Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials
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| Cite this: | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials/ F. Butin // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1321–1332. — Бібліогр.: 7 назв. — англ. |
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| citation_txt | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials/ F. Butin // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1321–1332. — Бібліогр.: 7 назв. — англ. |
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UDC 512.5
F. Butin (Univ. Lyon, France)
BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C
AND THE RELATED GENERALIZED POINCARÉ POLYNOMIALS
ЗАКОН ГАЛУЖЕННЯ ДЛЯ СКIНЧЕННИХ ПIДГРУП SL4C
ТА ВIДПОВIДНI УЗАГАЛЬНЕНI ПОЛIНОМИ ПУАНКАРЕ
Within the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finite-
dimensional irreducible representations of SL4C decompose under the action of Γ.
Let h be a Cartan subalgebra of sl4C and let$1, $2, $3 be the corresponding fundamental weights. For (p, q, r) ∈ N3,
the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p$1 + q$2 + r$3 of SL4C decomposes
as πp,q,r|Γ =
⊕l
i=0 mi(p, q, r)γi, where {γ0, . . . , γl} is the set of equivalence classes of irreducible finite-dimensional
complex representations of Γ. We determine the multiplicities mi(p, q, r) and prove that the series
PΓ(t, u, w)i =
∞∑
p=0
∞∑
q=0
∞∑
r=0
mi(p, q, r)t
puqwr
are rational functions.
This generalizes the results of Kostant for SL2C and the results of our preceding works for SL3C.
У рамках вiдповiдностi Маккея для кожної скiнченної пiдгрупи Γ групи SL4C визначено, яким чином скiнченно-
вимiрне незвiдне зображення SL4C розкладається пiд дiєю Γ.
Нехай h — картанова пiдалгебра sl4C, а $1, $2, $3 — вiдповiднi фундаментальнi ваги. Для (p, q, r) ∈ N3
звуження πp,q,r|Γ незвiдного зображення πp,q,r найбiльшої ваги p$1 + q$2 + r$3 в SL4C розкладається у виглядi
πp,q,r|Γ =
⊕l
i=0 mi(p, q, r)γi, де {γ0, . . . , γl} — множина класiв еквiвалентностi незвiдних скiнченновимiрних
комплексних зображень Γ. Визначено кратностi mi(p, q, r) та доведено, що ряди
PΓ(t, u, w)i =
∞∑
p=0
∞∑
q=0
∞∑
r=0
mi(p, q, r)t
puqwr
є рацiональними функцiями.
Це є узагальненням результатiв Костанта для SL2C, а також результатiв наших попереднiх робiт для SL3C.
1. Introduction and results. Let Γ be a finite subgroup of SL4C and {γ0, . . . , γl} the set of
equivalence classes of irreducible finite dimensional complex representations of Γ, where γ0 is the
trivial representation. The character associated to γj is denoted by χj .
Consider γ : Γ → SL4C the natural 4-dimensional representation, and γ∗ its contragredient
representation. The character of γ is denoted by χ. By complete reducibility we get the decompositions
∀ j ∈ [[0, l]] : γj ⊗ γ =
l⊕
i=0
a
(1)
ij γi, γj ⊗ (γ ∧ γ) =
l⊕
i=0
a
(2)
ij γi and γj ⊗ γ∗ =
l⊕
i=0
a
(3)
ij γi.
This defines the three following square matrices of Ml+1N:
A(1) :=
(
a
(1)
ij
)
(i,j)∈[[0, l]]2
, A(2) :=
(
a
(2)
ij
)
(i,j)∈[[0, l]]2
and A(3) :=
(
a
(3)
ij
)
(i,j)∈[[0, l]]2
.
Let h be a Cartan subalgebra of sl4C and let $1, $2, $3 be the corresponding fundamental
weights, and V (p$1 + q$2 + r$3) the simple sl4C-module of highest weight p$1 + q$2 + r$3
c© F. BUTIN, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10 1321
1322 F. BUTIN
with (p, q, r) ∈ N3. Then we get an irreducible representation πp,q,r : SL4C → GL(V (p$1 +
+ q$2 + r$3)). The restriction of πp,q,r to the subgroup Γ is a representation of Γ, and by complete
reducibility, we get the decomposition
πp,q,r|Γ =
l⊕
i=0
mi(p, q, r)γi,
where the mi(p, q, r)’s are non negative integers. Let E := (e0, . . . , el) be the canonical basis of
Cl+1, and
vp,q,r :=
l∑
i=0
mi(p, q, r)ei ∈ Cl+1.
