Table Of ContentA canonical system of basic invariants of a finite
3
reflection group
1
0
2
n Norihiro Nakashima∗and Shuhei Tsujie†
a
J
4
1
] Abstract
T
R A canonical system of basic invariants is a system of invariants satisfying
. certain differential equations. Some researchers studied properties of a canon-
h
t ical system related to the mean value property for polytopes. In this article,
a
m we naturally identify the vector space spanned by a canonical system of basic
invariants with an invariant space determined by a fundamental alternating
[
polynomial. From this identification, we give explicit formulas of canonical
3
v systems of basic invariants. Theconstruction of theformulas does notdepend
6 on the classification of irreducible finite reflection groups.
2
0
6 Key Words: basic invariants, invariant theory, finite reflection groups.
.
1 2010 Mathematics Subject Classification: Primary 13A50; Secondary
1
20F55.
2
1
:
v
1 Introduction
i
X
r
a Let V be a real n-dimensional Euclidean space, and W ⊆ O(V) a finite reflection
group. Let S denote the symmetric algebra S(V∗) of the dual space V∗, and S the
k
vector space consisting of homogeneous polynomials of degree k. Then W naturally
acts on S and S . According to Chevalley [2], the subalgebra R = SW of W-
k
invariant polynomials of S is generated by n algebraically independent homogeneous
polynomials. A system of such generators is called a system of basic invariants
of R. It is easy to construct systems of basic invariants for reflection groups of
∗email: naka [email protected]
†email: [email protected]
1
types A , B , D and I . Many researchers constructed an explicit system of basic
n n n 2
invariants for a reflection group of each type (Coxeter [3], Mehta [12], Saito, Yano,
and Sekiguchi [14], and Sekiguchi and Yano [15, 16])
Let v ,...,v be an orthonormal basis for V, x ,...,x the basis for V∗ dual
1 n 1 n
to v ,...,v , and ∂ ,...,∂ the basis for V∗∗ dual to x ,...,x . Since V is a
1 n 1 n 1 n
inner product space, we can identify V∗∗ with V naturally. To avoid a confusion, we
distinguish the basis ∂ ,...,∂ from the basis v ,...,v . We define an inner product
1 n 1 n
h·,·i : S ×S → R by
hf,gi = f(∂)g| (f,g ∈ S), (1.1)
x=0
where x = (x ,...,x ) and ∂ = (∂ ,...,∂ ). Define a bilinear map (·,·) : S×S → S
1 n 1 n
by
(f,g) = f(∂)g (f,g ∈ S). (1.2)
Two systems g ,...,g andh ,...,h ofbasicinvariantsaresaidtobeequivalent
1 n 1 n
if there exists A ∈ GL (R) such that
n
[h ,...,h ] = [g ,...,g ]A.
1 n 1 n
Definition 1.1. A system f ,...,f of basic invariants is said to be canonical if it
1 n
satisfies the following system of partial differential equations:
(f ,f ) = hf ,f iδ (1.3)
i j i j ij
for i,j = 1,...,n where δ is the Kronecker symbol.
ij
A canonical system was introduced by Flatto [4, 5] and Flatto and Wiener [6] for
determining the structure of the linear space consisting of P(0)-harmonic functions,
where P(k) is a k-skeleton of an n-dimensional polytope P for a nonnegative integer
0 ≤ k ≤ n−1. (An R-valued continuous function is called a P(k)-harmonic function
if it satisfies the mean value property on P(k).) Flatto [4, 5] and Flatto and Wiener
[6] verified that there exists a unique (up to equivalence) canonical system of basic
invariants for any finite reflection group, and gave an algorithm. In [4, 5, 6], a
canonical system of basic invariants was treated as a solution of a certain system of
partial differential equations. The formulation above is due to Iwasaki [9].
Explicit formulas of canonical systems play an important part in the study for
determining the structure of the linear space consisting of P(k)-harmonic functions.
