arXiv:1303.5532v1 [] 22 Mar 2013

N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS
THANH VU
Abstract. The rational homology groups of the matching complexes are
closely related to the syzygies of the Veronese embeddings. In this paper we
will prove the vanishing of certain rational homology groups of matching complexes, thus proving that the third Veronese embeddings satisfy the property
N6 . This settles the Ottaviani-Paoletti conjecture for third Veronese embeddings. This result is optimal since ν3 (Pn ) does not satisfy the property N7 for
n ≥ 2 as shown by Ottaviani-Paoletti in [OP].

1. Introduction
Let k be a field of characteristic 0. Let V be a finite dimensional vector space
over k of dimension n + 1. The projective space P(V ) has coordinate ring naturally
isomorphic to Sym V . For each natural number d, the d-th Veronese embedding
of P(V ), which is naturally embedded into the projective space P(Symd V ) has
kd
V . For each set of integers p, q, b, let
coordinate ring Ver(V, d) = ⊕∞
k=0 Sym
d
Kp,q (V, b) be the associated Koszul cohomology group defined as the homology of
the 3 term complex
p+1
^

Symd V ⊗ Sym(q−1)d+b V →
→

p−1
^

p
^

Symd V ⊗ Symqd+b V

Symd V ⊗ Sym(q−1)d+b V.

d
(V, b) is the space of minimal p-th syzygies of degree p + q of the GL(V )
Then Kp,q
kd+b
d
(b) : Vect → Vect for the functor on finite
V. We write Kp,q
module ⊕∞
k=0 Sym
dimensional k-vector spaces that assigns to a vector space V the corresponding
d
(V, b).
syzygy module Kp,q

Conjecture 1 (Ottaviani-Paoletti).
d
Kp,q
(V, 0) = 0 for q ≥ 2 and p ≤ 3d − 3.

The conjecture is known for d = 2 by the work of Josefiak-Pragacz-Weyman
[JPW] and also known for dim V = 2 and dim V = 3 by the work of Green [G]
and Birkenhake [B]. All other cases are open. Also, Ottaviani and Paoletti in [OP]
d
showed that Kp,2
(V, 0) 6= 0 for p = 3d − 2, when dim V ≥ 3, d ≥ 3. In other words,
the conjecture is sharp. Recently, Bruns, Conca, and Römer in [BCR] showed that
d
Kd+1,q
(V, 0) = 0 for q ≥ 2. In this paper we will prove the conjecture in the case
d = 3. Thus the main theorem of the paper is:
Date: February 28, 2013.
2010 Mathematics Subject Classification. Primary 13D02, 14M12, 05E10.
Key words and phrases. Syzygies, Veronese varieties, matching complexes.
1

2

THANH VU

Theorem 3.6. The third Veronese embeddings of projective spaces satisfy property
N6 .
To prove Theorem 3.6 we prove that vanishing results hold for certain rational homology groups of matching complexes (defined below), and then use [KRW,
d
(V, b) and the homology
Theorem 5.3] to translate between the zyzygy modules Kp,q
groups of matching complexes.
Definition 2 (Matching Complexes). Let d > 1 be a positive integer and A a
d
finite set. The matching complex CA
is the simplicial complex whose vertices are
all the d-element subsets of A and whose faces are {A1 , ..., Ar } so that A1 , ..., Ar
are mutually disjoint.
d
The symmetric group SA acts on CA
by permuting the elements of A making
d
the homology groups of CA representations of SA . For each partition λ, we denote
by V λ the irreducible representation of S|λ| corresponding to the partition λ, and
Sλ the Schur functor corresponding to the partition λ. For each vector space V ,
Sλ (V ) is an irreducible representation of GL(V ). The relation between the syzygies
of the Veronese embeddings and the homologies of matching complexes is given by
the following theorem of Karaguezian, Reiner and Wachs.

Theorem 1.1. [KRW, Theorem 5.3] Let p, q be non-negative integers, let d be
a positive integer and let b be a non-negative integer. Write N = (p + q)d + b.
d
(b) coincides with
Consider a partition λ of N . Then the multiplicity of Sλ in Kp,q
λ
d
the multiplicity of the irreducible SN representation V in H̃p−1 (CN
).
This correspondence makes Conjecture 1 equivalent to the following conjecture.
d
for n = 1, ..., 3d − 1 is
Conjecture 3. The only non-zero homology groups of Cnd
H̃n−2 .

