HILBERT SERIES AND DEGREE BOUNDS FOR MATRIX
(SEMI-)INVARIANTS

arXiv:1510.08420v1 [math.RT] 28 Oct 2015

VISU MAKAM
Abstract. We study the ring R(n, m) of invariants for the left-right action of SLn × SLn
on m-tuples of n × n complex matrices. We show that R(3, m) is generated by invariants
of degree ≤ 309 for all m. Then, we use a combinatorial description of the invariants to
show that R(n, m) cannot be generated by invariants of degree < n2 for large m. We also
compute the Hilbert series for several cases.

1. Introduction
We fix our ground field to be C, the field of complex numbers.
1.1. The ring R(n, m). Let Matn,n be the space of n × n matrices. We consider the action
of SLn × SLn on m-tuples of matrices, i.e, Matm
n,n given as follows:
For (A, B) ∈ SLn × SLn , and (X1 , X2 , . . . , Xm ) ∈ Matm
n,n , we have
(A, B) ◦ (X1 , X2 , . . . , Xm ) = (AX1 B −1 , AX2 B −1 , . . . , AXm B −1 ).
SLn × SLn
We are interested in the ring of polynomial invariants C[Matm
, which we denote
n,n ]
].
by R(n, m). This is a graded subring of the polynomial ring C[Matm
n,n

1.2. The quiver perspective. The m-Kronecker quiver is the quiver with 2 vertices x and
y with m-arrows from x to y :
a1
..
y .
x
.
am
The ring R(n, m) is the ring of semi-invariants for the m-Kronecker quiver for the dimension vector (n, n). Semi-invariants for quivers have been studied in [5],[9],[22], and [23],
where they exhibit a C-linear spanning set of the semi-invariants.
Unraveling the theory for our situation, we get a determinantal description of the semiinvariants by the following construction. Let {tpi,j |1 ≤ i, j ≤ k, 1 ≤ p ≤ m} be a set of k 2 m
indeterminates. We describe a kn × kn matrix χk by splitting
P it pinto blocks of size n × n
and describing each block individually. The (i, j)th block is m
p=1 ti,j Xp .
For example,
m
X
χ1 =
tp1,1 Xp
p=1

and

The author was partially supported by NSF grant DMS-1361789.
1



m
P



χ2 =  p=1
m
 P

p=1

tp1,1 Xp
tp2,1 Xp

m
P

p=1
m
P
p=1

tp1,2 Xp
tp2,2 Xp





.


View det(χk ) as a polynomial in the indeterminates tpi,j . The coefficients are expressions in
the entries of the matrices. Further these coefficients are homogeneous polynomials (of degree
kn) in the entries of the matrices, and they are invariant under the action of SLn × SLn .
We get the following :
(1) There are semi-invariants in degree d only if n|d.
(2) The coefficients of det(χk ) form a C-linear spanning set for the invariants of degree
kn.
Remark 1.1. The spanning set described above is usually not a basis, and it is difficult to
extract a basis.
1.3. Degree bounds and Hilbert series.
Definition 1.2. For a rational finite dimensional representation V of a linearly reductive
group G, we define β(C[V ]G ) to be the smallest integer N such that the invariants of degree ≤ N generate C[V ]G .
Computing β(C[V ]G ) is a hard problem in general. There is a general method to get upper
bounds, but it is far from being satisfactory in our case. See [4, Theorem 4.7.4, Proposition
4.7.12, Proposition 4.7.14].
A theorem of Weyl (see [17, Section 7.1,Theorem A]) tells us that β(R(n, m)) ≤ β(R(n, n2 )).
So we make the following definition.
Definition 1.3. βU (n) := β(R(n, n2 )) is called the universal bound.
In some small cases, tight upper bounds have been computed.
• m = 1 : It is easy to see that R(n, 1) = C[det(X1 )], i.e a polynomial ring in one
variable.
• m = 2 : The coefficients of det(t1 X1 + t2 X2 ) span the invariants of degree n. It turns
out that these n + 1 coefficients are algebraically independent and generate R(n, 2).
Hence R(n, 2) is in fact a polynomial ring! These results can be found in [11], [12],
and [18].
The above two situations deal with finite and tame quivers as the 1-Kronecker quiver is
finite and the 2-Kronecker quiver is tame. For m ≥ 3, the situation gets considerably more
complicated. The rings are not polynomial rings anymore.
Before we discuss more previous results, we discuss a few general definitions and results.
Details can be found in [4]. The ring R(n, m) is a finitely generated graded algebra, and
hence has a homogeneous system of parameters (hsop), i.e a set of algebraically independent homogeneous elements f1 , . . . , fd such that R(n, m) is a finitely generated module over
2