Especially, from the argument using explicit formulas of canonical systems, the
2
structure of the linear space consisting of P(k)-harmonic functions was determined
for any 0 ≤ k ≤ n−1 when P is a regular convex polytope (see [8, 10, 11]). (When
P is a regular convex polytope, the symmetry group of P is a finite reflection group
of type A, B, F, H, or I.)
However, the algorithm for constructing a canonical system (given by Flatto
[4, 5] and Flatto and Wiener [6]) does not seem to be effective in practice. (It is
hard to give an explicit formula of a canonical system from the algorithm.) It is
an interesting problem to determine canonical systems. Iwasaki [9] gave explicit
formulas of canonical systems for reflection groups of types A , B , D and I .
n n n 2
Iwasaki, Kenma and Matsumoto [11] gave explicit formulas of canonical systems for
the irreducible finite reflection groups of types F , H and H . The problem for
4 3 4
determining canonical systems of basic invariants for the remaining types (E , E
6 7
and E ) has been open. In this article, we explicitly construct canonical systems
8
of basic invariants from an arbitrary system of basic invariants (Theorem 3.5 and
Theorem 3.11).
In the case of the types except D with even n, we can uniformly construct the
n
canonical systems from an arbitrary system of basic invariants (Theorem 3.5). We
need to consider the case of type D with even n separately, because the degrees of
n
two invariants in a system meet at n. Iwasaki [9] constructed a canonical system of
basic invariants containing the monomial x ···x . By using the monomial x ···x ,
1 n 1 n
wecanalso construct acanonical system fromanarbitrarysystem ofbasicinvariants
(Theorem 3.11).
Let Φ be the root system associated with W, and Φ+ a positive system. For α ∈
Φ, fix a homogeneous polynomial L of degree 1 defining the reflecting hyperplane
α
H (i.e., kerL = H ). A polynomial f ∈ S is said to be alternating if wf =
α α α
(detw)f for all w ∈ W. Set ∆ = L , the product of polynomials L , then
α∈Φ+ α α
∆ is an alternating polynomial. We call ∆ the fundamental polynomial. Set
Q
R[∂]∆ := {f(∂)∆ | f ∈ S}.
We naturally identify the vector space spanned by a canonical system of basic in-
variants with (R[∂]∆ ⊗R V∗)W (Lemma 2.3). To determine canonical systems, we
investigate R[∂]∆. It is known that the graded vector space R[∂]∆ affords the reg-
ular representation of W (see Bourbaki [1] and Steinberg [17]). This is one of the
keys for determining canonical systems.
3
2 Characterization of the canonical systems
Let R be the ideal of R generated by homogeneous elements of positive degrees,
+
and I = SR the ideal of S generated by R .
+ +
The following lemma is obtained by Steinberg [17].
Lemma 2.1. Let f ∈ S be a homogeneous polynomial. Then we have the following:
(1) f ∈ I if and only if f(∂)∆ = 0,
(2) g(∂)f = 0 for any g ∈ I if and only if f ∈ R[∂]∆,
(3) I is the orthogonal complement of R[∂]∆, and S = I ⊕R[∂]∆.
It is known that a W-stable graded subspace U of S such that S = I ⊕ U is
isomorphic to the regular representation (see Bourbaki [1, Chap. 5 Sect. 5 Theorem
2]). ByLemma 2.1, thegradedvector space R[∂]∆ affordsthe regular representation
of W.
Assume that W is irreducible and V is generated by the roots. Then any endo-
morphism of V is a multiplicative map with a constant in R, and V is absolutely
irreducible (see Bourbaki [1, Chap. 5, Sect. 2, Proposition 1] and [1, Chap. 5, Sect.
3, Proposition 5]). Therefore, by the general theory of group representations, we
have that the multiplicity of V in the regular representation is dimV = n. Note
that there exist a W-module U′ such that U = VdimV ⊕U′.