We will prove this conjecture for d = 3 by computing the homology groups of
Cn3 by induction. To compute the homology groups of the matching complexes
inductively, the following equivariant long-exact sequence introduced by Raicu in
[R] is useful. Let A be a finite set with |A| ≥ 2d. Let a ∈ A be an element of A.
Let α be a d-element subset of A such that a ∈ α. Let β = α \ a, and let C = A \ α,
B = A \ {a}. Then we have the following long-exact sequence of representations of
SB .

(1.1)

d
d
d
· · · → IndSSB
(H̃r (CC
) ⊗ 1) → H̃r (CB
) → ResSSB
(H̃r (CA
))
C ×Sβ
A
d
(H̃r−1 (CC
) ⊗ 1) → · · ·
→ IndSSB
C ×Sβ

kd+b
Morever, for each b, 0 ≤ b ≤ d − 1, the GL(V ) module ⊕∞
V is a
k=0 Sym
Cohen-Macaulay module in the coordinate ring of the projective space P(Symd V ).
The dual of the resolution of each of these modules is the resolution of another
d
such module giving us the duality among the Koszul homology groups Kp,q
(V, b).
For simplicity, since we deal with third Veronese embeddings, we will from now on
assume that d = 3. To compute the homology groups of the matching complexes
Cn3 for n ≤ 10 we use the duality in the case dim V = 2, and to compute the
homology groups of the matching complexes Cn3 for n ≥ 14, we use the duality in
the case dim V = 3, so we will make them explicit here. In the case dim V = 2, the

N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS

3

canonical module of Ver(V, 3) as representation of GL(V ) is Sλ (V ) with λ = (5, 5).
From [W, Chapter 2],
∼ S(5−b,5−a)
Hom(S(a,b) , S(5,5) ) =
and the correspondence in Theorem 1.1, we have
Proposition 1.2. The multiplicity of V λ with λ = (λ1 , λ2 ) in the homology group
3
H̃p−1 (CN
) coincides with the multiplicity of V µ with µ = (5 − λ2 , 5 − λ1 ) in the
3
homology group H̃1−p (C10−N
).
Similarly, in the case dim V = 3, the canonical module of Ver(V, 3) as representation of GL(V ) is Sλ (V ) with λ = (9, 9, 9). From [W, Chapter 2],
Hom(S(a,b,c) , S(9,9,9) ) ∼
= S(9−c,9−b,9−a)
and the correspondence in Theorem 1.1, we have
Proposition 1.3. The multiplicity of V λ with λ = (λ1 , λ2 , λ3 ) in the homology
3
group H̃p−1 (CN
) coincides with the multiplicity of V µ with µ = (9−λ3 , 9−λ2 , 9−λ1 )
3
in the homology group H̃6−p (C27−N
).
3
Finally, to determine the homology groups of the matching complex CN
, we
apply the equivariant long exact sequence (1.1) to derive equalities and inequalities
for the multiplicities of the possible irreducible representations in the unknown
3
homology groups of CN
. This is carried out with the help of our Macaulay2 package
MatchingComplex.m2 that we will explain in the appendix. We then use Maple to
solve this system of equalities and inequalities.
The paper is organized as follows. In the second section, using the equivariant
long exact sequence (1.1) and the duality in Proposition 1.2 we compute homology
groups of matching complexes Cn3 for n ≤ 13. In the third section, using the results
in the second section and the duality in Proposition 1.3 we derive all irreducible
representations whose corresponding partitions have at most 3 rows in the homology
groups of Cn3 for n ≥ 14. The systems of equalities and inequalities from exact
sequences obtained by applying the equivariant long exact sequence (1.1) with |A| =
3
3
) uniquely. Thus we need
) and H̃5 (C23
20 and |A| = 23 will not determine H̃4 (C20
to compute the dimensions of Kp,1 (V, 2) with p = 5, 6 and dim V = 4. This is
done by using Macaulay2 to compute the dimensions of the spaces of minimal p-th
3k+2
syzygies of degree p + 1 for p = 6, 7 of the module ⊕∞
V with dim V = 4.
k=0 Sym
3
We then state the results for the homology groups of Cn with 14 ≤ n ≤ 24 and
finish the proof of our main theorem. The proof of Proposition 3.5 illustrates the
computation of the homology groups of the matching complexes Cn3 dealing with
3
the most complicated matching complex C23
in the series. In the appendix we
explain the ideas behind our package leading to the computation of the homology
groups of the matching complexes.