C[f1 , . . . , fd ]. Further the Hochster-Roberts theorem (see [4, Theorem 2.5.5],[13]) tells us
that R(n, m) is Cohen-Macaulay, and hence is a free module over any hsop. The free module
generators can be chosen to be homogeneous. The elements of the hsop are called primary
invariants and the free module generators are called secondary invariants. In particular, this
gives a generating set for R(n, m) and hence an upper bound for β(R(n, m)).
Definition 1.4. Hilbert series: The Hilbert series of a graded C-algebra R = ⊕d∈N Rd is
defined as
X
H(R, t) =
dim(Rd )td .
d

In the above definition, N denotes the set of non-negative integers. Observe that a set of
primary and secondary invariants for R(n, m) suffice to compute its Hilbert series, and write
it as a rational function.

Definition 1.5. Multi-graded Hilbert series: If a C-algebra R has a multi-grading, i.e.
R = ⊕d∈Nl Rd , then we define its multi-graded Hilbert series
X
H(R, t1 , t2 , . . . , tl ) =
dim(Rd )td11 td22 . . . tdl l .
d=(d1 ,d2 ,...,dl )∈Nl

Remark 1.6. Any multi-graded algebra can be viewed as a (singly) graded algebra by
considering the total degree. Then one can recover the Hilbert series from the multi-graded
Hilbert series by specializing all the variables ti = t, 1 ≤ i ≤ l, i.e,
H(R, t) = H(R, t, . . . , t).
Example 1.7. Let R = C[x1 , x2 , . . . , xl ] be the polynomial ring in l variables with the
natural multi-grading. Then
H(R, t1 , t2 , . . . , tl ) =

l
Y
i=1

and

H(R, t) = H(R, t, . . . , t) =

1
,
1 − ti
1
.
(1 − t)l

m
The ring R(n, m) is a multi-graded subring of C[Matm
n,n ] and hence has an N -grading.
One can then take total degree to get an N-grading.
• n = 2 : In [7], Domokos computes the multi-graded Hilbert series of R(2, m) by
computing it as an integral. See [4] for a general method to compute multi-graded
Hilbert series by computing integrals. He uses this to deduce βU (2) ≤ 4, and is able
to show that the bound is tight, and hence we get βU (2) = 4.
• n = 3, m = 3 : In [8], Domokos computes the multi-graded Hilbert series of R(3, 3).
This time, he makes use of the fact that R(n, m) ։ S(n, m − 1), where S(n, m − 1)
is the invariant ring of (m − 1)-tuples of n × n matrices under the conjugation action
of GLn . He then uses results of [26] on S(3, 2) to compute the multi-graded Hilbert
series for R(3, 3), by explicitly finding primary and secondary invariants. He gets

3

1 + t31 t32 t33
Q
H(R(3, 3), t1 , t2 , t3 ) = Q
.
( i (1 − ti )3 )( i6=j (1 − t2i tj ))(1 − t1 t2 t3 )(1 − t21 t22 t23 )
On specializing, we get the Hilbert series
1 + t9
.
(1 − t3 )10 (1 − t6 )
From this he deduces that β(R(3, 3)) ≤ 9. Further this set of primary and secondary invariants turns out to be a minimal generating set, and he is able to argue
that β(R(3, 3)) = 9.
The best known general bound is β(R(n, m)) ≤ βU (n) ≤ O(n4 · ((n + 1)!)2 ) (see [14],[15]).
At this point, it is unclear whether one would expect polynomial bounds (in n and m) for
β(R(n, m)).
H(R(3, 3), t) =

