Differential Operators and Invariant Theory
Will Traves
Department of Mathematics
United States Naval Academy
SIAM Conference on Applied Algebraic Geometry
Colorado State University
04 August 2013
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
1 / 16
Outline
Background on invariant theory and Grassmannians
Differential operators on Grassmannians
Conjecture inspired by a Hilbert series calculation
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
2 / 16
Invariant Theory
Given a k × n matrix M filled with variables,


x11 x12 . . . x1n

..
..  ,
M =  ...
.
. 
xk1 xk2 . . . xkn
we let SLk act on the left by matrix multiplication.
C[Mk,n ] = C[x11 , . . . , xkn ]
C[Mk,n ]SLk = polys invariant on SLk orbits of full rank matrices
= C[G(k, n)]
G(k , n): affine cone over variety param k-planes in n-space
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
3 / 16
Some obvious invariants
SLk acts by multiplication on the left so it acts on each column of


x11 . . . x1k . . . x1n

..
..  ,
M =  ...
.
. 
xk1 . . . xkk
. . . xkn
simultaneously.
The k × k subdeterminants are invariant because
det(σN) = det(σ) det(N) = 1 det(N).
Denote the k × k minor involving columns I1 , . . . , Ik by
[I] = [I1 . . . Ik ].
This is a polynomial of degree k in the variables xij .
det(N) = [12 . . . k]
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
4 / 16
First Fundamental Theorem of Invariant Theory
The First Fundamental Theorem of Invariant Theory
The k × k minors [I] generate the ring of invariants C[Mk,n ]SLk .
The coordinate ring of the Grassmannian C[G(k, n)] = C[Mk,n ]SLk
is called the bracket algebra.
The variables (the minors [I]) are the Plücker coordinates.
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
5 / 16
Second Fundamental Theorem of Invariant Theory
The 2nd Fundamental Theorem of Invariant Theory
The generators of C[Mk,n ]SLk satisfy the Plücker relations – quadratic
relations involving the brackets.
Example
When k = 2 and n = 4 all the Plücker relations are multiples of
[12][34] − [13][24] + [14][23] = 0.
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
6 / 16
SLk invariants of V n and its dual
Shift from C[Mk,n ] = C[V n ] to C[V n ⊕ (V ∗ )n ]
Coordinates on (V ∗ ) are ξ1` , . . . , ξk` with hξi` , xj`0 i = δij δ``0 .
SLk acts on the ξij by the contragredient representation
If σ ∈ SLk acts on C[V ] by A then σ acts on C[V ∗ ] by (A−1 )T
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
7 / 16
Some invariants




x11 . . . x1n
ξ11 . . . ξ1n




..
..
M=
 M∗ = 

.
.
xk1 . . . xkn
ξk1 . . . ξkn
Define: |J| = |J1 . . . Jk | = k × k minor of columns J1 . . . Jk of M ∗
Inner product of a vector in V and a vector in V ∗ is SLk -invariant
New invariants: hiji = inner product of i th column of M with j th
column of M ∗ .
In coordinates: hiji =
Will Traves (U.S. Naval Academy)
Pk
`=1 x`i ξ`j
Invariants & Differential Operators
AG-SIAM 2013
8 / 16
First Fundamental Theorem of Invariant Theory
First Fundamental Theorem of Invariant Theory
The invariants [I], |J| and hiji generate the ring C[V n ⊕ (V ∗ )n ]SLk .
The theory of invariants came into
existence about the middle of the
nineteenth century somewhat like
Minerva: a grown-up virgin, mailed in
the shining armor of algebra, she
sprang forth from Cayley’s Jovian
head. (Weyl, 1939)
H. Weyl
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
9 / 16
Second Fundamental Theorem of Invariant Theory
Second Fundamental Theorem of Invariant Theory
The relations among the [I], |J| and hiji are:
1.
2.
3.
4.
5.
Plücker relations on V : [I] [I 0 ]
Plücker relations
on V ∗ : |J| |J 0 |
P
hiji w/ [I] : Pi [Ii ]hiji
hiji w/ |J|: j hiji|Jj |
dethIJi = [I]|J|.
Example
If [I] = [134], |J| = |235| then