We have in particular v0,0,0 = e0 as γ0 is the trivial representation. Let us consider the vector
PΓ(t, u, w) :=
∞∑
p=0
∞∑
q=0
∞∑
r=0
vp,q,rt
puqwr ∈ (C[[t, u, w]])l+1,
and denote by PΓ(t, u, w)j its j th coordinate in the basis E , which is an element of C[[t, u, w]].
Note that PΓ(t, u, w) can also be seen as a formal power series with coefficients in Cl+1. The aim
of this article is to prove the following theorem.
Theorem 1. The coefficients of PΓ(t, u, w) are rational fractions in t, u, w, i.e., the formal
power series PΓ(t, u, w)i are rational functions
PΓ(t, u, w)i =
NΓ(t, u, w)i
DΓ(t, u, w)
, i ∈ [[0, l]],
where the NΓ(t, u, w)i’s and DΓ(t, u, w) are elements of Q[t, u, w].
The proof of this theorem uses a key-relation satisfied by PΓ(t, u, w) as well as a so-called
inversion formula. Two essential ingredients are the decomposition of the tensor product of πp,q,r
with the natural representation of SL4C and the simultaneous diagonalizability of certain matrices.
The effective calculation of PΓ(t, u, w) then reduces to matrix multiplication.
In [2] we applied a similar method for SL2C — recovering thereby in a quite easy way the results
obtained by Kostant in [6, 7], and by Gonzalez – Sprinberg and Verdier in [4] — and for SL3C in
order to get explicit computations of the series for every finite subgroup of SL3C.
The general framework of that study is the construction of a minimal resolution of singularities of
the orbifold Cn/Γ. It is related to the McKay correspondence (see [1, 3, 4]). For example, Gonzalez –
Sprinberg and Verdier use in [4] a Poincaré series to construct explicitly minimal resolutions for
singularities of V = C2/Γ when Γ is a finite subgroup of SL2C. To go further in this approach, our
results for SL4C could be used to construct an explicit synthetic minimal resolution of singularities
for orbifolds of the form C4/Γ where Γ is a finite subgroup of SL4C.
2. Properties of the matrices A(1), A(2), A(3). In order to compute the series PΓ(t, u, w),
we first establish here some properties of the matrices A(1), A(2), A(3). The first proposition essen-
tially follows from the uniqueness of the decomposition of a representation as sum of irreducible
representations.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C AND THE RELATED . . . 1323
Proposition 1. (i) A(3) = tA(1). (ii) A(2) is a symmetric matrix. (iii) A(1), A(2) and A(3)
commute. In particular, A(1) is a normal matrix.
Proof. Since a(1)
ij = (χi |χγ⊗γj ) =
1
|Γ|
∑
g∈Γ
χi(g)χ(g)χj(g), we have γ ⊗ γj =
⊕l
i=0 a
(1)
ij γi.
In the same way, (γ ∧ γ)⊗ γj =
⊕l
i=0 a
(2)
ij γi and γ∗ ⊗ γj =
⊕l
i=0 a
(3)
ij γi.
Then
a
(3)
ij = (χi |χγj⊗γ∗) =
1
|Γ|
∑
g∈Γ
χi(g)χj(g)χγ∗(g) =
1
|Γ|
∑
g∈Γ
χi(g)χj(g)χ(g−1) =
=
1
|Γ|
∑
g∈Γ
χi(g−1)χj(g
−1)χ(g) =
1
|Γ|
∑
g∈Γ
χi(g)χj(g)χ(g) = a
(1)
ji ,
hence A(3) = tA(1).