1.4. Null Cone. We introduce the null cone, which is an important tool in computational
invariant theory. A good understanding of the null cone could lead to strong degree bounds.
Definition 1.8. Null Cone : For a rational representation V of a linear reductive group G,
the null cone NV is the zero set of all homogeneous invariant polynomials of positive degree
NV = {v ∈ V |f (v) = 0 f or all f ∈ C[V ]G
+ }.
Definition 1.9. γ(C[V ]G ) is defined as the smallest integer D such the invariants of degree ≤ D define the null cone NV .
One can use bounds on γ(R(n, m)) to give bounds on β(R(n, m)) (see [4, Theorem 4.7.4])
and consequently bounds on βU (n) as well. In particular, a polynomial bound on γ(R(n, m))
would give us polynomial bounds for β(R(n, m)) and βU (n).
1.5. Organisation and Results. In Section 2, we exhibit invariants that define the null
cone for R(3, m). We have
Proposition 1.10. The null cone for R(3, m) is defined by a finite set of invariants of degree
≤ 6, namely
• A set of ≤ 9m−16 degree 3 invariants that define the same subvariety as the vanishing
of all degree 3 invariants.
!
Xj Xi
• The degree 6 invariants gi,j,k := det
for 1 ≤ i < j < k ≤ m.
Xi Xk
We can deduce from the proof of Proposition 1.10 that the invariants of degree 6 are
necessary to define the null cone if m ≥ 3.
Corollary 1.11. For m ≥ 3, we have γ(R(3, m)) = 6.
This in turn gives us a hsop.
Proposition 1.12. For m ≥ 3, there exists a set of 9m − 16 invariants of degree 6 that form
a hsop for R(3, m).
4

We then use Proposition 1.10 and Proposition 1.12 to find upper bounds for R(3, m) for
various m in Section 3. In particular,
Proposition 1.13. The ring R(3, m) is generated by invariants of degree ≤ 309 for all m,
i.e, βU (n) ≤ 309.
We introduce a combinatorial description of the invariants in Section 4. This description
helps us get a formula to compute the dimensions of the graded pieces of R(n, m).
Lemma 1.14. The dimenson of R(n, m)kn is given by the computable formula
X
akn ,kn,λ (dim Sλ (Cm )).
dim(R(n, m)kn ) =
λ⊢kn

Here Sλ is the Schur functor corresponding to the partition λ, and aλ,µ,ν denote Kronecker
coefficients, which are known to be invariant under permutation of λ, µ and ν. In this paper,
we use several well known results on Schur functors and symmetric functions, and these can
be found in standard books such as [10], [21], and [24].
We further analyze this combinatorial description in Section 5 to prove that invariants of
degree < n2 cannot generate R(n, m) for m ≥ n2 , i.e,
Theorem 1.15. Suppose m ≥ n2 , then β(R(n, m)) ≥ n2 . In particular βU (n) ≥ n2 .
Finally in Section 6, we use the results in [6] to get denominators of low degree for the
Hilbert series, which makes computations more feasible, and give explicit computations.
2. Null cone for R(3, m)
2.1. Krull dimension of R(n, m). There is a formula for the dimension of the ring of semiinvariants of a quiver for a given dimension vector in terms of the canonical decomposition
of the dimension vector (see [16, Proposition 4]). In the case of the m-Kronecker quiver, the
canonical decomposition of the dimension vector (n, n) is the following :
• m = 1, 2 : The canonical decomposition is (n, n) = (1, 1)⊕n ;
• m ≥ 3 : (n, n) is an imaginary Schur root and its canonical decomposition is trivial,
i.e, (n, n) = (n, n).
The cases for m = 1, 2 have already been dealt with, so we only apply Kac’s formula for
m ≥ 3 to get the following lemma.
Lemma 2.1. If m ≥ 3, then we have dim R(n, m) = mn2 − 2(n2 − 1).
2.2. Invariants defining the null cone. Proposition 1.10 gives a finite set of invariants
that define the null cone. We rely heavily on the results in [8] for proving it.
Proof of Proposition 1.10. Let Z denote the vanishing set of all the degree 3 invariants.
Note that the dimension of R(3, m) is 9m − 16, by Lemma 2.1. Hence, there is a set of
≤ 9m − 16 degree 3 invariants that defines Z, by the Noether normalization lemma (see [4,
Lemma 2.4.7]).
In [8], Domokos analyses the maximal singular matrix spaces in order to compute a hsop
for R(3, 3). We quickly summarize the results which we’ll use.
A singular matrix space is a linear subspace of the space of matrices which does not contain
an invertible matrix. The m-tuples in Z are precisely the m-tuples whose span is a singular
matrix space, by the determinantal description in Section 1.2.
5