2. Homology of matching complexes
In this section, using the equivariant long exact sequence (1.1) we compute the
homology groups of the matching complexes Cn3 for n ≤ 13. In the following, we
denote the partition λ with row lengths λ1 ≥ λ2 ≥ ... ≥ λk ≥ 0 by the sequence
(λ1 , λ2 , ..., λk ) and we use the same notation for the representation V λ . To simplify
notation, we omit the subscript and superscript when we use the operators Ind and

4

THANH VU

Res. It is clear from the context and the equivariant long exact sequence what the
induction and restriction are. From the definition of the matching complexes, it is
not hard to see the following.
Proposition 2.1. The only non-vanishing homology groups of Cn3 with n = 4, 5, 6
are respectively
H̃0 C43 = (3, 1), H̃0 C53 = (4, 1) ⊕ (3, 2), H̃0 C63 = (4, 2).
Together with Proposition 1.2 we have
Proposition 2.2. The only irreducible representations whose corresponding partitions have at most 2 rows in the homology groups of the matching complexes Cn3 ,
7 ≤ n ≤ 10 are
3
)2 = (5, 5)
(H̃1 C83 )2 = (5, 3), (H̃1 C93 )2 = (5, 4), (H̃1 C10

where (H̃i Cn3 )2 denote the subrepresentation of H̃i Cn3 consisting of all irreducible
representations of H̃i Cn3 whose corresponding partitions have at most 2 rows.
Proposition 2.3. The only non-vanishing homology group of C73 is
H̃1 C73 = (5, 1, 1) ⊕ (3, 3, 1).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 7 we have
exact sequences
0 → Res H̃i C73 → 0
for i 6= 0, 1, and an exact sequence
0 → Res H̃1 C73 → Ind H̃0 C43 → H̃0 C63 → Res H̃0 C73 → 0.
Therefore, H̃i C73 = 0 for i 6= 0, 1. To show that H̃0 C73 is zero, note that H̃0 C63 maps
surjectively onto Res H̃0 C73 . Since H̃0 C63 = (4, 2) and by Proposition 2.2, H̃0 C73 does
not contain any irreducible representations whose corresponding partitions have at
most 2 rows, it must be zero. Therefore, we know that Res H̃1 C73 as representation
of S6 is equal to
Ind H̃0 C43 − H̃0 C63 = (5, 1) ⊕ (4, 1, 1) ⊕ (3, 3) ⊕ (3, 2, 1).
Moreover, by Proposition 2.2, H̃1 C73 does not contain any irreducible representations whose corresponding partitions have at most 2 rows, thus H̃1 C73 must consist
(5, 1, 1) and (3, 3, 1) as its restriction contains (5, 1) and (3, 3). But the restriction

of (5, 1, 1) ⊕ (3, 3, 1) is equal to Res H̃1 C73 so we have the desired conclusion.
Proposition 2.4. The only non-vanishing homology group of C83 is
H̃1 C83 = (6, 1, 1) ⊕ (5, 2, 1) ⊕ (5, 3) ⊕ (4, 3, 1) ⊕ (3, 3, 2).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 8 we have
exact sequences
0 → Res H̃i C83 → 0
for i 6= 1, and an exact sequence
0 → H̃1 C73 → Res H̃1 C83 → Ind H̃0 C53 → 0.
Therefore, H̃i C83 = 0 for i 6= 1. To compute H̃1 C83 , note that from the exact
sequence
Res H̃1 C83 = H̃1 C73 + Ind H̃0 C53

N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS

5

which consists of irreducible representations whose corresponding partitions have
at most 3 rows, therefore H̃1 C83 contains only irreducible representations whose
corresponding partitions have at most 3 rows. Moreover, by Proposition 2.2, the
only irreducible representation whose corresponding partition has at most 2 rows
in H̃1 C83 is (5, 3). Therefore, the restrictions of irreducible representations whose
corresponding partitions have 3 rows in H̃1 C83 is equal to
(3, 2, 2) ⊕ 2 · (3, 3, 1) ⊕ 2 · (4, 2, 1) ⊕ (4, 3) ⊕ 2 · (5, 1, 1) ⊕ (5, 2) ⊕ (6, 1).
It is easy to see that the set of irreducible representations whose corresponding
partitions have 3 rows in H̃1 C83 has to be equal to (6, 1, 1) ⊕ (5, 2, 1) ⊕ (4, 3, 1) ⊕
(3, 3, 2).

