ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
ADAM S. SIKORA
Abstract. We find finite, reasonably small, explicit generator sets of the
coordinate rings of G-character varieties of finitely generated groups for all
classical groups G. This result together with the method of Gröbner basis
gives an algorithm for describing character varieties by explicit polynomial
equations.
For a reductive group G over a field K, denote the G-character variety of a
finitely generated group Γ by XG (Γ). For every γ ∈ Γ and for every representation φ : G → GL(n, K), there is a regular map τφ,γ : XG → K sending the
equivalence class of ρ : Γ → G to tr(φρ(γ)). We prove that functions of this
form do not generate the coordinate rings of SO(2n, K)-character varieties.
We discuss the generating sets over fields of arbitrary characteristics.
1. Character Varieties
We will assume throughout the paper that G is an affine reductive group over
an algebraically closed field K of characteristic zero and that Γ is a (discrete) group
generated by γ1 , ..., γN . The space of all G-representations of Γ forms an algebraic
subset, Hom(Γ, G), of GN , called the G-representation variety of Γ. The group G
acts on this set by conjugating representations and the categorical quotient of that
action
XG (Γ) = Hom(Γ, G)//G
is the G-character variety of Γ, c.f. [S2] and the references within.
With Hom(Γ, G) and XG (Γ), there are naturally associated algebraic schemes
Hom(Γ, G) and XG (Γ) = Hom(Γ, G)//G such that the coordinate rings, K[Hom(Γ, G)]
and K[XG (Γ)], are nil-radical quotients of the algebras of global sections K[Hom(Γ, G)]
and K[XG (Γ)], c.f. [S2].
Due to ubiquitous applications of character varieties in low-dimensional topology, geometry, gauge theory, and quantum field theories one is interested in an
explicit description of them by polynomial equations, or, equivalently a description
of K[X(Γ, G)] and of K[X (Γ, G)] by generators and relations.
In this paper we describe generating sets of these rings for all classical G and all
Γ.
We do not discuss here the second part of the problem: finding complete sets of
relations between generator sets. An algorithmic solution to this problem is given
by the theory of Gröbner basis. (However, due to its computational complexity,
this method may be difficult to be employed in practice.)
Let T be a maximal torus in G.
Key words and phrases. character variety, moduli space.
1
2
ADAM S. SIKORA
Example 1. ([St, 6.4], [S2, Example 42]) If G is connected then XG (Z) = T /W,
where T /W is the quotient of T by the action of the Weyl group of G. Consequently,
K[XG (Z)] is isomorphic to the representation ring of G.
For a matrix group G ⊂ GL(n, K) we define τγ : XG (Γ) → K by τγ (ρ) = tr(ρ(γ)).
Throughout this paper we will consider the classical groups as specific matrix
groups:
O(n, K) = {A ∈ GL(n, K) : AAT = I},
SO(n, K) = O(n, K) ∩ SL(n, K),
and Sp(n, K) = {A ∈ GL(n, K) : AJAT =J} where n is even and J is a non0
In/2
degenerate skew-symmetric matrix,eg. J =
.
−In/2
0
Our main results are Theorems 6,8,9, 10 giving efficient descriptions of coordinate
rings of character varieties. In particular, they imply the following:
Theorem 2. (1) For G = SL(n, K), Sp(n, K), O(n, K), for any n and for G =
SO(n, K) for n odd, K[XG (Γ)] is generated by τγ , where subscript gammas are
±1
words in γ1±1 , ..., γN
length at most n2 . (For G = O(n, K), SO(n, K), the choice of
specific J as above, does not matter.)
(2) For all n for which Kuzmin’s conjecture holds, c.f. [Ku], (in particular, for
n ≤ 4), it is enough to consider such words of length at most n+1
2 .
(3) If Γ is abelian, then it is sufficient to consider above words of length at most n
only.
(4) For G = SL(n, K) it is enough to consider words γ of length specified above,
composed of positive powers of γ1 , ..., γN only.
Because of the epimorphism
K[XG (Γ)] K[XG (Γ)]
any generating set of K[XG (Γ)] is a generating set of K[XG (Γ)] as well.
For example, parts (3) and (4) of the above theorem imply that K[XSL(2,K) (Γ)]
is generated by
τγi , τγi γj , τγi γj γk ,
for i, j, k ∈ {1, ..., N }.