In [8], Domokos classifies the maximal singular 3 × 3 spaces as being equivalent to one of
4 types, which are denoted Hi , i = 1, 2, 3, 4 (see [8, Proposition 2.2]). A triple of matrices
belongs to the null cone if and only if it belongs to a maximal singular space of type H1 , H2 ,
or H3 , by [8, Proposition 3.2]. The same proof goes through for an m-tuple of matrices for
any m ≥ 3. Domokos remarks after [8, Proposition 2.2] that any 2-dimensional singular space
is contained in H1 , H2 , or H3 . H4 is the space of skew-symmetric matrices, and in!particular
X2 X1
is a 3-dimensional space. In [8], Domokos shows that the invariant det
does not
X1 X3
vanish on a triple of matrices (X1 , X2 , X3 ) if the span of the triple is equivalent to H4 .
Suppose an m-tuple (X1 , X2 , . . . , Xm ) is in Z, but not in the null cone, then the span
of the m-tuple is equivalent to H4 , and hence 3-dimensional. Hence, there exist 3 matrices
Xi , Xj , Xk which span the space. Hence gi,j,k is an invariant that does not vanish on the
given m-tuple.

Proof of Corollary 1.11. By Proposition 1.10, the invariants of degree ≤ 6 define the null
cone. We observe from the proof of Proposition 1.10, that the degree 3 invariants are not
sufficient to define the null cone if m ≥ 3.

Remark 2.2. The set of invariants in Proposition 1.10 forms a hsop for m = 3, but not for
m ≥ 4 as the number of invariants is larger than the dimension of the ring.
2.3. A hsop for R(3, m), m ≥ 3.
Proof of Proposition 1.12. Recall that dim(R(3, m)) = 9m − 16. Since invariants of degree 3
and degree 6 define the null cone, it is clear that just the set of invariants of degree 6 define
the null cone. By the Noether normalization lemma (see [4, Lemma 2.4.7]), we conclude
that there exists 9m − 16 degree 6 invariants that form a hsop.

3. Upper bounds for β(R(3, m))
We want to bound the degrees of primary and secondary generators, in order to obtain
upper bounds on β(R(3, m)). The following result of Knop is very useful in that regard.
Theorem 3.1 ([20]). Let V be a rational representation of a semisimple connected group G.
Let r be the Krull dimension of C[V ]G , then
deg(H(C[V ]G )) ≤ −r.
In [4], this is used to get the following result.
Proposition 3.2 ([4]). Let V and G be as in the Theorem 3.1. Suppose f1 , f2 , . . . , fl ∈ C[V ]G
are homogeneous invariants that define the null cone. Let di = deg fi . Then
β(C[V ]G ) ≤ max{d1 , d2 , . . . , dl , d1 + d2 + · · · + dl − l}.
There is a stronger result by Knop on the degree of the Hilbert series.
Theorem 3.3 ([19]). Let V be a rational representation of a semisimple connected group G.
Let Z = {v ∈ V | dim Gv > 0}. Then
deg(H(C[V ]G )) = − dim V ⇐⇒ codim(Z) ≥ 2.
6

In [7], the codimension condition was proved for R(n, m) for m ≥ 3, and n ≥ 2. Since
this stronger result on the degree of the Hilbert series holds, we can get a stronger result
by repeating the proof of Proposition 3.2 (see the proof of [4, Corollary 4.7.7]). Lemma 2.1
2
implies that for m ≥ 3, the difference between dim R(n, m) and dim Matm
n,n is 2n − 2. So,
we get
Proposition 3.4. Let m ≥ 3 and n ≥ 2. Suppose f1 , f2 , . . . , fl ∈ R(n, m) are homogeneous
invariants that define the null cone. Let di = deg fi . Then
β(R(n, m)) ≤ max{d1 , d2 , . . . , dl , d1 + d2 + · · · + dl − l − 2n2 + 2)}.
For computing upper bounds for β(R(3, m)), we can apply Proposition 3.4 to the set of
invariants defining the null cone given by either Proposition 1.10 or Proposition 1.12. For
m ≤ 3, tight upper bounds have already been computed. For 4 ≤ m ≤ 6, Proposition 1.10
gives better bounds, whereas for m ≥ 7, Proposition 1.12 gives better bounds. So we get
the following table.
m Upper bound for β(R(3, m))
1

3

2

3

3

9

4

44

5

92

6

160

7

219

8

264

9

309

Proof of Proposition 1.13. As remarked in the introduction, a theorem of Weyl (see [17,
Section 7.1,Theorem A]) gives us β(R(3, m)) ≤ β(R(3, 9)) ≤ 309.