We also have (γj ⊗ γ) ⊗ γ∗ =
(⊕l
i=0 a
(1)
ij γi
)
⊗ γ∗ =
⊕l
i=0 a
(1)
ij
(⊕l
k=0 a
(3)
ki γk
)
=
=
⊕l
k=0
(∑l
i=0
a
(3)
ki a
(1)
ij
)
γk and γ⊗(γj⊗γ∗)=γ⊗
(⊕l
i=0 a
(3)
ij γi
)
=
⊕l
i=0 a
(3)
ij
(⊕l
k=0 a
(1)
ki γk
)
=
=
⊕l
k=0
(∑l
i=0
a
(1)
ki a
(3)
ij
)
γk, hence A(3)A(1) = A(1)A(3). The proofs of the other statements are
the same.
SinceA(1), A(2), A(3) are normal, we know that they are diagonalizable with eigenvectors forming
an orthogonal basis. Now we will diagonalize these matrices by using the character table of the group
Γ. Let us denote by {C0, . . . , Cl} the set of conjugacy classes of Γ, and for any j ∈ [[0, l]], let gj be
an element of Cj . So the character table of Γ is the matrix TΓ ∈Ml+1C defined by (TΓ)i,j := χi(gj).
Proposition 2. (i) For k ∈ [[0, l]], set wk := (χ0(gk), . . . , χl(gk)) ∈ Cl+1. Then wk is an
eigenvector of A(3) associated to the eigenvalue χ(gk). Similarly, wk is an eigenvector of A(1)
associated to the eigenvalue χ(gk).
(ii) For k ∈ [[0, l]], wk is an eigenvector of A(2) associated to the eigenvalue
1
2
(
χ(gk)
2 + χ(g2
k)
)
.
Proof. From the relation γi ⊗ γ =
∑l
j=0
a
(1)
ji γj , we get χiχ = χγi⊗γ =
∑l
j=0
a
(1)
ji χj . By
evaluating this on gk, we obtain χi(gk)χ(gk) =
∑l
j=0
a
(1)
ji χj(gk) =
∑l
j=0
a
(3)
ij χj(gk) according to
Proposition 1. So wk is an eigenvector of A(3) associated to the eigenvalue χ(gk). The method is
similar for the other results.
As the wj’s are the columns of TΓ, which are always orthogonal, the matrix TΓ is invertible
and the family W := (w0, . . . , wl) is a common basis of eigenvectors of A(1), A(2) and A(3). Then
Λ(1) := T−1
Γ A(1) TΓ, Λ(2) := T−1
Γ A(2) TΓ and Λ(3) := T−1
Γ A(3) TΓ are diagonal matrices, with
Λ
(1)
jj = χ(gj), Λ
(2)
jj =
1
2
(χ(gj)
2 − χ(g2
j )) and Λ
(3)
jj = χ(gj).
Now, we make use of the Clebsch – Gordan formula
π1,0,0 ⊗ πp,q,r = πp+1,q,r ⊕ πp,q,r−1 ⊕ πp−1,q+1,r ⊕ πp,q−1,r+1,
π0,1,0 ⊗ πp,q,r = πp,q+1,r ⊕ πp,q−1,r ⊕ πp+1,q−1,r+1 ⊕ πp−1,q+1,r−1 ⊕ πp−1,q,r+1 ⊕ πp+1,q,r−1, (1)
π0,0,1 ⊗ πp,q,r = πp,q,r+1 ⊕ πp−1,q,r ⊕ πp,q+1,r−1 ⊕ πp+1,q−1,r.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1324 F. BUTIN
Proposition 3. The vectors vp,q,r satisfy the following recurrence relations:
A(1)vp,q,r = vp+1,q,r + vp,q,r−1 + vp−1,q+1,r + vp,q−1,r+1,
A(2)vp,q,r = vp,q+1,r + vp,q−1,r + vp+1,q−1,r+1 + vp−1,q+1,r−1 + vp−1,q,r+1 + vp+1,q,r−1,
A(3)vp,q,r = vp,q,r+1 + vp−1,q,r + vp,q+1,r−1 + vp+1,q−1,r.