The following is left as an exercise for the reader:
Proposition 3. If Γ is generated by γ1 , ..., γN and K[XSL(n,K) (Γ)] is generated
by τγ , for gammas in some set B ⊂ Γ then K[XGL(n,K) (Γ)] is generated by τγ ,
for γ ∈ B and by the functions σ1 , ..., σN , where σi (ρ) = det(ρ(γi ))−1 . (Since the
algebraic group GL(n, K) is composed of elements of (M, d) ∈ M (n, K) × K such
that det(M ) · d = 1, the function d = det−1 is regular on GL(n, K).)
For even n, the generators of XSO(n,K) (Γ) are more difficult to describe. Consider
a function Qn : M (n, K)n/2 → K, on the Cartesian product of n/2 n × n matrix
algebras, given for n/2 matrices A, B, ..., Z by
P
Qn (A, B, ..., Z) = σ∈Sn sn(σ)(Aσ(1),σ(2) −Aσ(2),σ(1) )(Bσ(3),σ(4) −Bσ(4),σ(3) )...
(1)
(Zσ(n−1),σ(n) − Zσ(n),σ(n−1) ),
where Mj,k is the (j, k)-th entry of a matrix M and sn(σ) = ±1 is the sign of σ.
ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
3
Observe that Qn is a multi-linear, symmetric function such that Qn (X, ..., X) is
2n/2 (n/2)! times the Pfaffian of X −X T . A function with these properties is unique.
It is the “full polarization” of 2n/2 (n/2)!P f (X − X T ).
The function Hom(Γ, SO(n, K)) → K sending ρ to Qn (ρ(γ1 ), ..., ρ(γn )) is a polynomial function invariant under conjugation by elements of SO(n, K). Therefore, it
factors to a function on XSO(n,K) (Γ) which we denote by Qn (γ1 , ..., γn/2 ).
Theorem 4. For n even, K[XSO(n,K) (Γ)] is generated by τγ where γ’s are words
of length at most n2 and by Qn (w1 , ..., wn/2 ), for all possible words w1 , ..., wn/2 of
length at most n2 − 1 in which the number of inverses is not larger than half the
length of the word.
Functions τγ do not generate K[XSO(n,K) (Γ)] alone. Consider, for example, m =
1 and Γ = Z. Let ρ, ρ0 : Z → SO(2, K) send 1 to
1
x + x−1
i(x − x−1 )
A=
∈ SO(2, K)
−i(x − x−1 )
x + x−1
2
and to AT , respectively, for some x 6= ±1. Since Q2 (ρ) = 4(x − x−1 ) 6= Q2 (ρ0 ), [ρ]
and [ρ0 ] are distinct points of XSO(2) (Z). However [ρ] and [ρ0 ] are not distinguished
by τγ for any γ ∈ Z.
In fact, we believe that a much stronger result holds. For every representation
φ : G → GL(n, K) and for every γ ∈ Γ,
τγ,φ ([ρ]) = tr(φρ(γ))
defines a regular function τγ,φ : XG (Γ) → K.
Conjecture 5. For G = SO(2m, K), m > 1, and every free group Γ of rank ≥ 1,
K[XG (Γ)] is not generated by τγ,φ for all γ ∈ Γ and all G-representations φ.
Acknowledgments: We would like to thank S. Lawton for helpful conversations. He has proved versions of some of the results of this paper as well, independently of us.
2. SL(n)-Character Varieties
The result of this section (albeit technical) provides a very efficient system of
generators of K[XSL(n,K) (Γ)].
We say that a set of generators s1 , ..., sN of a semigroup S is homogenous if there
is a homomorphism of semigroups
deg : S → N = {1, 2, ...}
called the degree map such that deg(si ) = 1 for all i. (Note that such S never has
an identity.)
Let Sd = {γ ∈ S : deg(γ) ≤ d}. Let I be the two-sided ideal in KS generated by
elements z n for all z ∈ KS. Let B ⊂ S be such that B ∩ Sd spans KSd /KSd ∩ I for
all d ∈ N. Let Br ⊂ B be the set of those elements of B which are words in s1 , ..., sr
for 1 ≤ r ≤ N.
Theorem 6. Let S be a semigroup with a homogeneous generating set s1 , ..., sN .
If η : S → Γ is an epimorphism of semigroups, then K[XSL(n,K) (Γ)] is generated by
4
ADAM S. SIKORA
τγ , for γ ∈ η(B), where
B = {s1 , ..., sN } ∪
N
[
Br · sr ,
r=1
for any B as above.