4. Combinatorial description of R(n, m)
In this section, we introduce a combinatorial description of the invariants. This description
has been studied before (see [1] and [2]). We write λ ⊢ d to denote that λ is a partition of
d. We denote by Sλ , the Schur functor corresponding to the partition λ. We identify Matn,n
n
n
m
with Cn ⊗ Cn , and consequently identify Matm
n,n with C ⊗ C ⊗ C . Thus
C[Matm
n,n ]

n

n

m

= C[C ⊗ C ⊗ C ] =

∞
M

Sd (Cn ⊗ Cn ⊗ Cm ).

d=0

Let λ ⊢ d. We have the decomposition
M
(Sµ (V ) ⊗ Sν (W ))aλ,µ,ν ,
Sλ (V ⊗ W ) =
µ,ν⊢d

7

where aλ,µ,ν are known as the Kronecker coefficients. A particular case is the Cauchy
formula,
M
Sλ (V ) ⊗ Sλ (W ).
Sd (V ⊗ W ) =
λ⊢d

Applying the above two decompositions, we get
Sd (V ⊗ W ⊗ Z) =

M

Sλ (V ⊗ W ) ⊗ Sλ (Z)

λ⊢d

=

M

M

(Sµ (V ) ⊗ Sν (W ))

µ,ν⊢d

λ⊢d

=

M

aλ,µ,ν

!

⊗ Sλ (Z)

(Sµ (V ) ⊗ Sν (W ) ⊗ Sλ (Z))aλ,µ,ν .

λ,µ,ν⊢d

This shows in particular that the Kronecker coefficients are invariant under permutating
λ, µ and ν. The above is essentially a decomposition of Sd (V ⊗ W ⊗ Z) as a direct sum of
irreducible representations of GL(V ) × GL(W ) × GL(Z).
Proposition 4.1. The invariants of R(n, m) have the following description.
(1) R(n, m)d = 0 if n ∤ d.


L
n
n
n
m
a
n
Sλ (C ) k ,k ,λ .
(2) R(n, m)kn = Skn (C ) ⊗ Skn (C ) ⊗
λ⊢kn

Proof. We want the polynomials which are invariant under the action of SLn × SLn . SLn
acts trivially on the irreducible representations of GLn corresponding to the rectangular
partitions of length n (i.e the powers of the determinant representation). On all other
irreducible representations, SLn acts with no invariants.
Thus the SLn × SLn invariants of degree d are the summands (Sµ (Cn )⊗Sν (Cn )⊗Sλ (Cm ))aλ,µ,ν
in the decomposition of Sd (Cn ⊗ Cn ⊗ Cm ) where µ, ν are rectangular partitions of length n,
i.e, µ = ν = k n for some k. So, in particular, unless d is a multiple of n, we cannot have any
invariants. This proves (1). For (2),
R(n, m)kn =

M

(Skn (Cn ) ⊗ Skn (Cn ) ⊗ Sλ (Cm ))akn ,kn ,λ

λ⊢kn

= Skn (Cn ) ⊗ Skn (Cn ) ⊗

M

Sλ (Cm )akn ,kn ,λ

λ⊢kn

!

.

Proof of Lemma 1.14. Since Skn (Cn ) is 1-dimensional, as GLm representations, we have
M
R(n, m)kn =
Sλ (Cm )akn ,kn ,λ .
λ⊢kn

Hence we get the formula

dim(R(n, m)kn ) =

X

λ⊢kn

8

akn ,kn ,λ(dim Sλ (Cm )).