Proposition 2.5. The only non-vanishing homology group of C93 is
H̃1 C93 = (6, 2, 1) ⊕ (5, 4) ⊕ (5, 3, 1) ⊕ (4, 3, 2).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 9 we have
exact sequences
0 → Res H̃i C93 → 0
for i 6= 1, and an exact sequence
0 → H̃1 C83 → Res H̃1 C93 → Ind H̃0 C63 → 0.
Therefore, H̃i C93 = 0 for i 6= 1, and
Res H̃1 C93 = H̃1 C83 + Ind H̃0 C63 .
Moreover, by Proposition 2.2, the only irreducible representation whose corresponding partition has at most 2 rows in H̃1 C93 is (5, 4). Therefore, the restrictions of
irreducible representations whose corresponding partitions have 3 rows in H̃1 C93 is
equal to
(3, 3, 2) ⊕ (4, 2, 2) ⊕ 2 · (4, 3, 1) ⊕ 2 · (5, 2, 1) ⊕ (5, 3) ⊕ (6, 1, 1) ⊕ (6, 2).
It is easy to see that the set of irreducible representations whose corresponding
partitions have 3 rows in H̃1 C93 has to be equal to (6, 2, 1) ⊕ (5, 3, 1) ⊕ (4, 3, 2). 
3
Proposition 2.6. The only non-vanishing homology groups of C10
are
3
3
H̃2 C10
= (7, 1, 1, 1) ⊕ (5, 3, 1, 1) ⊕ (3, 3, 3, 1), H̃1 C10
= (5, 5).

Proof. Applying the equivariant long exact sequence (1.1) with |A| = 10 we have
exact sequences
3
0 → Res H̃i C10
→0
for i 6= 1, 2, and an exact sequence
3
3
0 → Res H̃2 C10
→ Ind H̃1 C73 → H̃1 C93 → Res H̃1 C10
→ 0.
3
3
Therefore, H̃i C10
= 0 for i 6= 1, 2. Since H̃1 C93 maps surjectively onto Res H̃1 C10
,
3
and by Proposition 2.2, H̃1 C10 contains the irreducible representation (5, 5), (6, 2, 1)⊕
(5, 3, 1)⊕(4, 3, 2) maps surjectively onto the restrictions of the other irreducible rep3
resentations of H̃1 C10
. Moreover, there is no irreducible representation of S10 whose

6

THANH VU

3
restriction is a subrepresentation of (6, 2, 1)⊕(5, 3, 1)⊕(4, 3, 2), thus H̃1 C10
= (5, 5).
3
Therefore, Res H̃2 C10 is equal to
3
Ind H̃1 C73 + Res H̃1 C10
− H̃1 C93 = (3, 3, 2, 1) ⊕ (3, 3, 3) ⊕ (4, 3, 1, 1)

⊕ (5, 2, 1, 1) ⊕ (5, 3, 1) ⊕ (6, 1, 1, 1) ⊕ (7, 1, 1).
3
Moreover, since H̃1 C10
= (5, 5), by Theorem 1.1, K2,1 (V, 1) = Sλ (V ) with λ =
∞
(5, 5). Since ⊕k=0 Sym3k+1 (V ) with dim V = 3 is a Cohen-Macaulay module of
codimension 7 with h-vector (3, 6),
 
 
7
7
dim K3,0 (V, 1) = 6 ·
+ dim K2,1 (V, 1) − 3 ·
= 0.
2
3
3
By Theorem 1.1, H̃2 C10
does not contain any irreducible representations whose
corresponding partitions have at most 3 rows, therefore it must contain irreducible
representations (3, 3, 3, 1), (5, 3, 1, 1), (7, 1, 1, 1) each with multiplicity 1. But the
sum of the restrictions of these irreducible representations is equal to the restriction
3
of H̃2 C10
already, thus we have the desired conclusion.

3
Proposition 2.7. The only non-vanishing homology group of C11
is
3
H̃2 C11
= (7, 3, 1) ⊕ (6, 4, 1) ⊕ (6, 3, 2) ⊕ (5, 4, 2) ⊕ (5, 3, 3)

⊕ (6, 3, 1, 1) ⊕ (8, 1, 1, 1) ⊕ (7, 2, 1, 1) ⊕ (5, 4, 1, 1)
⊕ (5, 3, 2, 1) ⊕ (4, 3, 3, 1) ⊕ (3, 3, 3, 2).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 11 we have
exact sequences
3
0 → Res H̃i C11
→0