In practice, B can be constructed inductively by taking B0 = ∅ and Bd+1 to be
a minimal set of elements of Sd+1 containing Bd and spanning KSd+1 /KSd+1 ∩ I.
Every group Γ is an image of a semigroup S with a homogenous generating set.
Indeed, one can take S to be a free semigroup on s1 , ..., s2N and send s1 , ..., sN
to the generators of Γ and sN +1 , ..., s2N to their inverses. However, sometimes a
smaller S exists.
The nilpotency index, ν(A), of an algebra A is the smallest integer r such that the
product of every r elements of A vanishes. By Nagata-Higman theorem, ν(KS/I)
is finite. Furthermore, by [Ra] (page 759 in English translation),
ν(KS/I) ≤ n2 .
Therefore, one can take B to be the sets of all elements in S of degree at most
n2 − 1. This implies part (3) of Theorem 2 and, hence, part (1) for G = SL(n, K)
as well. Kuzmin’s conjecture, stating that
n+1
ν(KS/I) =
2
for all n, [Ku], implies part (2) of Theorem 2 for G = SL(n, K). His conjecture
holds for n ≤ 4, [Du, VL].
Corollary 7. (1) For every Γ generated by γ1 , ..., γN , K[XSL(2,K) (Γ)] is generated
by τγi , for i = 1, .., N, τγi γj for i < j and τγi γj γk , for i < j < k.
(2) If Γ is an abelian then K[XSL(2,K) (Γ)] is generated by τγi , for i = 1, .., N, and
τγi γj for i < j.
Proof. (1) Let S be the free semigroup on s1 , ..., s2N and let η : S → Γ send
−1
s1 , ..., sN to γ1 , ..., γN and s1 , ..., sN to γ1−1 , ..., γN
. Since the nilpotency index of
KS/I is 3 for n = 2, we can take B = {s1 , ..., s2N , si sj , 1 ≤ i, j ≤ 2N }.
si sj + sj si = s2i + si sj + sj si + s2j = (si + sj )2 = 0
implies sj si = −si sj in KS/I. Therefore, we can assume i < j for si sj in B and,
hence,
B = {si , for i = 1, ..., 2N, si sj , for i ≤ j, si sj sk , for i < j ≤ k}.
τα−1
Since
= τα and ταβ −1 = τα τβ − ταβ , Theorem 6 yields the generators of the
Corollary, with the possibility of j = k. However, since ταβ 2 = τβ ταβ + τα for all
α, β ∈ Γ, the statement follows.
(2) Let S be the free abelian semigroup on generators s1 , ..., s2N and let η : S → Γ
be as above. Since
s2i + si sj + sj si + s2j = (si + sj )2 = 0,
sj si = 0 in KS/I. By taking B = {s1 , ..., s2N }, we get
B = {si , i = 1, .., N, si sj , for i ≤ j}.
Since τγ 2 =
τγ2 − 2
we can assume that i < j in η(B) and the statement follows.
ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
5
3. Sp(n, K)- and O(n, K)-character varieties
We say that a semigroup S has an involution ∗ : S → S iff s∗∗ = s and (st)∗ =
t∗ s∗ for all s, t ∈ S. We say that s1 , ..., sN in S are homogeneous generators of a
semigroup with involution S iff s1 , ..., sN , s∗1 , ..., s∗N are homogeneous generators of
S as a semigroup. In particular, deg(s) = deg(s∗ ) then for every s ∈ S.
Let S be a semigroup with involution with homogeneous generators s1 , ..., sN .
As before, let I be the two sided ideal in KS generated by the n-th powers of all
elements. Let Sd = {s ∈ S : deg(s) ≤ d}. Let B ⊂ S be such that B ∩ Sd spans
KSd /KSd ∩ I for all d ∈ N. Let Br ⊂ B be the set of those elements of B which are
words in s1 , ..., sr , s∗1 , ..., s∗r for 1 ≤ r ≤ N.
Every group Γ is a semigroup with an involution γ ∗ = γ −1 .
Theorem 8. Let G = Sp(n, K), O(n, K) for any n or G = SO(n, K) for odd n.
Then for every epimorphism of semigroups with involution η : S → Γ, K[XG (Γ)] is
generated by τγ , for γ ∈ η(B), where
B = {s1 , ..., sN } ∪
N
[
Br · sr ,
r=1
for any B as above.