Proof. The definition of vp,q,r reads vp,q,r =
∑l
i=0
mi(p, q, r)ei, thus A(1)vp,q,r =
=
∑l
i=0
(∑l
j=0
mj(p, q, r)a
(1)
ij
)
ei. Now
(π1,0,0 ⊗ πp,q,r)|Γ = πp,q,r|Γ ⊗ γ =
l∑
j=0
mj(p, q, r)γj ⊗ γ =
l∑
i=0
l∑
j=0
mj(p, q, r)a
(1)
ij
γi
and
πp+1,q,r|Γ + πp,q,r−1|Γ + πp−1,q+1,r|Γ + πp,q−1,r+1|Γ =
=
l∑
i=0
(mi(p+ 1, q, r) +mi(p, q, r − 1) +mi(p− 1, q + 1, r) +mi(p, q − 1, r + 1)) γi.
By uniqueness,
l∑
j=0
mj(p, q, r)a
(1)
ij = mi(p+ 1, q, r) +mi(p, q, r − 1)+
+mi(p− 1, q + 1, r) +mi(p, q − 1, r + 1).
3. The series PΓ(t, u, w) is a rational function. This section is mainly devoted to the proof
of Theorem 1.
3.1. A key-relation satisfied by the series PΓ(t, u, w).
Proposition 4. Set
J(t, u, w) := (1− u2)((1 + ut2)(1 + uw2)− tw(1 + u2))In + twu(1− u2)A(2)−
−tu(1 + uw2)(A(3) − uA(1))− wu(1 + ut2)(A(1) − uA(3)).
Then the series PΓ(t, u, w) satisfies the following relation:
J(t, u, w) v0,0,0 =
=
(
1− tA(1) + t2A(2) − t3A(3) + t4
)(
1− wA(3) + w2A(2) − w3A(1) + w4
)
×
×
(
(1 + u2)(1− u2)2 − u(1− u2)2A(2) + u2(A(1) − uA(3))(A(3) − uA(1))
)
PΓ(t, u, w).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C AND THE RELATED . . . 1325
Proof. Set x := PΓ(t, u, w). Set also vp,q,−1 := 0, vp,−1,r := 0 and v−1,q,r := 0 for (p, q, r) ∈
∈ N3, such that, according to the Clebsch – Gordan formula, the formulae of the preceding corollary
are still true for (p, q, r) ∈ N3. So we have
(
by denoting
∑∞
p=0
∑∞
q=0
∑∞
r=0
by
∑
pqr
)
(1− wA(3) + w2A(2) − w3A(1) + w4)x =
=
∑
pqr
vp,q,rt
puqwr −
∑
pqr
(vp,q,r+1 + vp−1,q,r + vp,q+1,r−1 + vp+1,q−1,r)t
puqwr+1+
+
∑
pqr
(vp,q+1,r + vp,q−1,r + vp+1,q−1,r+1 + vp−1,q+1,r−1 + vp−1,q,r+1 + vp+1,q,r−1)tpuqwr+2−
−
∑
pqr
(vp+1,q,r + vp,q,r−1 + vp−1,q+1,r + vp,q−1,r+1)tpuqwr+3 +
∑
pqr
vp,q,rt
puqwr+4,
hence
(1− wA(3) + w2A(2) − w3A(1) + w4)x =
= (1− tw + uw2 − t−1uw)
∞∑
p=0
∞∑
q=0
vp,q,0t
puq + t−1uw
∞∑
q=0
v0,q,0u
q. (2)
In the same way
(
by denoting
∑∞
p=0
∑∞
q=0
by
∑
pq
)
(1− tA(1) + t2A(2) − t3A(3) + t4)
∞∑
p=0
∞∑
q=0
vp,q,0t
puq =
=
∑
pq
vp,q,0t
puq −
∑
pq
(vp+1,q,0 + vp−1,q+1,0 + vp,q−1,1)tp+1uq+