for i 6= 1, 2 and an exact sequence
3
3
3
3
0 → H̃2 C10
→ Res H̃2 C11
→ Ind H̃1 C83 → H̃1 C10
→ Res H̃1 C11
→ 0.
3
3
3
,
is zero, note that H̃1 C10
= 0 for i 6= 1, 2. To show that H̃1 C11
Therefore, H̃i C11
3
3
consists of partition (5, 5) only, maps surjectively onto Res H̃1 C11 , so H̃1 C11 = 0.
Thus
3
3
3
Res H̃2 C11
= H̃2 C10
+ Ind H̃1 C83 − H̃1 C10
.
3
Moreover, H̃2 C11
does not contain any irreducible representations whose corre3
contains (6, 4) and (7, 3),
sponding partitions have at most 2 rows, but Res H̃2 C11
3
thus H̃2 C11 must contain (6, 4, 1) and (7, 3, 1) each with multiplicity 1. The restrictions of the remaining irreducible representations is equal to
3
3
A = H̃2 C10
+ Ind H̃1 C83 − H̃1 C10
− Res(6, 4, 1) − Res(7, 3, 1).

To find a set of irreducible representations whose sum of the restrictions is A, we
first induce A. We then eliminate all irreducible representations whose restrictions
are not contained in A. After that, we have a set of partitions B. Then we write
down the set of equations that make the restriction of B equal to A. Then we solve
for the non-negative integer solutions of that system. For this problem, it is easy
to see that we have a unique solution as in the statement of the proposition.


N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS

7

3
Proposition 2.8. The only non-vanishing homology group of C12
is
3
H̃2 C12
= (7, 4, 1) ⊕ (7, 3, 2) ⊕ (6, 5, 1) ⊕ (6, 4, 2) ⊕ (6, 3, 3) ⊕ (5, 5, 2)

⊕ (5, 4, 3) ⊕ (8, 2, 1, 1) ⊕ (7, 3, 1, 1) ⊕ (6, 4, 1, 1) ⊕ (6, 3, 2, 1)
⊕ (5, 4, 2, 1) ⊕ (5, 3, 3, 1) ⊕ (4, 3, 3, 2).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 12 we have
exact sequences
3
0 → Res H̃i C12
→0
for i 6= 2 and an exact sequence
3
3
0 → H̃2 C11
→ Res H̃2 C12
→ Ind H̃1 C93 → 0.
3
Therefore, H̃i C12
= 0 for i 6= 2 and
3
3
+ Ind H̃1 C93 .
Res H̃2 C12
= H̃2 C11
3
Moreover, H̃2 C12
does not contain any irreducible representations whose corre3
sponding partitions have at most 2 rows, but Res H̃2 C12
contains (6, 5) and (7, 4),
3
thus H̃2 C12 must contain (6, 5, 1) and (7, 4, 1) each with multiplicity 1. Therefore,
3
the sum of the restrictions of other irreducible representations in H̃2 C12
is equal to
3
A = H̃2 C11
+ Ind H̃1 C93 − Res(6, 5, 1) − Res(7, 4, 1).

Let B be the set containing all irreducible representations that appears in the
induction of A whose restrictions are contained in A. Write down the equations
that make the restriction of B equal to A. Then we solve for the non-negative
integer solutions of that system. For this problem, it is easy to see that we have a
unique solution as in the statement of the proposition.

3
are
Proposition 2.9. The only non-vanishing homology groups of C13
3
H̃3 C13
= (9, 1, 1, 1, 1) ⊕ (7, 3, 1, 1, 1) ⊕ (5, 5, 1, 1, 1) ⊕ (5, 3, 3, 1, 1)

⊕ (3, 3, 3, 3, 1)
3
H̃2 C13

= (7, 5, 1) ⊕ (7, 3, 3) ⊕ (6, 5, 2) ⊕ (5, 5, 3).

Proof. Applying the equivariant long exact sequence (1.1) with |A| = 13 we have
exact sequences
3
0 → Res H̃i C13
→0
for i 6= 2, 3 and an exact sequence
3
3
3
3
3
0 → Res H̃3 C13
→ Ind H̃2 C10
→ H̃2 C12
→ Res H̃2 C13
→ Ind H̃1 C10
→ 0.
3
3
Therefore, H̃i C13
= 0 for i 6= 2, 3. From the sequence, Res H̃3 C13
is contained in
3
3
3
Ind H̃2 C10 and contains Ind H̃2 C10 − H̃2 C12 . There is a unique solution as indicated
in the proposition. Therefore,
3
3
3
3
3
Res H̃2 C13
= Res H̃3 C13
− Ind H̃2 C10
+ H̃2 C12
+ Ind H̃1 C10
.
3
Moreover, Res H̃2 C13
does not contain any irreducible representations whose corresponding partitions have at most two rows, but its restriction contains (7, 5), thus
it must contain (7, 5, 1) with multiplicity 1. Therefore, the sum of the restrictions
3
of the other irreducible representations in H̃2 C13
is equal to

(5, 4, 3) ⊕ 2 · (5, 5, 2) ⊕ (6, 3, 3) ⊕ (6, 4, 2) ⊕ (6, 5, 1) ⊕ (7, 3, 2).
It is easy to see that there is a unique solution as stated in the proposition.