Remark 4.2. Let λ, µ ⊢ d. If Tλ (resp. Tµ ) denotes the irreducible representation of the
symmetric group on d letters corresponding to the partition λ (resp. µ), then
M a
Tλ ⊗ Tµ =
Tν λ,µ,ν .
ν

Example 4.3. T1n ⊗ T1n = Tn . Therefore by Lemma 1.14,


m+n−1
m
.
dim(R(n, m)n ) = dim(Sn (C )) =
n

Example 4.4. T23 ⊗ T23 = T6 + T(4,2) + T23 + T(3,1,1,1) . Therefore by Lemma 1.14,
dim(R(3, m)6 ) = dim S6 (Cm ) + dim S(4,2) (Cm ) + dim S23 (Cm ) + dim S(3,1,1,1) (Cm ).
5. Lower bounds for β(R(n, m))
Let R ⊂ C[W ⊗V ] be a GL(V ) stable graded subring. Then each Rd is a finite dimensional
GL(V ) representation, and we can decompose it as a direct sum of irreducibles, i.e,
M
(Sλ (V ))nλ , nλ ∈ N.
Rd =
λ⊢d

Note here that as GL(V ) representations, the k th exterior power
positive integers k.

Vk

(V ) is S1k (V ) for all

Proposition 5.1. Let R ⊂ C[W ⊗ V ] be a GL(V ) stable graded subring. Assume
V
(1) Vi (V ) does not occur in the decomposition of Ri , for i = 1, 2, ..., t − 1 ;
(2) t (V ) occurs in the decomposition of Rt at least once ;
(3) dim V ≥ t.
Then β(R) ≥ t.
Proof. We have a GL(V ) equivariant map Ri ⊗ Rt−i → Rt given by multiplication. We can
collect the maps for various i to get a map
⌊t/2⌋

ϕ:

M

Ri ⊗ Rt−i → Rt .

i=1

It is clear that if R is generated by invariants of degree < t, then ϕ is surjective.
Let λ ⊢ a and µ ⊢ b. Recall the well known identity
ν

Sλ (V ) ⊗ Sµ (V ) = ⊕ν (Sν (V ))cλ,µ ,
where cνλ,µ are the Littlewood-Richardson coefficients. By the Littlewood-Richardson rule,
V
if a+b (V ) = S1a+b (V ) appears in the decomposition for Sλ (V ) ⊗ Sµ (V ), then λ = 1a and
µ = 1b .
Vi
We assume that
for
1
≤
i
≤
t
−
1,
(V ) = S1i (V ) does not occur in the decomposition
Vt
for Ri . Hence, (V ) does not occur in the decomposition for Ri ⊗ Rt−i , and hence does not
L
of its
occur in the decomposition for ⌊t/2⌋
i=1 Ri ⊗ Rt−i , and thus not in the decomposition
Vt
image under ϕ. But in the decomposition
of Rt , there is at least one copy of (V ). Since
V
dim V ≥ t, we are guaranteed that t (V ) is non-empty. So ϕ cannot be surjective.
9

Thus R cannot
V be generated in degree < t as the invariants corresponding to the isotypic
component for t (V ) cannot be generated by smaller degree invariants.

Proof of Theorem 1.15. We want to apply Proposition 5.1 to the ring R(n, m), via the combinatorial description in Section 4. Take W = Cn ⊗ Cn and V = Cm . Then by the results in
Section 4, we have that
L

(Sλ (V ))akn ,kn ,λ if i = kn,
Ri = λ⊢kn
0
otherwise.
From the representation theory of the symmetric group, we know that Tkn ⊗ T1kn = Tnk .
Moreover, since the Kronecker coefficients are invariant under permutations, we have
(
0 if k 6= n,
akn ,kn ,1kn =
1 if k = n.

This gives the first two conditions required for Proposition 5.1 (for t = n2 ) . Since we
assume dim V = m ≥ n2 , the third condition holds as well. Hence we have β(R(n, m)) ≥ n2 .

In fact, we can describe explicitly these invariants in degree n2 . For a matrix M, denote
by M , a column matrix obtained by stacking the columns of M.
 
a
 
!
c
a b
 
, then M =   .
Example 5.2. If M =
b
c d
 
d
Define a function f on n2 -tuples of n × n matrices by


f (X1 , X2 , . . . , Xn2 ) = det X1 X2 . . . Xn2 .

Then f ∈ R(n, n2 )n2 is the unique invariant (upto scalars) in the isotypic component
V 2
2
corresponding to n (Cn ). For n = 2, this is the invariant of degree 4 constructed in [7].