+
∑
pq
(vp,q+1,0 + vp,q−1,0 + vp+1,q−1,1 + vp−1,q,1)tp+2uq−
−
∑
pq
(vp,q,1 + vp−1,q,0 + vp+1,q−1,0)tp+3uq +
∑
pq
vp,q,0t
p+4uq,
hence
(1− tA(1) + t2A(2) − t3A(3) + t4)
∞∑
p=0
∞∑
q=0
vp,q,0t
puq =
= (1 + t2u)
∞∑
q=0
v0,q,0u
q − tu
∞∑
q=0
v0,q,1u
q. (3)
Moreover, we have
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1326 F. BUTIN
(1− tA(1) + t2A(2) − t3A(3) + t4)
∞∑
q=0
v0,q,0u
q =
=
∞∑
q=0
v0,q,0u
q −
∞∑
q
(v1,q,0 + v0,q−1,1)tuq+
∞∑
q
(v0,q+1,0 + v0,q−1,0 + v1,q−1,1)t2uq−
−
∞∑
q
(v0,q,1 + v1,q−1,0)t3uq +
∞∑
q
v0,q,0t
4uq,
hence
(1− tA(1) + t2A(2) − t3A(3) + t4)
∞∑
p=0
v0,q,0u
q =
= (1 + t4 + t2u−1 + t2u)
∞∑
q=0
v0,q,0u
q − t2u−1v0,0,0 − (t+ t3u)
∞∑
q=0
v1,q,0u
q−
−(tu+ t3)
∞∑
q=0
v0,q,1u
q + t2u
∞∑
q=0
v1,q,1u
q. (4)
By combining equations (2), (3) and (4), we get
(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(3) + w4)x =
= (1− tA(1) + t2A(2) − t3A(3) + t4)×
×
(1− tw + uw2 − t−1uw)
∑
pq
vp,q,0t
puq + t−1uw
∞∑
q=0
v0,q,0u
q
=
= (1− tw + uw2 − t−1uw)
(1 + t2u)
∞∑
q=0
v0,q,0u
q − tu
∞∑
q=0
v0,q,1u
q
+
+(1 + t4 + t2u−1 + t2u)t−1uw
∞∑
q=0
v0,q,0u
q − twv0,0,0 − (1 + t2u)uw
∞∑
q=0
v1,q,0u
q−
−(u+ t2)uw
∞∑
q=0
v0,q,1u
q + tu2w
∞∑
q=0
v1,q,1u
q,
hence
(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)x =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C AND THE RELATED . . . 1327
= (1 + ut2)(1 + uw2)
∞∑
q=0
v0,q,0u
q − tu(1 + uw2)
∞∑
q=0
v0,q,1u
q−
−wu(1 + ut2)
∞∑
q=0
v1,q,0u
q − twv0,0,0 + tu2w
∞∑
q=0
v1,q,1u
q. (5)
Besides, we have the two following equations:
A(1)
∞∑
q=0
v0,q,0u
q =
∞∑
q=0
v1,q,0u
q + u
∞∑
q=0
v0,q,1u
q, (6)
and
A(3)
∞∑
q=0
v0,q,0u
q =
∞∑
q=0
v0,q,1u
q + u
∞∑
q=0
v1,q,0u
q. (7)
From these two equations, we deduce
∞∑
q=0
v0,q,1u
q = (1− u2)−1(A(3) − uA(1))
∞∑
q=0
v0,q,0u
q. (8)
Now, we have
A(1)
∞∑
q=0
v0,q,1u
q =
∞∑
q=0
v1,q,1u
q +
∞∑
q=0
v0,q,0u
q + u
∞∑
q=0
v0,q,2u
q, (9)
and
A(3)
∞∑
q=0
v0,q,1u
q =
∞∑
q=0
v0,q,2u
q + u−1
∞∑
q=0
v0,q,0u
q + u
∞∑
q=0
v1,q,1u
q − u−1v0,0,0, (10)
therefore
∞∑
q=0
v1,q,1u
q = (1− u2)−1(A(1) − uA(3))
∞∑
q=0
v0,q,1u
q − (1− u2)−1v0,0,0.