Let M be a finite dimensional W-module, and set U := U ∩S for any nonneg-
k k
ative integer k. Let {v′,...,v′} be a basis of M, and {x′,...,x′} the basis of M∗
1 r 1 r
dual to {v ,...,v }. Define a linear map
1 r
θ : (Uk ⊗R M∗)W −→ HomW(M,Uk) (2.1)
by θ( u ⊗ x)(v) = x(v)u for any u ⊗ x ∈ (Uk ⊗R M∗)W and v ∈ M, and
define a linear map
P P P
η : HomW(M,Uk) −→ (Uk ⊗R M∗)W (2.2)
by η(φ) = r φ(v′)⊗x′ for φ ∈ Hom (M,U ). It immediately follows from the
j=1 j j W k
correspondence above that
P
(Uk ⊗R M∗)W ≃ HomW(M,Uk). (2.3)
In section 3, we use the isomorphisms θ (2.1) and η (2.2).
4
Lemma 2.2. The vector space (U ⊗R V∗)W has dimension n.
Proof. Since V is absolutely irreducible, the proof goes similarly to [13, Lemma
6.45].
Lemma 2.2 gives us a method for constructing a basis for (U ⊗ V∗)W by the
representation theory. Ifwe completely determine homomorphisms inHom (V,U ),
W k
then we have an R-basis for (U ⊗R V∗)W.
To determine canonical systems of basic invariants, we characterize the systems.
Let f ,...,f be a canonical system of basic invariants (Flatto [4, 5] and Flatto and
1 n
Wiener [6]). Since {f1,...,fn}islinearlyindependent, thevector space hf1,...,fniR
has dimension n. The vector space hf1,...,fniR does not depend on a choice of a
canonical system, since a canonical system isuniquely determined upto equivalence.
Lemma 2.3. Let f ,...,f be a canonical system of basic invariants. Define a
1 n
linear map
ρ : hf1,...,fniR −→ (R[∂]∆⊗V∗)W (2.4)
by ρ(f ) = n ∂ f ⊗x for i = 1,...n. Then the map ρ is an isomorphism from
i j=1 j i j
the vector sPpace hf1,...,fniR to the vector space (R[∂]∆⊗V∗)W:
hf1,...,fniR ≃ (R[∂]∆⊗V∗)W.
Proof. For f ∈ SW, r ∂ f ⊗x is W-invariant.
j=1 j j
For i,j,k = 1,...,n, we have f (∂)∂ f = 0. Then we have by Lemma 2.1
k j i
that ∂ f ∈ R[∂]∆ foPr i,j = 1,...,n. By Lemma 2.2, dim(R[∂]∆ ⊗ V∗)W = n =
j i
dimhf1,...,fniR. To prove that ρ is an isomorphism, we only need to prove that ρ
is injective.
Let f be an R-linearly combination of f ,...,f with ρ(f) = 0. Since ρ(f) =
1 n
n ∂ f⊗x , we have ∂ f = 0 for all j = 1,...,n. The degree of f is greater than
j=1 j j j i
0 for i = 1,...,n. This implies f = 0. Hence ρ is injective.
P
3 Determining canonical systems
In this section, we assume that W is an irreducible finite reflection group and V is
generated by the root system. Then the representation V is absolutely irreducible.
Let v ,...,v be an orthonormal basis for V, x ,...,x the basis for V∗ dual to
1 n 1 n
v ,...,v , and ∂ ,...,∂ the basis for V∗∗ dual to x ,...,x .
1 n 1 n 1 n
Let f ,...,f be a canonical system of basic invariants.
1 n
5
Proposition 3.1. The W-module
Mi := h∂1fi,...,∂nfiiR ⊆ R[∂]∆ (3.1)
is isomorphic to V as a W-module for i = 1,...,n.