8

THANH VU

3. Proof of the main theorem
In this section we determine the homology groups of the matching complexes Cn3
for 14 ≤ n ≤ 24 using the equivariant long exact sequence (1.1) and the duality
3
as stated in Proposition 1.3. Note that to determine the homology groups of C20
3
and C23
we need to compute the dimensions of Kp,0 (V, 2) for p = 6, 7. In the
following, for a representation W of the symmetric group SN and a positive number
k, we denote by Wk the direct sum of all irreducible representations of W whose
corresponding partitions have at most r rows and let W k = W − Wk .
Proposition 3.1. The only homology groups of Cn3 for 14 ≤ n ≤ 24 containing
irreducible representations whose corresponding partitions have at most 3 rows are
3
3
3
3
3
3
3
3
3
3
H̃3 C14
, H̃3 C15
, H̃3 C16
, H̃4 C17
, H̃4 C18
, H̃4 C19
, H̃4 C20
, H̃5 C21
, H̃5 C22
and H̃5 C23
.
Moreover,
3
3
)3 = (9, 8, 6).
(H̃4 C20
)3 = (8, 8, 4) ⊕ (8, 6, 6), and (H̃5 C23

Proof. This follows from the results of the homology of matching complexes Cn3 for
n ≤ 13 in section 2 and the duality in Proposition 1.3.

Proposition 3.2. The only non-vanishing homology groups of Cn3 for 14 ≤ n ≤ 19
3
3
3
3
3
3
3
3
are H̃3 C14
, H̃3 C15
, H̃3 C16
, H̃4 C16
, H̃4 C17
, H̃4 C18
, H̃4 C19
, H̃5 C19
.
Proof. The computational proof is given in our Macaulay2 package MatchingComplex.m2.

3
3
Proposition 3.3. The only non-vanishing homology groups of C20
are H̃5 C20
and
3
H̃4 C20
= (8, 8, 4) ⊕ (8, 6, 6).

Proof. Applying the equivariant long exact sequence (1.1) with |A| = 20 we have
exact sequences
3
0 → Res H̃i C20
→0
for i 6= 4, 5 and an exact sequence
3
3
3
3
3
0 → H̃5 C19
→ Res H̃5 C20
→ Ind H̃4 C17
→ H̃4 C19
→ Res H̃4 C20
→ 0.
3
= 0 for i 6= 4, 5. By Theorem 1.1, K4,2 (2) = 0. To determine
Therefore, H̃i C20
3
H̃4 C20 , note that by Proposition 3.1 and Theorem 1.1, K5,1 (V, 2) contains M =
Sλ (V ) ⊕ Sµ (V ) where λ = (8, 8, 4) and µ = (8, 6, 6). Moreover, using Macaulay2 to
3k+2
compute the dimensions of minimal linear syzygies of the module ⊕∞
(V )
k=0 Sym
3k+2
∞
with dim V = 4, we get dim K6,0 (V, 2) = 14003. Since ⊕k=0 Sym
(V ) is a
Cohen-Macaulay module of codimension 16 with h-vector (10, 16, 1),
 
   
16
16
16
dim K5,1 (V, 2) = 14003 − 10 ·
+ 16 ·
−
= 1991.
6
5
4
∼ M . By Theorem 1.1,
Since dim M = 1991, K5,1 (V, 2) =
3
H̃4 C20
= (8, 8, 4) ⊕ (8, 6, 6).

Finally,
3
3
3
3
3
Res H̃5 C20
= H̃5 C19
+ Ind H̃4 C17
− H̃4 C19
+ Res H̃4 C20
.
3
This determines H̃5 C20 as given in our package MatchingComplex.m2.



3
3
Proposition 3.4. The only non-vanishing homology groups of C21
and C22
are
3
3
3
H̃5 C21 , H̃5 C22 , H̃6 C22 .

N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS

9

Proof. The computational proof is given in our Macaulay2 package MatchingComplex.m2.