h12i h13i h15i
det h32i h33i h35i = [134] |235|
h42i h43i h45i
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
10 / 16
The Weyl algebra
The ring of differential operators on Ck is the Weyl algebra
D(Ck ) = D(C[x1 , . . . , xk ]) = Chx1 , . . . , xk , ∂1 , . . . , ∂k i
Product Rule:
∂i = ∂/∂xi ⇒ ∂i xj = xj ∂i + δij
Filtration by order: x1 ∂1 ∂2 + ∂1 − 3x2 has order 2
Symbol map: D(Ck ) −→ GrD(Ck ) = C[x1 , . . . , xk , ξ1 , . . . , ξk ]
θ 7−→ θ mod lower order terms
∂i
Will Traves (U.S. Naval Academy)
7−→ ξi
Invariants & Differential Operators
AG-SIAM 2013
11 / 16
The group action on D(C[Mk ,n ])
We are interested in the operators on Mk,n = V n .
Write R = C[Mk,n ] = C[V n ]
SLk acts on each variable ∂ij in the same way it acts on ξij (A−1 )T
Gr [D(R)SLk ] = [GrD(R)]SLk = C[V n ⊕ (V ∗ )n ]SLk
Write D = D(R). Lifts of generators for Gr [D SLk ] generate D SLk .
1st Fundamental Theorem for D SLk [T-]
The generators of D SLk have the form [I], |J| and hiji, where ξij is
replaced by ∂ij .
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
12 / 16
2nd Fundamental Theorem for D SLk
2nd Fundamental Theorem for D SLk [T-]
The relations among the [I], |J| and hiji are:
1.
2.
3.
4.
5.
Plücker relations on V : [I] [I 0 ]
Plücker relations
on V ∗ : |J| |J 0 |
P
hiji w/ [I] : Pi hiji[Ii ]
hiji w/ |J|: j hiji|Jj |
dethIJi = [I]|J|.
Example (Capelli’s Relation)
The fifth relation needs to be interpreted as Capelli’s relation:
replace hkki with hkki+ (index of k in [I] - 1)


h12i h13i
h15i
det  h32i h33i+1 h35i  = [134]|235|.
h42i h43i
h45i
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
13 / 16
Differential Operators on Grassmannians
Rings of differential operators on singular varieties are often hard to
describe.
SLk
Restriction map π∗ : D SLk = D(Mk,n )SLk −→ D(Mk,n
) = D(G(k , n))
Theorem (T-, based on work of Schwarz)
The map π∗ is surjective. So D(G(k , n)) is generated by the restriction
of the operators [I], |J|, and hiji.
Theorem (Schwarz)
The kernel of π∗ is (Dslk )SLk , a two-sided ideal in D SLk .
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
14 / 16
Example: D(G(2, 4))
The kernel of π∗ is just (D(C[M2,4 ])slk )SLk
sl2 = Trace zero matrices (generated by E12 , E21 , E11 − E22 ).
Each acts as a derivation on C[M2,4 ]: g12 , g21 , g11 − g22 .
These derivations are eigenvectors of the torus action for the
maximal torus sitting in SL2 . Their images form a regular
sequence in GrD(G(2, 4)).
Reduction to torus action (technical) gives
H(GrD(G(2, 4)), t) =
Will Traves (U.S. Naval Academy)
1 + 18t 2 + 65t 4 + 65t 6 + 18t 8 + t 10
.
(1 − t 2 )10
Invariants & Differential Operators
AG-SIAM 2013
15 / 16
Gorenstein?
Theorem (T-)
GrD(G(2, 4)) is Gorenstein. Moreover, GrD(G(2, n)) is Gorenstein for
n ≤ 16.
Stanley: Enough to check that GrD(G(2, 4)) is Cohen-Macaulay.
Hochster-Roberts Theorem: GrD SL2 is CM.
GrD(G(2, 4)) is CM since it is a quotient of GrD SL2 by a regular
sequence.
Conjecture
The graded rings GrD(G(k, n)) are always Gorenstein.
Will Traves (U.S. Naval Academy)
Invariants & Differential Operators
AG-SIAM 2013
16 / 16