Theorem 8 implies Theorem 2 for G = Sp(n, K), O(n, K) and G = SO(n, K) for
n odd. Indeed, if S is the free semigroup on s1 , ..., sN , s∗1 , ..., s∗N then, as before,
one can take B to be the sets of all elements in S of degree at most n2 − 1.
Theorem 8 can be strengthened further for symplectic groups. For n even, let
I s be the two sided ideal in KS generated by the n/2-th powers of all symmetric
elements, (x+x∗ )n/2 , x ∈ KS. Let B s ⊂ S be such that B s ∩Sd spans KSd /KSd ∩I s
for all d ∈ N. Let Brs ⊂ B be the set of those elements of B which are words in
s1 , ..., sr , s∗1 , ..., s∗r .
Theorem 9. For every epimorphism of semigroups with involution η : S → Γ,
K[XSp(n,K) (Γ)] is generated by τγ , for γ ∈ η(B s ), where
B s = {s1 , ..., sN } ∪
N
[
Brs · sr ,
r=1
for any B s as above.
By [P2, Thm 8.10.2], I ⊂ I s . Therefore, any set B as above is also a B s set.
For G = SO(n, K) for n even the description of generators is more complicated.
Let M ⊂ B be such that {s − s∗ : s ∈ M} spans the space {s − s∗ : s ∈ KB} In
particular, one can take M to be the subset of B composed of elements s0i1 ....s0id
where s0i is either si or s∗i and the number of stared si ’s is not larger than the
number of the non-stared ones.
Theorem 10. Let n be even. For every epimorphism of semigroups with involution
η : S → Γ, K[XSO(n,K) (Γ)] is generated by τγ for γ ∈ η(B) defined in Theorem 8
and by Qn (µ1 , ..., µn/2 ), for all possible µ1 , ..., µn/2 ∈ η(M). (Qn was defined by
(1)).
Since KS/I is spanned by monomials in s1 , ..., sN , s∗1 , ..., s∗N of degree at most
n − 1, Theorem 4 follows.
Recall that there are two representations D+ , D− of SO(n, K) which ...
2
6
ADAM S. SIKORA
The representation ring of SO(n, K) is generated by the defining rep and D+ , S− .
They satisfy the following relations.
Q is a multilinearization or polarization of D+ − D− .
4. Proof of Theorem 6
L∞
Recall that a K algebra R is graded if R =
k=0 Rk as a vector space and
Rk · Rl ⊂ Rk+l . An element of R is homogeneous if it belongs to Rk for some k. All
graded algebras in this paper are connected, i.e. R0 = K. An element r ∈ R has
Ld
L
degree d if d is the smallest index such that r ∈ k=0 Rk . We will denote k>0 Rk
by R+ . The following lemma will be useful
Lemma 11. If R is a graded ring with a given set of its generators which are
homogeneous elements, then every element of degree d in R+ · R+ is a polynomial
in generators of degree < d.
Proof. By splitting elements of R+ into
Psums of homogenous elements, every element of R+ · R+ can be written as r = i si · ti , where all si , ti are homogeneous of
positive degree. After
P eliminating all summands such that deg si + deg ti > deg r,
the equality r = i si · ti still holds and deg si , deg ti < deg r for all i. Now the
statement follows from the fact that each homogeneous element x is a polynomial
in homogeneous generators of degree ≤ deg(x).
Remark 12. Throughout the paper we will often use the following fact: If a reductive group G acts on K-algebras A and B such that an epimorphism φ : A → B
is G-equivariant then φ restricts to a G-equivariant epimorphism AG → B G . This
follows from the existence of Reynolds operators ∇ : A → AG and ∇ : B → B G
which are are onto and from the commutativity of the following diagram:
φ
A →
∇↓
AG
φ
→
B
↓∇
BG
Let Cn,N = K[ai,j,k : i = 1, ..., N, j, k = 1, ..., n]. (The letter “C” stands for the
fact that this will be our ring of coefficients.) Let F SG(s1 , ..., sN ) be the free semigroup on these generators. By “abstract nonsense” argument there exists a unique
universal quotient, P (S), of Cn,N such that the homomorphism of semi-groups
Ψ : F SG(s1 , ..., sN ) → M (n, Cn,N )
sending s1 , ..., sN to
A1 = (a1,∗,∗ ), ..., AN = (aN,∗,∗ ) ∈ M (n, Cn,N )
composed with the natural projection M (n, Cn,N ) → M (n, P (S)) factors to a homomorphism
(2)
Ψ : S → M (n, P (S)),
which we denote by the same letter, Ψ. (“Universal” means that every other such
quotient factors through this one.) Here is a concrete construction of P (S) : as
every semi-group, S has a presentation
S = hs1 , ..., sN | r1,i = r2,i , i ∈ Ii .