Remark 5.3. Theorem 1.15 gives a tight lower bound for n ≤ 2, i.e βU (n) = n2 for n ≤ 2.
6. Computing Hilbert series

We have seen in Section 1 that for the cases m = 1, 2, R(n, m) is a polynomial ring. It
is also clear that R(1, m) is a polynomial ring since SL1 is trivial. For R(2, m), the Hilbert
series has already been computed by Domokos in [7]. So, we restrict to the cases m ≥ 3,
n ≥ 3. Notice that for these cases, we have deg H(R(n, m), t) = − dim Matm
n,n , as discussed
in Section 3.
Remark 6.1. If we can compute a denominator for the Hilbert series of R(n, m), then we
can compute the polynomial in the numerator once we know the dimensions of the graded
pieces of R(n, m) upto the degree of the numerator, which is given by
deg(Numerator) = deg(Denominator) − n2 m.
10

Remark 6.2. Computing dim R(n, m)kn is a hard task even with a computer, and is the
bottleneck for these computations. So, it is desirable to minimize the degree of the numerator
as much as possible, and hence it is desirable to minimize the degree of the denominator.
Fortunately, the theory of universal denominators (see [3], [6]) gives us strong results in
our case. We first renormalize our grading to agree with the grading in [6].
Definition 6.3. The renormalized Hilbert series is defined as
∞
X
e
H(R(n,
m), t) =
dim R(n, m)kn tk .
k=0

Remark 6.4. The usual Hilbert series and the renormalized Hilbert series are related by
e
H(R(n,
m), tn ) = H(R(n, m), t).
The most relevant result is [6, Corollary 1]. We restate it for our situation.

Proposition 6.5 ([6]). Let r be the Krull dimension of R(n, m). Then
P (t)
e
H(R(n,
m), t) =
,
(1 − t)r
where P (t) is a polynomial with integer coefficients.
This gives us denominators of the lowest degree possible, making several computations
accessible. Domokos proved a functional equation for the Hilbert series of R(n, m) for m ≥ 3,
n ≥ 2 in [7]. This implies that when we use the universal denominator, the coefficients of
the polynomial in the numerator are palindromic, so we need to compute only half the
coefficients. In view of Remarks 6.1-6.2, this makes a few more computations feasible.
We give a few explicit computations that we are able to compute.
1 − t + t2
e
. This was already computed by Domokos in [8]. We remark
(1) H(R(3,
3), t) =
(1 − t)11
that even though β(R(3, 3)) = 9, we only needed the dim(R(3, 3)3) to compute the
Hilbert series.
1 + 20t2 + 20t3 + 55t4 + 20t5 + 20t6 + t8
e
(2) H(R(3,
4), t) =
.
(1 − t)20

P (t)
, where the coefficients of P (t) are 1, 6, 141, 931, 4816,
(1 − t)29
13916, 27531, 33391, 27531, 13916, 4816, 931, 141, 6, 1.

e
(3) H(R(3,
5), t) =

P (t)
, where the coefficients of P (t) are 1, 18, 626, 10246, 114901,
(1 − t)38
830484, 4081260, 13763184, 32507115, 54176230, 64224060, 54176230, 32507115,
13763184, 4081260, 830484, 114901, 10246, 626, 18, 1.

e
(4) H(R(3,
6), t) =

P (t)
, where the coefficients of P (t) are 1, 37, 2033, 62780,
(1 − t)47
1301634, 18067706, 173883458, 1186198090, 5851715254, 21192401230, 57013957462,
114926408114, 174616665986, 200665719450, 174616665986, 114926408114, 57013957462,

e
(5) H(R(3,
7), t) =

11

21192401230, 5851715254, 1186198090, 173883458, 18067706, 1301634, 62780, 2033,
37, 1.
1 − 3t + 9t2 + 8t3 + 9t4 − 3t5 + t6
e
(6) H(R(4,
3), t) =
.
(1 − t)18

P (t)
, where the coefficients of P (t) are 1, 1, 141, 981, 8534,
(1 − t)34
39193, 139348, 325823, 556368, 652716, 556368, 325823, 139348, 39193, 8534, 981,
141, 1, 1.

e
(7) H(R(4,
4), t) =

P (t)
, where the coefficients of P (t) are 1, -6, 36, -70, 231, -189,
(1 − t)27
419, -189, 231, -70, 36, -6, 1.

e
(8) H(R(5,
3), t) =

Acknowledgements. I would like to thank my advisor Harm Derksen for suggesting the
problem and his guidance throughout the project. I would also like to thank J. Stembridge
for useful conversations. For the computations in this paper, I used J. Stembridge’s SF
package (see [25]).
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