Proof. Define a W-homomorphism
σi : V = hv1,...,vniR −→ Mi = h∂1fi,...,∂nfiiR (3.2)
by σ (v ) = ∂ f for j = 1,...,n. We only need to prove kerσ = {0}.
i j j i i
Assume that σ (v ) = 0 for j = 1,...,n. This implies that ∂ f = 0 for all
i j j i
j = 1,...,n. Then we have that the polynomial f is a constant. This contradicts
i
degf > 0. Therefore we have kerσ = {0}.
i i
Let m ≤ m ≤ ··· ≤ m be the degrees of elements of a system of basic
1 2 n
invariants (which do not depend on a choice of a system of basic invariants). If W
is not of type D with even n (n ≥ 4), then the degrees are distinct: m < m <
n 1 2
··· < m . If W is of type D with even n (n ≥ 4), then the degrees are the numbers
n n
2,4,...,n,n,...,2n−2 (see [7, sect. 3.7 Table 1]).
3.1 Types except D with even n
n
In this subsection, we assume that W is not of type D with even n (n ≥ 4). Then
n
m < m < ··· < m . We may assume degf = m for i = 1,...,n.
1 2 n i i
Put (R[∂]∆)k := R[∂]∆∩Sk, and hf1,...,fnik := hf1,...,fniR∩Sk for any non-
negative integer k. Since the graded vector space R[∂]∆ is isomorphic to the regular
representation, the multiplicity of V in R[∂]∆ is dimV = n. We can find n W-
submodules which are isomorphic to the W-module V in homogeneous components
of R[∂]∆. By the correspondence (2.3) and Lemma 2.3, we have
Rf if k = m −1,
Hom (V,(R[∂]∆) ) ≃ hf ,...,f i = i i
W k 1 n k+1
({0} otherwise.
Hence the multiplicity of V in the W-module (R[∂]∆) is equal to 1 if k = m −1,
k i
and 0 if k 6= m − 1 (i = 1,...,n). By Lemma 2.1 and Proposition 3.1, we have
i
V ≃ M ⊆ (R[∂]∆) .
i mi−1
Define a W-homomorphism
φ : S −→ R[∂]∆, φ(f) := ((f,∆),∆) for f ∈ S (3.3)
where (·,·) is the bilinear map (1.2). Then kerφ = I by Lemma 2.1, and hence the
map φ induces an automorphism of R[∂]∆ by Lemma 2.1 again.
6
Lemma 3.2. The map φ induces an automorphism of M .
i
Proof. There exists a W-submodule M′ such that (R[∂]∆) = M ⊕ M′ by
i mi−1 i i
Mascke’s theorem. The projection π : (R[∂]∆) −→ M′ is a W-homomorphism.
mi−1 i
The multiplicity of V in (R[∂]∆) is 1, hence π ◦φ| = 0 by Schur’s lemma.
mi−1 Mi
Let u ∈ M , and write φ(u) = x + y for some x ∈ M and y ∈ M′. Since
i i i
y = π(x + y) = π ◦ φ(u) = 0, we have φ(u) = x ∈ M . Therefore the image
i
φ(M ) is included in M . Since the restriction φ| is injective, then we have
i i R[∂]∆
φ(M ) = M .
i i
By Lemma 3.2, we may regard φ| as the automorphism φ| : M −→ M .
Mi Mi i i
Hence, by [1, Chap. 5, Sect. 2, Proposition 1], there exists a scalar c ∈ R× such
i
that φ| = c 1. The argument above give us the following proposition.
Mi i
Proposition 3.3. Fix i = 1,...,n. Every nonzero element in M is an eigenvector
i
of φ with a nonzero eigenvalue. The eigenvalue does not depend on a choice of the
elements, and depends only on index i.
Theorem 3.4. Let h ,...,h bean arbitrarysystem ofbasic invariantswithdegh =
1 n i
m . Then there exists a scalar λ ∈ R× such that
i i
φ(∂ h ) = λ ∂ f
j i i j i
for j = 1,...,n. Therefore we have
φ(h∂1hi,...,∂nhiiR) = Mi.
Proof. Since h ,...,h and f ,...,f are systems of basic invariants, we may write
1 n 1 n
h = d f +P(f ,...,f ) (3.4)
i i i 1 i−1
for i = 1,...,n, where d ∈ R× and P(f ,...,f ) is a polynomial in f ,...,f .
i 1 i−1 1 i−1
Note that each term of P is a product of at least two polynomials of f ,...,f .
