Equivariant resolutions
of De Concini-Procesi ideals
Federico Galetto
AMS Fall Eastern Sectional Meeting 2014
Rn “ Crx1 , . . . , xn s;
µ partition of n;
Iµ Ď Rn (homogeneous) DCP ideal associated to µ.
(Garsia-Procesi 1992) Rn {Iµ is a finite dimensional graded
representation of Sn whose graded character is related to
Kostka-Foulkes polynomials.
(De Concini-Procesi 1981) Rn {Iµ is the cohomology of the
Springer fiber of type A associated to µ.
(Kraft 1981) SpecpRn {Iµ q is the scheme theoretic intersection of
diagonal matrices with nilpotent matrices of type µ.
(Geramita-Hoefel-Wehlau 2014) Rn {Iµ appears in the
computation of certain Hilbert series.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
Questions
Can we describe a minimal generating set of Iµ ?
What are the graded Betti numbers of Iµ ?
(Biagioli-Faridi-Rosas 2007) provides complete answers when µ is
a hook, i.e. µ “ pa, 1b q.
In general, a minimal generating set of Iµ can be quite
complicated.
Goals
Describe an Sn -stable minimal generating set of Iµ as an
Sn -representation to link “complexity” of these ideals to
representation theory.
Describe an Sn -equivariant minimal free resolution of Iµ .
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
Proposition (Biagioli-Faridi-Rosas)
For 1 ď d ď n and µ “ pn ´ d ` 1, 1d´1 q,
Iµ “ pe1 , . . . , ed´1 q ` pxi1 . . . xid q “ En,d ` In,d ;
te1 , . . . , ed´1 u Y txi1 . . . xid u is a minimal generating set of Iµ ;
e1 , . . . , ed´1 is a maximal regular sequence on Rn {In,d .
0
Rn {In,d
F‚
¨ei
α‚
Rn {In,d
Rn {pIn,d ` ei q
F‚
conepα‚ q
0
Facts
If F‚ is a minimal free resolution of Rn {In,d , then conepα‚ q is
a minimal free resolution Rn {pIn,d ` ei q.
Since ei is Sn -invariant, conepα‚ q is Sn -equivariant.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
In,d has a linear Sn -equivariant minimal free resolution:
0
In,d
F0
F1
...
Fn´d
0
so Fi – Uin,d bC Rn p´d ´ iq.
Question
How does Sn act on Uin,d ?
Examples
For d “ 1, F‚ is a Koszul complex, so
Ź
Uin,1 – i`1 Cn – pn ´ i ´ 1, 1i`1 q ‘ pn ´ i ` 2, 1i´1 q.
For d “ n ´ 1,
0
In,n´1
pnq ‘ pn ´ 1, 1q
pn ´ 1, 1q
Federico Galetto
0
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
We have two short exact sequences:
one of Rn -modules with an Sn´1 -action
0
In´1,d´1 {In´1,d
xi1 . . . xid´1
¨xn
Rn {In´1,d
Rn {In,d
0
xi1 . . . xid´1 xn
xi1 . . . xid
xi1 . . . xid
one of Rn´1 -modules with an Sn´1 -action
0
In´1,d
In´1,d´1
In´1,d´1 {In´1,d
0
We will use these exact sequences to assemble a resolution of
Rn {In,d as an Rn -module inductively.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
0
In´1,d´1 {In´1,d
¨xn
n´1,d
U‚´1
‘ U‚n´1,d´1
0
Rn {In´1,d
Rn {In,d
Rn
Rn
U‚n´1,d
???
In´1,d
In´1,d´1
In´1,d´1 {In´1,d
U‚n´1,d
U‚n´1,d´1
n´1,d
U‚´1
‘ U‚n´1,d´1
0
0
Proposition
n´1,d
??? “ U‚´1
‘ U‚n´1,d´1 ‘ U‚n´1,d
is a minimal free resolution of In,d with an Sn´1 -action.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
Question
How does Sn act on a minimal free resolution U‚n,d of In,d ?
The i-th term should satisfy:
´
¯
n,d
n´1,d
n
ResS
U
– Ui´1
‘ Uin´1,d´1 ‘ Uin´1,d .
i
Sn´1
Conjecture
`
˘
Sn
i
Uin,d – IndS
pd,
1
q
b
pn
´
d
´
iq
ˆS
d`i
n´d´i
Verified computationally up to n “ 8.
The dimension matches the Betti numbers.
The conjectured Uin,d is multiplicity free.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
Proposition
`
˘
i
n
IndS
Sd`i ˆSn´d´i pd, 1 q b pn ´ d ´ iq has a basis consisting of
standard Young tableaux on pd, 1i q with entries from t1, . . . , nu.
Example (n=5,d=2,i=2)
U24,2
1 2
3
4
1 3
2
4
1 4
2
3
U24,1
1 5
2
3
1 5
2
4
1 5
3
4
2 5
3
4
U14,2
1 2
3
5
1 3
2
5
1 2
4
5
1 4
2
5
1 3
4
5
1 4
3
5
Federico Galetto
2 3
4
5
2 4
3
5
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals
Example (Going from n “ 4 to n “ 5)
1
0
5
0
10
4
10
6
10
0
5
4
20
6
4
3
15
4
1
1
15
8
10
0
6
3
5
4
1
1
The Betti numbers (and the representations Uin,d ) can be arranged
in a Pascal-like tetrahedron.
Federico Galetto
Queen’s University
Equivariant resolutions of De Concini-Procesi ideals