Hilbert Functions of Artinian Gorenstein
Ideals Stabilized by the Symmetric Group
David Wehlau
October 18, 2014
Hilbert Functions
slide 1
Co-authors
Co-authors
Anthony Geramita (Queen’s University, Kingston)
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 2
Co-authors
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Anthony Geramita (Queen’s University, Kingston)
Andrew Hoefel (Google, Paradise)
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 2
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of
Orbits
Artinian Gorenstein Quotients
of the Polynomial Ring
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 3
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
R = C[x1 , x2 , . . . , xn ]
I is an Artinian Gorentstein homogeneous ideal of R.
The natural action of the symmetric group, Sn on R stabilizes I .
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 4
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
R = C[x1 , x2 , . . . , xn ]
I is an Artinian Gorentstein homogeneous ideal of R.
The natural action of the symmetric group, Sn on R stabilizes I .
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
Problem: Describe the Hilbert Function of A = R/I
HA (k) := dimC Ak = dimC Rk /Ik
slide 4
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Macaulay’s Theorem
Notation
Sn -modules
Factorization
Submodules of the Apolarity
Module
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 5
Macaulay’s Theorem
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Macaulay’s Theorem
Notation
There exists a homogeneous function F ∈ R such that
M := {F }∂ satisfies
HM (k) = HA (k) for all k
where {F }∂ is the minimal vector subspace of R containing F and
∂
for all j .
closed under ∂x
j
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 6
Notation
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Macaulay’s Theorem
Notation
Sn -modules
α
α
n
If (α1 , α2 , . . . , αn ) ∈ Nn then xα := x1 1 x2 2 · · · xα
n and
|α| = α1 + α2 + · · · + αn .
∂ |α| (F )
hF, x i :=
∂xα1 1 · · · ∂xαnn
α
and extend this linearly to define hF, f i fo any f, F ∈ R.
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 7
Notation
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Macaulay’s Theorem
Notation
Sn -modules
α
α
n
If (α1 , α2 , . . . , αn ) ∈ Nn then xα := x1 1 x2 2 · · · xα
n and
|α| = α1 + α2 + · · · + αn .
∂ |α| (F )
hF, x i :=
∂xα1 1 · · · ∂xαnn
α
and extend this linearly to define hF, f i fo any f, F ∈ R.
Factorization
Coordinate Rings of
Orbits
Then h·, ·i is a non-degenerate bilinear form on Rk for all k .
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 7
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Group Action
F alternating
Sn -modules
F symmetric
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 8
Group Action
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Group Action
Sn acts on R by permuting the xi .
In order to have Sn act on {F }∂ we need the line C · F to be
Sn -stable.
Two Cases
F alternating
F symmetric
Factorization
Coordinate Rings of
Orbits
1.
F is an alternating function.
2.
F is a symmetric function
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 9
F alternating
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
The alternating case was solved by Bergeron, Garcia and Tesler:
The Hilbert series of A satisfies
HM (t) = g(t)HRG (t)
Sn -modules
Group Action
F alternating
F symmetric
where RG = C[x1 , x2 , . . . , xn ]/(e1 , e2 , . . . , en ) is the ring of
coinvariants and g(t) is a certain polynomial.
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 10
F alternating
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
The alternating case was solved by Bergeron, Garcia and Tesler:
The Hilbert series of A satisfies
HM (t) = g(t)HRG (t)
Sn -modules
Group Action
F alternating
F symmetric
where RG = C[x1 , x2 , . . . , xn ]/(e1 , e2 , . . . , en ) is the ring of
coinvariants and g(t) is a certain polynomial.
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
Mike Roth extended this by proving it remains true without the
assumption that A be Gorenstein.
slide 10
F symmetric
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
We will assume that F is symmetric. Then we may write
Pr P
F = i=1 σ∈Sn σ(Li )d where each Li ∈ R1 is a linear form.
Submodules of the
Apolarity Module
Sn -modules
Group Action
F alternating
F symmetric
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 11
F symmetric
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
We will assume that F is symmetric. Then we may write
Pr P
F = i=1 σ∈Sn σ(Li )d where each Li ∈ R1 is a linear form.
d
We treat the case F =
σ∈Sn (σL) .