Then P (S) is the quotient of Cn,N by the ideal generated by n2 polynomial entries
of the matrix Ψ(r1,i ) − Ψ(r2,i ) taken for every i ∈ I.
ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
7
Since s1 , ..., sN are homogeneous generators of S, deg(r1,i ) = deg(r2,i ) for each i.
Therefore, the grading on Cn,N descends to a grading on P (S). (Hence the image
of every ai,j,k in P (S) has degree 1.) This grading will be important later. Indeed,
its existence is the reason we considered homogenous generators of Γ in the first
place.
In [S2], we have defined the universal representation algebras R(Γ, G) for algebraic groups G. (See also [S1], for G = SL(n, K).) By their definition,
R(Γ, G) = K[Hom(Γ, G)].
By the universality of P (S), we have a natural homomorphism P (S) → R(Γ, G).
Let G = SL(n, K). Since Cn,N is the coordinate ring of M (n, K)N (the Cartesian
product of N copies of M (n, K)), the SL(n, K) action on M (n, K)N by conjugation,
induces an SL(n, K) action on Cn,N . This action descends to an action on P (S)
and on R(Γ, G). Since by the definition of categorical quotient
R(Γ, G)G = K[Hom(Γ, G)]G = K[XG (Γ)],
we have an epimorphism
P (S)G → R(Γ, G)G = K[XG (Γ)].
by Remark 12.
By abuse of notation, denote the images of A1 , ..., AN under the projection
M (n, Cn,N ) → M (n, P (S)) by the same symbols. Let T (S) be the subalgebra
of P (S) generated by the traces of monomials in A1 , ..., AN ∈ M (n, P (S)). Clearly,
T (S) ⊂ P (S)G .
Lemma 13. T (S) = P (S)G .
G
Proof. By [P1, P2], Cn,N
is generated by the traces of monomials in A1 , ..., AN ∈
M (n, Cn,N ). Now the statement follows from the fact that P (S) is a G-equivariant
quotient of Cn,N and from Remark 12.
By abuse of notation, denote the set of monomials in A1 , ..., AN in M (n, P (S))
corresponding to the elements of B ⊂ S via the map (2) by the same symbol, B.
In order for the above discussion to imply Theorem 6, we need to show that
T (S) is generated by traces of monomials in B. We are going to do it by induction
on the degree.
Proposition 14. For every monomial M of degree d > 1 in variables A1 , ..., AN ,
tr(M ) belongs to the K-subalgebra of P (S)G generated by the traces of monomials
in Ψ(B) and by the traces of monomials in A1 , ..., AN of degree < d.
Proof. Identify P (S) with the scalar matrices in M (n, P (S)). Let S(S) be the
T (S)-subalgebra of M (n, P (S)) generated by the matrices A1 , ..., AN .
Note that T (S) is a graded subalgebra of P (S). Additionally M (n, P (S)) is a
graded algebra, with a matrix having degree k iff all its entries are of degree k in
P (S). In particular, deg Ai1 ...Aik = k. S(S) is a graded subalgebra of M (n, P (S)).
Define T (S)+ ⊂ T (S) and S(S)+ ⊂ S(S) to be the subalgebras without identity
spanned by the elements of positive degree. Hence,
T (S) = T (S)+ ⊕ K and S(S) = S(S)+ ⊕ K.
The homomorphism (2) restricts to an epimorphism
Ψ : KS → S(S)+
8
ADAM S. SIKORA
sending si to Ai .
Lemma 15.
Ψ(I) ⊂ T (S)+ S(S).
Proof. Every element of Ψ(I) is a sum of elements XY n Z, where X, Z ∈ S(S) and
Y ∈ S(S)+ . The matrix Y ∈ S(S)+ satisfies its characteristic polynomial
n
Y +
n−1
X
ci Y i = 0,
i=0
with c0 , ..., cn−1 ∈ T (S), since they are conjugation invariant. Furthermore, they
belong to T (S)+ since each ci is homogeneous of degree n − i in the entries of
Y ∈ S(S)+ . Hence,
XY n Z = −
n−1
X
ci XY i Z ∈ T (S)+ S(S).
i=0
Let M = Ai1 ....Aid , for some i1 , .., id ∈ {1, ..., N }, d > 1. Let r = M ax{i1 , ..., id }.
