146 x2127 ´ x2345 x136 x246 x2127 ` x245 x345 x136 x346 x2127 ´
x145 x345 x236 x346 x2127 ´ x235 x345 x146 x346 x2127 `
x135 x345 x246 x346 x2127 ` x235 x145 x2346 x2127 ´ x135 x245 x2346 x2127 `
x234 x345 x146 x356 x2127 ´ x134 x345 x246 x356 x2127 ´
x234 x145 x346 x356 x2127 ` x134 x245 x346 x356 x2127 ´
x234 x345 x136 x456 x2127 ` x134 x345 x236 x456 x2127 `
x234 x135 x346 x456 x2127 ´ x134 x235 x346 x456 x2127 ´
2x245 x345 x236 x146 x127 x137 ` x2345 x126 x246 x127 x137 `
x245 x345 x136 x246 x127 x137 ` x145 x345 x236 x246 x127 x137 `
x235 x345 x146 x246 x127 x137 ´ x135 x345 x2246 x127 x137 ´
x245 x345 x126 x346 x127 x137 ´ x2245 x136 x346 x127 x137 `
x145 x245 x236 x346 x127 x137 ` x235 x245 x146 x346 x127 x137 ´
2x235 x145 x246 x346 x127 x137 ` x135 x245 x246 x346 x127 x137 ´
x125 x345 x246 x346 x127 x137 ` x125 x245 x2346 x127 x137 ´
x234 x345 x146 x256 x127 x137 ` x134 x345 x246 x256 x127 x137 ` . . .
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
15 / 20
Proving the rings are reduced
Our methods produce a minimal free resolution
0
A{I
F‚
0
for some ideal I with zero set O. To ensure A{I – CrOs we need
to show I is radical or equivalently A{I is reduced.
Proposition (Serre)
A Noetherian ring R is reduced if and only if it satisfies:
R0: the localization of R at each prime of height 0 is regular;
S1: all primes associated to zero have height 0.
A{I satisfies S1 because O is irreducible;
if O is non singular at its generic point, then A{I satisfies R0.
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
16 / 20
Resolution of non normal orbit closures
0
C‚´1
F‚
CrOs
CrN pOqs
π̃
G‚
π
C
0
F‚ from geometric technique/interactive method
G‚ from F‚ by dropping rows/columns of the first differential
lift π to π̃ in M2 using extend or // (step by step)
C‚ “ conepπ̃q and H1 pC‚ q – CrOs
minimize and resolve H1 pC‚ q in M2
Since we only need H1 pC‚ q, it is enough to build a truncated cone
C‚t : C0
C1
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
C2
Northeastern University
17 / 20
pE6 , 3q: the representation
Example (E6 , 3)
1
3
4
5
6
2
g “ g´2 ‘ g´1 ‘ g0 ‘ g1 ‘ g2
G0 “ C˚ ˆ SL2 pCq ˆ SL5 pCq
Ź
g1 “ C 2 b 2 C 5
The action G0 ý g1 has eight orbits:
O0 , O1 , O2 , O3 , O4 , O5 , O6 , O7
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
18 / 20
pE6 , 3q: the resolution of CrO6 s
A “ SympC2 b
Ź2
C5 q “ Crxi;jk | i “ 1, 2; 1 ď j ă k ď 5s
resolution of CrN pO6 qs
A ‘ Ap´2q5
Ap´3q10
Ap´5q4
0
resolution of the cokernel Cp6q
A5
Ap´1q10
Ap´3q4 ‘ Ap´4q10
Ap´5q10
Ap´8q
0
resolution of CrO6 s
A
Ap´6q10
Ap´7q10
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Ap´10q
0
Northeastern University
19 / 20
pE6 , 3q: orbit closures and singularities
O7
O6
O5
O4
O3
O2
O1
O0
O1
O2
O3
O4
O5
O6
O1
12
O2
9
O3
8
O4
5
O5
4
O6
2
0
12
0
9
9
0
6
0
0
4
2
5
0
0
4
2
4
4
0
0
0
0
0
0
2
8
O0
Federico Galetto
Free resolutions of orbit closures for representations with finitely many orbits
Northeastern University
20 / 20