1 i−1
Applying ∂ on the equality (3.4), we have
j
∂ h = d ∂ f +∂ P.
j i i j i j
Since ∂ is a derivation, for each term T of ∂ P, there exists f (i = 1,...,i − 1)
j j k
and a polynomial F such that T = Ff . Then the polynomial ∂ P is in the ideal I,
k j
and we have φ(∂ P) = 0 by Lemma 2.1. Hence we obtain
j
φ(∂ h ) = d φ(∂ f ) = d c ∂ f
j i i j i i i j i
by Proposition 3.3. We complete the assertion by setting λ := d c
i i i
7
Let h ,...,h be a system of basic invariants with degh = m . By Theorem
1 n i i
3.4, for i = 1,...,n, there exists a nonzero scalar λ ∈ R× such that
i
φ(∂ h ) = λ ∂ f . (3.5)
j i i j i
Recall the W-isomorphism (3.2)
σ : V −→ M , σ (v ) = ∂ f (j = 1,...,n)
i i i j j i
for i = 1,...,n. Since the W-module M is a unique submodule of (R[∂]∆)
i mi−1
which is isomorphic to V, we have
Hom (V,M ) ≃ Hom (V,(R[∂]∆) ).
W i W mi−1
We also recall the isomorphisms η (2.2) and ρ (2.4):
HomW(V,(R[∂]∆)mi−1) −−η−→ ((R[∂]∆)mi−1 ⊗R V∗)W −−ρ−−1→ Rfi.
Then we have the following correspondence:
HomW(V,Mi) −−η→ (Mi ⊗R V∗)W −−ρ−−1→ Rfi,
σ ←→ n ∂ f ⊗x ←→ f = 1 n x ∂ f .
i j=1 j i j i mi j=1 j j i
P P
Hence we obtain the following explicit formula of a canonical system of basic invari-
ants by (3.5).
Theorem 3.5. Let h ,...,h be a system of basic invariants with degh = m .
1 n i i
Then the system
n n
x φ(∂ h ),..., x φ(∂ h )
j j 1 j j n
j=1 j=1
X X
form a canonical system of basic invariants, where φ(∂ h ) = (((∂ h ),∆),∆) for
j i j i
i = 1,...,n with the bilinear map (·,·) (1.2).
8
3.2 Type D with even n (n ≥ 4)
n
Let n = 2ℓ (ℓ ≥ 2). In this subsection, let W be the irreducible finite reflection
group of type D . We may write that
n
∆ = x2i −x2j = caxa (3.6)
1≤i<j≤n a∈Nn
Y (cid:0) (cid:1) X
for some coefficients ca ∈ R. If a multi-index a has a component with odd number,
then the coefficient ca in (3.6) is equal to 0.
Iwasaki [9] constructed a canonical system of basic invariants which contains the
monomial f = x ···x . We choice a canonical system f ,...,f ,f ,...,f of
ℓ+1 1 n 1 ℓ ℓ+1 n
basic invariants which contains the monomial f = x ···x . We may assume that
ℓ+1 1 n
degf < ··· < degf = degf < ··· < degf . Then we have degf = degf =
1 ℓ ℓ+1 n ℓ ℓ+1
ℓ. Note that the W-modules
Mℓ = h∂1fℓ,...,∂nfℓiR and Mℓ+1 = h∂1fℓ+1,...,∂nfℓ+1iR. (3.7)
are included in the same homogeneous component (R[∂]∆) .
ℓ−1
We denote |a| := a +···+a for a multi-index a = (a ,...,a ) ∈ Nn. We write
1 n 1 n
a > b if a > b for all k = 1,...,n. Put 1 := (1,...,1) ∈ Nn. Let e ∈ Nn be the
k k j
j-th unit vector of Nn.