Since T r(Ai1 ...Aid ) is invariant under a cyclic permutation of Ai1 , ...., Aid , we can
assume that id = r. By the definition of B,
X
(3)
si1 ...sid−1 =
cs · s + C,
s∈B, deg s≤d−1
where cs ∈ K and C ∈ I.
By multiplying both sides of (4) by sid , applying Ψ and then taking trace, we
get
X
T r(M ) = T r(Ai1 ...Aid ) =
cs · T r(Ψ(s)Ar ) + T r(Ψ(C)Ar ).
s∈B, deg s≤d−1
Note that Ψ(s)Ar ∈ B for s ∈ B and that T r(Ψ(C)Ar ) is an element of degree
≤ d in T (S). Furthermore, by Lemma 15,
T r(Ψ(C)Ar ) ∈ T r(T (S)+ S(S)Ar ) = T r(T (S)+ S(S)+ ) =
T (S)+ · T r(S(S)+ ) = T (S)+ · T (S)+ .
Since the traces of monomials in A1 , ..., AN are homogeneous generators of T (S),
Lemma 11 implies that T r(Ψ(C)Ar ) is polynomial in traces of monomials of degree
< d. This completes the proof of Theorem 14.
5. Proof of Theorem 8
The proof is an adaptation of the proof of Theorem 6. Let G = Sp(n, K),
O(n, K), for any n or G = SO(n, K) for odd n.
As before, let F SG(s1 , ..., sN , s∗1 , ..., s∗N ) be the free semigroup on s1 , ..., sN , s∗1 , ..., s∗N
(and, hence, the free semigroup with involution on s1 , ..., sN ). Consider M (n, Cn,N )
as a semigroup with the symplectic involution, A∗ = JAT J −1 , for G symplectic and
with the orthogonal involution, A∗ = AT , for G orthogonal and special orthogonal.
Let
Ψ : F SG(s1 , ..., sN , s∗1 , ..., s∗N ) → M (n, Cn,N )
ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
9
be a homomorphism of semigroups with involution sending si to Ai = (ai,∗,∗ ) Let
P (S) be the universal quotient of Cn,N such that Ψ composed with M (n, Cn,N ) →
M (n, P (S)) factors to
Ψ : S → M (n, P (S)).
Since S has a homogenous set of generators, the grading on Cn,N descends to a
grading on P (S).
As before, we have a natural epimorphism P (S) → R(Γ, G) = K[Hom(Γ, G)].
The G action on M (n, K)N by conjugation defines a G action on Cn,N , which
descends to an action on P (S) and on Rep(Γ, G). By Remark 12,
P (S)G → R(Γ, G)G = K[XG (Γ)].
is an epimorphism.
By abuse of notation, denote the images of A1 , ..., AN , A∗1 , ..., A∗N ∈ M (n, Cn,N )
in M (n, P (S)) (under the natural projection) by the same symbols. Let T (S) be
the subalgebra of P (S) generated by the traces of monomials in these matrices.
Then T (S) = P (S)G , by [P1, Thm 10.1], [P2, Sec 11.8.2].
Therefore, in order to establish Theorem 8 it is enough to prove that T (S) is
generated by traces of monomials in B. That will follow by induction on the degree
of monomials, from the following proposition. (The degree of a monomial is the
number of components Ai and A∗i in it.)
Proposition 16. For every monomial M of degree d in variables A1 , ..., AN , A∗1 , ..., A∗N ,
tr(M ) belongs to the K-subalgebra of T (S) generated by the traces of monomials in
Ψ(B) and by the traces of monomials of degree < d.
The proof of this proposition is almost identical to that of Proposition 14.
Let M = A0i1 ....A0id , for some i1 , .., id ∈ {1, ..., N }, d > 1, where A0i is either Ai
or A∗i . Let r = M ax{i1 , ..., id }. Since tr(M ) = tr(M ∗ ), one can replace M with
M ∗ without loss of generality. Therefore, we can assume that there is at least one
component Ar (without the star) in that monomial. Since T r(Ai1 ...Aid ) is invariant
under a cyclic permutation of its components, we can assume that id = r. By the
definition of B,
X
(4)
si1 ...sid−1 =
cs · s + C,
s∈B, deg s≤d−1
where cs ∈ K and C ∈ I.