3
3
Proposition 3.5. The only non-vanishing homology groups of C23
are H̃6 C23
and
3
H̃5 C23
= (9, 8, 6) ⊕ (8, 6, 6, 3) ⊕ (8, 7, 6, 2) ⊕ (8, 8, 4, 3) ⊕ (8, 8, 5, 2)

⊕ (8, 8, 6, 1) ⊕ (9, 6, 6, 2) ⊕ (9, 7, 6, 1) ⊕ (9, 8, 4, 2) ⊕ (9, 8, 5, 1)
⊕ (10, 6, 6, 1) ⊕ (10, 8, 4, 1).
Proof. Applying the equivariant long exact sequence (1.1) with |A| = 23 we have
exact sequences
3
0 → Res H̃i C23
→0
for i 6= 5, 6, and an exact sequence
3
3
3
3
0 → H̃6 C22
→ Res H̃6 C23
→ Ind H̃5 C20
→ H̃5 C22
→

(3.1)

3
3
→ 0.
→ Res H̃5 C23
→ Ind H̃4 C20

3
3
Therefore, H̃i C23
= 0 for i 6= 5, 6. Moreover, by Proposition 3.3, (Ind H̃5 C20
)3 =
0. Therefore, we have an exact sequence
3
3
3
)3 → 0.
0 → (H̃5 C22
)3 → (Res H̃5 C23
)3 → (Ind H̃4 C20

(3.2)

By Proposition 3.1, we know that
3
(H̃5 C23
)3 = (9, 8, 6).

Let Y be the direct sum of irreducible representations whose corresponding parti3
tions have 4 rows in H̃5 C23
. Then from exact sequence (3.2), we have
3
3
(Res Y )3 = (H̃5 C22
)3 + (Ind H̃5 C20
)3 − Res(9, 8, 6)

= (8, 8, 6) ⊕ (9, 7, 6) ⊕ (9, 8, 5) ⊕ (10, 6, 6) ⊕ (10, 8, 4).
Therefore,
Y1 = (8, 8, 6, 1) ⊕ (9, 7, 6, 1) ⊕ (9, 8, 5, 1) ⊕ (10, 6, 6, 1) ⊕ (10, 8, 4, 1)
3
= (8, 8, 4) ⊕ (8, 6, 6), thus
is a subrepresentation of Y . By Proposition 3.3, H̃4 C20
3
Res Y1 ⊕ Res(9, 8, 6) maps surjectively onto Ind H̃4 C20 . Let
3
3
D = H̃5 C22
+ Ind H̃4 C20
− Res Y1 − Res(9, 8, 6).

Then the exact sequence (3.1) becomes
(3.3)

3
3
3
0 → H̃6 C22
→ Res H̃6 C23
→ Ind H̃5 C20
→ D → Res Z → 0

3
where Z = H̃5 C23
− Y1 − (9, 8, 6).
3
contains only irreducible representations whose corBy Proposition 3.4, H̃6 C22
3
responding partitions have 8 rows while H̃5 C22
contains only irreducible representations whose corresponding partitions have at most 6 rows. Thus we have
3
3 6
3 6
0 → H̃6 C22
→ (Res H̃6 C23
) → (Ind H̃5 C20
) →0
3 6
3
3
3 6
3
where (Res H̃6 C23
) = Res H̃6 C23
− (Res H̃6 C23
)6 and (Ind H̃5 C20
) = Ind H̃5 C20
−
3
(Ind H̃5 C20 )6 . This exact sequence determines the irreducible representations of
3
H̃6 C23
whose corresponding partitions have 7 or 8 rows as given in our package
MatchingComplex.m2. Let
3
3
3 6
C = Ind H̃5 C20
+ H̃6 C22
− Res(H̃6 C23
) ,

10

THANH VU

3 6
3
where (H̃6 C23
) is the direct sum of irreducible representations in H̃6 C23
whose corresponding partitions have 7 and 8 rows described above. Then the exact sequence
(3.3) becomes

(3.4)

3
)6 → C → D → Res Z → 0.
0 → Res(H̃6 C23

Finally, using Macaulay2 to compute the dimensions of minimal linear syzygies of
3k+2
the module ⊕∞
(V ) with dim V = 4, we get dim K7,0 (V, 2) = 5400. Morek=0 Sym
3k+2
3
(V ) is
over, H̃4 C23 = 0, thus by Theorem 1.1, K5,2 (V, 2) = 0. Since ⊕∞
k=0 Sym
a Cohen-Macaulay module of codimension 16 with h-vector (10, 16, 1),
 
   
16
16
16
dim K6,1 (V, 2) = 10 ·
− 16 ·
+
− 5400 = 14760.
7
6
5
Since the sum of dimensions of irreducible representations Sλ (V ) corresponding to
partitions λ in Y1 ⊕ (9, 8, 6) is equal to 11520, the sum of dimensions of irreducible
representations Sλ (V ) corresponding to partitions λ in Z is 14760 − 11520 = 3240.
3
This fact and exact sequence (3.4) determine H̃5 C23
as stated in the proposition
3
and H̃6 C23 as given in our package MatchingComplex.m2.