Theorem 3.6. The monomial ∂ f is an eigenvector of φ with a nonzero eigen-
j ℓ+1
value for j = 1,...,n. The eigenvalue does not depend on a choice of index j. In
particular, the map φ induces an automorphism of M .
ℓ+1
Proof. Let a and b be multi-indices with |a| = |b| = deg∆. Assume that all
components of the multi-indices a and b are even, and write
a = (2a ,...,2a ) and b = (2b ,...,2b ).
1 n 1 n
For j = 1,...,n, we obtain a sequence of equivalent conditions: ((∂ f ,xa),xb) 6=
j ℓ+1
0; if and only if a+e > 0 and 2b −2a + 1 ≥ 0 for k 6= j and 2b −2a ≥ 0; if
j k k j j
and only if a+e > 0 and b ≥ a for k = 1,...,n. It follows from |a| = |b| that
j k k
((∂ f ,xa),xb) 6= 0 if and only if a+e > 0 and a = b. Hence we conclude that
j ℓ+1 j
λ∂ f if a+e > 0, a = b,
a b j ℓ+1 j
((∂ f ,x ),x ) =
j ℓ+1
(0 otherwise,
for some λ ∈ R×.
9
By (3.6), the polynomial φ(∂ f ) = ((∂ f ,∆),∆) is an R-linearly combina-
j ℓ+1 j ℓ+1
tion of polynomials in
{((∂ f ,xa),xb) | a = (2a ,...,2a ),b = (2b ,...,2b ) ∈ Nn}.
j ℓ+1 1 n 1 n
If a 6= b, then the polynomial ((∂ f ,xa),xb) is equal to 0. Hence there exists a
j ℓ+1
scalar c ∈ R such that
φ(∂ f ) = c∂ f ∈ M .
j ℓ+1 j ℓ+1 ℓ+1
This implies φ(M ) ⊆ M .
ℓ+1 ℓ+1
The scalar c does not depend on achoice of j by [1, Chap. 5, Sect. 2, Proposition
1]. Since φ| : M −→ M is injective, then we have c 6= 0.
Mℓ ℓ+1 ℓ+1
To verify that the polynomials ∂ f (j = 1,...,n) are eigenvector of φ, we
j ℓ
introduce three lemmas.
Lemma 3.7. There exist scalars d ,d ∈ R such that
1 2
φ(∂ f ) = d ∂ f +d ∂ f
j ℓ 1 j ℓ 2 j ℓ+1
for j = 1,...,n. The scalars d ,d do not depend on a choice of j.
1 2
Proof. There exists a W-stable vector space M′ such that the W-module R[∂]∆
ℓ−1
is decomposed into the direct sum of W-modules M , M and M′:
ℓ ℓ+1
R[∂]∆ = M ⊕M ⊕M′. (3.8)
ℓ−1 ℓ ℓ+1
Let π be the projection from R[∂]∆ to M′. Since the multiplicity of V in
3 ℓ−1
R[∂]∆ is two, the homomorphism π ◦φ| is 0. Hence we have by the decompo-
ℓ−1 3 Mℓ
sition (3.8) that φ(M ) is included in M ⊕M .
ℓ ℓ ℓ+1
Let π be the projection from (R[∂]∆) to M . The map π ◦ φ| is en-
1 ℓ−1 ℓ 1 Mℓ
domorphism of M . Thus we have that there exists a scalar d ∈ R such that
ℓ 1
π ◦φ| = d ·1 .
1 Mℓ 1 Mℓ
Let us consider the projection π from R[∂]∆ to M . Recall maps σ :
2 ℓ−1 ℓ+1 ℓ
V −→ M and σ : V −→ M defined by
ℓ ℓ+1 ℓ+1
σ (v ) = ∂ f , σ (v ) = ∂ f
ℓ j j ℓ ℓ+1 j j ℓ+1
for j = 1,...,n. The maps σ and σ are W-isomorphisms. Any endomorphism
ℓ ℓ+1
of V is a multiplicative map. Thus there exists a scalar d ∈ R such that σ−1 ◦π ◦
2 ℓ+1 2
10