By multiplying both sides of (4) by sid , applying Ψ and then taking trace, we
get
X
T r(M ) = T r(A0i1 ...A0id ) =
cs · T r(Ψ(s)Ar ) + T r(Ψ(C)Ar ).
s∈B, deg s≤d−1
The completion of the proof is identical to that of Proposition 14.
6. Proof of Theorem 9
The proof is identical to that of Theorems 8, except that one replaces Lemma
15 with the following one:
Lemma 17.
Ψ(I s ) ⊂ T (S)+ S(S).
10
ADAM S. SIKORA
Proof. Every element of Ψ(I s ) is a sum of elements X(Y + Y ∗ )n/2 Z, where X, Z ∈
S(S) and Y ∈ S(S)+ . For n even and M ∈ M (n, K) invariant under the symplectic
involution,
P fM (λ) = P f ((λI − M )J)
is called the characteristic Pfaffian of M. (Here I is the identity matrix and J the
skew-symmetric matrix defined before.)
Let
n−1
X
n
P fM (λ) = λ +
ci λi
i=0
be the characteristic Pfaffian of M = X + X ∗ . Since every matrix invariant under
the symplectic involution satisfies its characteristic Pfaffian equation,
P fM (M ) = 0,
c.f. [P2, Sec 11.8.7]. The coefficients c0 , ..., cn−1 belong to P (S)G = T (S), since
they are conjugation invariant. Furthermore, they belong to T (S)+ since each ci is
homogeneous of degree n − i in the entries of M ∈ S(S)+ .
Hence,
n−1
X
X(Y + Y ∗ )n Z = −
ci X(Y + Y ∗ )i Z ∈ T (S)+ S(S).
i=0
7. Proof of Theorem 10
Let G = SO(2n). Let ... be defined us in the proof of Theorem 8. Let
T (S) be the subalgebra of P (S) generated by the traces of monomials in matrices A1 , ..., AN , A∗1 , ..., A∗N ∈ M (n, P (S)). Unlike for other classical groups, T (S) ⊂
P (S)G is usually a proper embedding.
The proof of Theorem 8 implies that T (S) is generated by the traces of matrices
in Ψ(B). (Note that Lemma 15 holds here – since c0 , ..., cn−1 are not only SO(n, K)
invariant but also O(n, K) invariant, they belong to T (S).)
Proposition 18. (1) P (S)G is generated by T (S) and by the values of Qn (M1 , ..., Mn/2 )
for all M1 , ..., Mn/2 in S(S).
(2) It is enough to consider M1 , ..., Mn/2 ∈ B only.
Proof. (1) By Remark 12, the natural projection M (n, Cn,N )G → P (S)G is onto.
Therefore, it is enough to prove that statement for the semigroup M (n, Cn,N ) with
involution M ∗ = M T . That was done in [P2, Sec 11.8.2].
We will prove (2) by contradiction: Denote by T 0 (S) the subalgebra of P (S)G
generated by T (S) and by the values of Qn (M1 , ..., Mn/2 ) for M1 , ..., Mn/2 ∈ B
only. Assume that T 0 (S) 6= T (S). Let (d1 , ..., dn/2 ) ∈ Nn/2 be the smallest element
in the lexicographic order such that there exist monomials M1 , ..., Mn/2 of degrees
d1 , ..., dn/2 such that Qn (M1 , ..., Mn/2 ) 6∈ T 0 (S). We can assume that at least one
Mi is not in Ψ(B). Let us assume that it is the first one for the simplicity of notation.
Abbreviate d1 to d. Then M1 = A0i1 ...A0id . By definition of B,
X
s0i1 ...s0id =
cs · s + C,
s∈B, deg s≤d
where C ∈ I.
ALGEBRAIC DESCRIPTION OF CHARACTER VARIETIES
11
By applying Ψ we get
(5)
M1 =
X
cs · Ψ(s) + Ψ(C).
s∈B, deg s≤d
By Lemma 15,
(6)
Ψ(C) =
X
Tj · Xj ,
j
where Tj ∈ T + (S), Xj ∈ S(S). Furthermore, by splitting each Tj , Xj into a sum of
homogeneous summands if necessary, we can assume that all Tj , Xj are homogeneous with respect to the grading on T (S) and on S(S). Finally, we can remove all
summands Tj Xj of degree greater than d from the right side of (6) without loss of
validity of that equation. Therefore, (5) combined with (6) expresses M1 as a sum
of terms of the form T X where T ∈ T (S), X ∈ S(S) and and either X ∈ Ψ(B)
or deg X < d. Since elements of T (S) are scalar matrices in S(S) and Qn is multilinear, Qn (M1 , ..., Mn/2 ) can be expressed as a sum of terms T ·Qn (X, M2 , ..., Mn/2 )
with deg X < deg M1 . Therefore, Qn (X, M2 , ..., Mn/2 ) 6∈ T 0 (S), for at least one such
X of degree < d – contradicting the initial assumption.