σ∈Sn σ(L ) =
Write L = a1 x1 + a2 x2 + · · · + an xn and a = (a1 , a2 , . . . , an ).
P
d
P
Group Action
F alternating
F symmetric
Factorization
For technical reasons, we will assume from now on, that
a1 + a2 + · · · + an 6= 0.
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 11
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 12
Maps
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Define θk : Rd−k → Rk by θk (f ) = hF, f i
so that Im(θk ) = Mk and rank(θk ) = HM (k).
Submodules of the
Apolarity Module
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 13
Maps
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Define θk : Rd−k → Rk by θk (f ) = hF, f i
so that Im(θk ) = Mk and rank(θk ) = HM (k).
Submodules of the
Apolarity Module
Sn -modules
Factorization
We need to know rank(θk ) for all k .
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 13
Factorization
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
θk = hF, · i
Rd−k
Rk
φk
ψd−k
Factorization
Maps
Factorization
Transposes
Perpendicular
CSn
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 14
Factorization
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
θk = hF, · i
Rd−k
Submodules of the
Apolarity Module
Rk
φk
ψd−k
Sn -modules
Factorization
Maps
Factorization
Transposes
CSn
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
α
ψk (x ) := σ σ(aα )σ
φk (σ) = σLk
P
P k α α
α
(φk ◦ ψd−k )(x ) = |α|=k σ α σ(a )x .
P
P d! k α α
α
α
θk (x ) = hF, x i = |α|=k σ k! α σ(a )x
P
slide 14
Transposes
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
α
α
ψk (x ) := σ σ(a
)σ
P
φk (σ) = σLk = |α|=k
P
k
α
σ(aα )xα .
Submodules of the
Apolarity Module
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 15
Transposes
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
α
α
ψk (x ) := σ σ(a
)σ
P
φk (σ) = σLk = |α|=k
P
k
α
σ(aα )xα .
We see that ψk and φk are (almost) transposes.
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 15
Perpendicular
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Lemma. If a ∈ Rn then
(Im ψk ) ⊥ (ker φk ) for all k
(w.r.t. the usual sesquilinear inner product on CSn ).
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 16
Perpendicular
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Maps
Lemma. If a ∈ Rn then
(Im ψk ) ⊥ (ker φk ) for all k
(w.r.t. the usual sesquilinear inner product on CSn ).
Corollary. If a ∈ Rn then
Factorization
Transposes
Perpendicular
(Im ψk )⊥ = (ker φk ) for all k
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
(w.r.t. the usual sesquilinear inner product on CSn ).
Thus
Im ψk ⊕ ker φk = CSn
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 16
Nesting
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Lemma.
ker φ1 ⊇ ker φ2 ⊇ · · ·
Submodules of the
Apolarity Module
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 17
Nesting
Co-authors
Lemma.
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
ker φ1 ⊇ ker φ2 ⊇ · · ·
Corollary.
Im ψ1 ⊆ Im ψ2 ⊆ · · ·
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 17
Transversality
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Lemma. If a ∈ Rn then
Im ψi meets ker φj transversely for all i and j.
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 18
Transversality
Co-authors
Lemma. If a ∈ Rn then
Artinian Gorenstein
Quotients of the
Polynomial Ring
Im ψi meets ker φj transversely for all i and j.
Submodules of the
Apolarity Module
Sn -modules
Factorization
Maps
Factorization
Transposes
Proof.
Im ψi ∩ ker φi = 0
If i < j then ker φj ⊆ ker φi and thus Im ψi ∩ ker φj = 0.
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Im ψj + ker φj = CSn
if i > j then Im ψi ⊇ Im ψj and thus Im ψi + ker φj = CSn .
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 18
rank(θk )
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Consequence:
rank θk = min{rank(ψd−k ), rank(φk )}
= min{rank(φd−k ), rank(φk )}.
Sn -modules
Factorization
Maps
Factorization
Transposes
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 19
rank(θk )
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Consequence:
rank θk = min{rank(ψd−k ), rank(φk )}
= min{rank(φd−k ), rank(φk )}.