Theorem 3.6. The third Veronese embeddings of projective spaces satisfy property
N6 .
Proof. It remains to prove that the only non-zero homology groups of the matching
3
3
complex C24
is H̃6 C24
. Applying the equivariant long exact sequence (1.1) with
|A| = 24 we have exact sequences
3
0 → Res H̃i C24
→0

for i 6= 5, 6, and an exact sequence
3
3
3
3
3
0 → H̃6 C23
→ Res H̃6 C24
→ Ind H̃5 C21
→ H̃5 C23
→ Res H̃5 C24
→ 0.
3
3
3
.
maps surjectively onto Res H̃5 C24
= 0 for i 6= 5, 6 and H̃5 C23
Therefore, H̃i C24
Moreover, by the result of Ottaviani-Paoletti [OP], the third Veronese embedding of
3
P3 satisfies property N6 . By Theorem 1.1, H̃5 C24
does not contain any irreducible
representations whose corresponding partitions have at most 4 rows. By Proposition
3
contains only irreducible representations whose corresponding partitions
3.5, H̃5 C23
3
is zero.

have at most 4 rows, thus H̃5 C24

Appendix
In this appendix we explain the ideas behind our Macaulay2 package Match3
ingComplex.m2. To determine the homology groups of CN
inductively we use the
equivariant long exact sequence (1.1) with |A| = N to determine two representations B and C of SN −1 satisfying the following property. B is a subrepresentation
3
3
3
of Res H̃i CN
and C is a superrepresentation of Res H̃i CN
. To determine H̃i CN
,
we need to find a representation X of SN satifying the property that Res X is a
subrepresentation of C and a superrepresentation of B. The function findEquation
in our package will first determine a list D of allPpossible partition λ of N such
that Res λ is a subrepresentation of C. Let X = λ∈D xλ · λ where xλ ≥ 0 is the
multiplicity of the partition λ ∈ D that we need to determine. Restricting X we
have inequalities (obtained by calling findEquation B and findEquation C) express3
ing the fact that Res H̃i CN
is a subrepresentation of C and a superrepresentation

N6 PROPERTY FOR THIRD VERONESE EMBEDDINGS

11

of B. We then use maple to solve for non-negative integer solutions of this system
of inequalities.
Acknowledgements
I would like to thank Claudiu Raicu for introducing the problem and its relation
to the homology of matching complexes and my advisor David Eisenbud for useful
conversations and comments on earlier drafts of the paper.
References
[B] C. Birkenhake, Linear systems on projective spaces, Manuscripta Math. 88 (1995), no.2, 177
- 184.
[BCR] W. Bruns, A. Conca and T. Römer, Koszul homology and syzygies of Veronese subalgebras,
Math. Ann. 351 (2011), no.4, 761-779.
[G] M. Green, Koszul cohomology and the geometry of projective varieties I, II, J. Differ. Geometry, 20 (1984), 125-171, 279-289.
[GS] D. Grayson, M. Stillman, Macaulay2, a software system for research in algebraic geometry,
Available at http://www.math.uiuc.edu/Macaulay2/.
[JPW] T. Josefiak, P. Pragacz and J. Weyman, Resolutions of determinantal varieties and tensor
complexes associated with symmetric and antisymmetric matrices, Asterisque, 87-88 (1981),
109-189.
[KRW] D. Karaguezian, V. Reiner and M. Wachs, Matching complexes, bounded degree graph
complexes and weight spaces of GLn complexes, Journal of Algebra 239 (2001), no.1, 77-92.
[OP] G. Ottaviani, R. Paoletti, Syzygies of Veronese embeddings, Compositio. Math. 125 (2001),
no. 1, 31-37.
[R] C. Raicu, Representation stability for syzygies of line bundles on Segre-Veronese varieties,
Arxiv 1209.1183v1 (2012).
[W] J. Weyman, Cohomology of vector bundles and zygyzies, Cambridge Tracts in Mathematics,
vol.149, Cambridge University Press, Cambridge, 2003.
Department of Mathematics, University of California at Berkeley, Berkeley, CA
94720
E-mail address: vqthanh@math.berkeley.edu