By the above proposition P (S)G is generated by T (S) and by Qn (M1 , ..., Mn/2 )
for M1 , ..., Mn/2 ∈ KB. Since the substitution of Mi by MiT changes sign in
Qn (x1 , ..., xn/2 , it is enough to consider M1 , ..., Mn/2 in the skew-symmetric part
of KB only. Elements {M − M T : M ∈ Ψ(M)} span that space. Finally, since
the substitution of Mi by Mi − MiT multiplies Qn (x1 , ..., xn/2 by 2, it is enough to
consider M1 , ..., Mn/2 ∈ Ψ(M).
References
[ADS] H. Aslaksen, V. Drensky, L. Sadikova, Defining relations of invariants of two 3×3 matrices,
J. of Algebra 298 (2006) 41–57.
[BD] F. Benanti, V. Drensky, Defining relations of minimal degree of the trace algebra of 3 × 3
matrices, arXiv: math/0701609
[Bo] A. Borel, Linear Algebraic Groups, 2nd enlarged ed., Springer 1991.
[D1] S. Donkin, Invariants of Unipotent Radicals, Math. Z. 198 (1988) 117–125.
[D2] S. Donkin, Invariants of several matrices, Invent. math. 110 (1992) 389–401.
[Dr] V. Drensky, Computing with matrix invariants, arXiv:math/0506614
[DS] V. Drensky, L. Sadikova, Generators of invariants of two 4 × 4 matrices,
arXiv:math/0503146v2
[Du] J. Dubnov, Sur une généralisation de léquation de Hamilton-Cayley et sur les invariants
simultanés de plusieurs affineurs, Proc. Seminar on Vector and Tensor Analysis, Mechanics
Research Inst., Moscow State Univ. 2/3 (1935), 351–367.
[Fo] J. Fogarty, Invariant Theory, W. A. Benjamin, Inc. 1969.
[FH] W. Fulton, J. Harris, Representation Theory, A First Course, Graduate Texts in Mathematics, Springer, 1991.
[Hu] J.E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Springer 1975.
[Ku] E.N. Kuzmin, On the Nagata-Higman theorem, in Mathematical Structures- Computational
Mathematics-Mathematical Modeling, Proceedings dedicated to the sixtieth birthday of academican L. Iliev, Sofia, 1975, 101–107.
[Lo] A.A. Lopatin, Relatively free algberas with the identity x3 = 0, Comm. in Algebra 33 3583–
3605.
[LM] A. Lubotzky, A. Magid, Varieties of representations of finitely generated groups, Memoirs
of the AMS 336 (1985).
[MFK] D. Mumford, J. Fogarty, F. Kirwan Geometric Invariant Theory, Springer-Verlag 1994.
[P1] C. Procesi, The invariant theory of n × n matrices, Advances in Math. 19 (1976), no. 3,
306–381.
12
ADAM S. SIKORA
[P2] C. Procesi, Lie Groups, An Approach through Invariants and Representations, Universitext,
Springer 2007.
[Ra] Y. P. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Izv.
Akad. Nauk SSSR, 38 (1974) 723-756; English transl., Math. USSR-Izv. 8 (1974) 727-760.
[S1] A. S. Sikora, SLn -Character Varieties as Spaces of Graphs, Trans. Amer. Math. Soc., 353
(2001), 2773–2804.
[S2] A. S. Sikora, Character Varieties, Trans. of AMS, to appear, arXiv:0902.2589
[St] R. Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ.
Math. 25 (1965), 49-80.
[VL] M. Vaughan-Lee, Michael, An algorithm for computing graded algebras, J. Symbolic Comput. 16 (1993), no. 4, 345-354.
Dept. of Mathematics, 244 Math. Bldg.
University at Buffalo, SUNY
Buffalo, NY 14260, USA
asikora@buffalo.edu