Sn -modules
Factorization
Maps
Factorization
Transposes
All we need do is determine rank(φk ) for all k .
Perpendicular
Nesting
Transversality
rank(θk )
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 19
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of Orbits
Coordinate Rings of
Orbits
φk
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 20
φk
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Recall φk (σ) = σLk . Define
Nk := Im φk = spanC {σLk | σ ∈ Sn }.
Sn -modules
Factorization
Coordinate Rings of
Orbits
φk
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 21
φk
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Recall φk (σ) = σLk . Define
Nk := Im φk = spanC {σLk | σ ∈ Sn }.
Sn -modules
Factorization
Coordinate Rings of
Orbits
If f ∈ Rk then hLk , f i = k!f (a) and hσLk , f i = k!f (σ(a))
where L = a1 x1 + a2 x2 + · · · + an xn .
φk
Kostka-Foulkes
Polynomials
Hilbert Functions
slide 21
φk
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Recall φk (σ) = σLk . Define
Nk := Im φk = spanC {σLk | σ ∈ Sn }.
Sn -modules
Factorization
Coordinate Rings of
Orbits
φk
Kostka-Foulkes
Polynomials
Hilbert Functions
If f ∈ Rk then hLk , f i = k!f (a) and hσLk , f i = k!f (σ(a))
where L = a1 x1 + a2 x2 + · · · + an xn .
Thus f ∈ (Nk )⊥ iff k!f (σ(a)) = 0 for all σ iff f ∈ Ik
where I = I(X) is the ideal corresponsing to the variety
X = {σ(a) | σ ∈ Sn } ⊂ Pn−1 .
slide 21
φk
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Recall φk (σ) = σLk . Define
Nk := Im φk = spanC {σLk | σ ∈ Sn }.
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of
Orbits
φk
Kostka-Foulkes
Polynomials
If f ∈ Rk then hLk , f i = k!f (a) and hσLk , f i = k!f (σ(a))
where L = a1 x1 + a2 x2 + · · · + an xn .
Thus f ∈ (Nk )⊥ iff k!f (σ(a)) = 0 for all σ iff f ∈ Ik
where I = I(X) is the ideal corresponsing to the variety
X = {σ(a) | σ ∈ Sn } ⊂ Pn−1 .
Therefore
rank(φk ) = dim Nk = dim Rk − dim(Nk )⊥
= dim Rk − dim I(X)k = dim(R/I(X))k .
Hilbert Functions
slide 21
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Sn -modules
Factorization
Kostka-Foulkes Polynomials
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Kostka-Foulkes
Polynomials
Example
Reference
Hilbert Functions
slide 22
Kostka-Foulkes Polynomials
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Fortunately for us, it is known that the Hilbert Function for R/I(X)
is entirely determined by certain Kostka-Foulkes Polynomials.
Submodules of the
Apolarity Module
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Kostka-Foulkes
Polynomials
Example
Reference
Hilbert Functions
slide 23
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
Hilbert Functions
slide 24
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
The Kostka-Foulkes polynomials Kλ,µ (t) for µ = (2, 1, 1) are
K4,µ (t) = t3 , K31,µ (t) = t + t2 , K22,µ (t) = t, K211,µ (t) = 1, K1111,µ (t) = 0.
Hilbert Functions
slide 24
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
The Kostka-Foulkes polynomials Kλ,µ (t) for µ = (2, 1, 1) are
K4,µ (t) = t3 , K31,µ (t) = t + t2 , K22,µ (t) = t, K211,µ (t) = 1, K1111,µ (t) = 0.
The graded character of Rµ = R/I(X) is, with n(µ) = 3
χRµ (t) =
Hilbert Functions
n(µ) λ
4
31
31
22 2
211 3
K
(1/t)t
χ
=
χ
+
χ
t
+
(χ
+
χ
)t
+
χ
t.
λ,µ
λ⊢n
P
slide 24
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
The Kostka-Foulkes polynomials Kλ,µ (t) for µ = (2, 1, 1) are
K4,µ (t) = t3 , K31,µ (t) = t + t2 , K22,µ (t) = t, K211,µ (t) = 1, K1111,µ (t) = 0.
The graded character of Rµ = R/I(X) is, with n(µ) = 3
χRµ (t) =
n(µ) λ
4
31
31
22 2
211 3
K
(1/t)t
χ
=
χ
+
χ
t
+
(χ
+
χ
)t
+
χ
t.
λ,µ
λ⊢n
P
The graded character of N is χN (t) =
χ4 + (χ4 + χ31 )t + (χ4 + 2χ31 + χ22 )t2 + (χ4 + 2χ31 + χ22 + χ211 )t3 (1 − t)−1 .
Hilbert Functions
slide 24
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
The Kostka-Foulkes polynomials Kλ,µ (t) for µ = (2, 1, 1) are
K4,µ (t) = t3 , K31,µ (t) = t + t2 , K22,µ (t) = t, K211,µ (t) = 1, K1111,µ (t) = 0.
The graded character of Rµ = R/I(X) is, with n(µ) = 3
χRµ (t) =
n(µ) λ
4
31
31
22 2
211 3
K
(1/t)t
χ
=
χ
+
χ
t
+
(χ
+
χ
)t
+
χ
t.
λ,µ
λ⊢n
P
The graded character of N is χN (t) =
χ4 + (χ4 + χ31 )t + (χ4 + 2χ31 + χ22 )t2 + (χ4 + 2χ31 + χ22 + χ211 )t3 (1 − t)−1 .
Thus, the graded character of M (or A) is
χM (t) = χ4 + (χ4 + χ31 )t + (χ4 + 2χ31 + χ22 )t2 + (χ4 + 2χ31 + χ22 + χ211 )t3 +
(χ4 + 2χ31 + χ22 + χ211 )t4 + (χ4 + 2χ31 + χ22 )t5 + (χ4 + χ31 )t6 + χ4 t7 .
Hilbert Functions
slide 24
Example
F :=
7
σ(x
+
x
+
2x
+
3x
)
with associated partition µ = (2, 1, 1).
1
2
3
4
σ
P
The Kostka-Foulkes polynomials Kλ,µ (t) for µ = (2, 1, 1) are
K4,µ (t) = t3 , K31,µ (t) = t + t2 , K22,µ (t) = t, K211,µ (t) = 1, K1111,µ (t) = 0.
The graded character of Rµ = R/I(X) is, with n(µ) = 3
χRµ (t) =
n(µ) λ
4
31
31
22 2
211 3
K
(1/t)t
χ
=
χ
+
χ
t
+
(χ
+
χ
)t
+
χ
t.
λ,µ
λ⊢n
P
The graded character of N is χN (t) =
χ4 + (χ4 + χ31 )t + (χ4 + 2χ31 + χ22 )t2 + (χ4 + 2χ31 + χ22 + χ211 )t3 (1 − t)−1 .
Thus, the graded character of M (or A) is
χM (t) = χ4 + (χ4 + χ31 )t + (χ4 + 2χ31 + χ22 )t2 + (χ4 + 2χ31 + χ22 + χ211 )t3 +
(χ4 + 2χ31 + χ22 + χ211 )t4 + (χ4 + 2χ31 + χ22 )t5 + (χ4 + χ31 )t6 + χ4 t7 .
Evaluating at the identity in S4 we obtain the Hilbert polynomial as
HA (t) = HM (t) = 1 + 4t + 9t2 + 12t3 + 12t4 + 9t5 + 4t6 + t7 .
Hilbert Functions
slide 24
Reference
Co-authors
Artinian Gorenstein
Quotients of the
Polynomial Ring
Submodules of the
Apolarity Module
Andrew Hoefel, Anthony Geramita and David Wehlau
Hilbert Functions of Sn -Stable Artinian Gorenstein Ideals
arXiv:1407.7228 [math.AC]
Sn -modules
Factorization
Coordinate Rings of
Orbits
Kostka-Foulkes
Polynomials
Kostka-Foulkes
Polynomials
Example
Reference
Hilbert Functions
slide 25