So, according to equation (8), we deduce
∞∑
q=0
v1,q,1u
q = (1− u2)−2(A(1) − uA(3))(A(3) − uA(1))
∞∑
q=0
v0,q,0u
q − (1− u2)−1v0,0,0. (11)
By using equation (11), we may write equation (5) as
(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)x =
=
(
(1 + ut2)(1 + uw2) + tu2w(1− u2)−2(A(1) − uA(3))(A(3) − uA(1))
) ∞∑
q=0
v0,q,0u
q−
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1328 F. BUTIN
−tu(1 + uw2)
∞∑
q=0
v0,q,1u
q − wu(1 + ut2)
∞∑
q=0
v1,q,0u
q − (tw + tu2w(1− u2)−1)v0,0,0. (12)
From equations (6) and (7), we also deduce
∞∑
q=0
v1,q,0u
q = (1− u2)−1(A(1) − uA(3))
∞∑
q=0
v0,q,0u
q. (13)
So, by using equations (8) and (13), we obtain
(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)x =
=
(
(1 + ut2)(1 + uw2)− tu(1 + uw2)(1− u2)−1(A(3) − uA(1))−
−wu(1 + ut2)(1− u2)−1(A(1) − uA(3))+
+tu2w(1− u2)−2(A(1) − uA(3))(A(3) − uA(1))
) ∞∑
q=0
v0,q,0u
q−
−(tw + tu2w(1− u2)−1)v0,0,0, (14)
i.e., by multiplying (14) by (1− u2)2,
(1− u2)2(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)x =
=
(
(1− u2)2(1 + ut2)(1 + uw2)− tu(1 + uw2)(1− u2)(A(3) − uA(1))−
−wu(1 + ut2)(1− u2)(A(1) − uA(3)) + tu2w(A(1) − uA(3))(A(3) − uA(1))
) ∞∑
q=0
v0,q,0u
q−
−(tw(1− u2)2 + tu2w(1− u2))v0,0,0. (15)
Consider now the following equation:
A(2)
∞∑
q=0
v0,q,0u
q = u−1
∞∑
q=0
v0,q,0u
q + u
∞∑
q=0
v0,q,0u
q + u
∞∑
q=0
v1,q,1u
q − u−1v0,0,0. (16)
Then, according to equation (11), we have
A(2)
∞∑
q=0
v0,q,0u
q = u−1
∞∑
q=0
v0,q,0u
q + u
∞∑
q=0
v0,q,0u
q+
+u(1− u2)−2(A(1) − uA(3))(A(3) − uA(1))
∞∑
q=0
v0,q,0u
q − u(1− u2)−1v0,0,0 − u−1v0,0,0,
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BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C AND THE RELATED . . . 1329
i.e., (
A(2) − u−1 − u− u(1− u2)−2(A(1) − uA(3))(A(3) − uA(1))
) ∞∑
q=0
v0,q,0u
q =
= −(u(1− u2)−1 + u−1)v0,0,0. (17)
This last equation reads
(
−u(1− u2)2A(2) + (1 + u2)(1− u2)2 + u2(A(1) − uA(3))(A(3) − uA(1))
) ∞∑
q=0
v0,q,0u
q =
= (1− u2)v0,0,0. (18)
Now, by using equations (15) and (18), we get
(1− u2)2(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)×
×
(
−u(1− u2)2A(2) + (1 + u2)(1− u2)2 + u2(A(1) − uA(3))(A(3) − uA(1))
)
x =
= −tw(1− u2)
(
−u(1− u2)2A(2) + (1 + u2)(1− u2)2+
+u2(A(1) − uA(3))(A(3) − uA(1))
)
v0,0,0+
+
(
(1− u2)2(1 + ut2)(1 + uw2)− tu(1 + uw2)(1− u2)(A(3) − uA(1))−
−wu(1 + ut2)(1− u2)(A(1) − uA(3)) + tu2w(A(1) − uA(3))(A(3) − uA(1))
)
(1− u2)v0,0,0, (19)
i.e., after simplification by (1− u2),
(1− tA(1) + t2A(2) − t3A(3) + t4)(1− wA(3) + w2A(2) − w3A(1) + w4)×
×
(
(1 + u2)(1− u2)2 − u(1− u2)2A(2) + u2(A(1) − uA(3))(A(3) − uA(1))
)
x =
=
(
(1− u2)((1 + ut2)(1 + uw2)− tw(1 + u2)) + twu(1− u2)A(2)−
−tu(1 + uw2)(A(3) − uA(1))− wu(1 + ut2)(A(1) − uA(3))
)
v0,0,0. (20)
Proposition 4 is proved.
3.2. An inversion formula. In order to inverse the relation obtained in Proposition 4 and get an
explicit expression for PΓ(t, u, w), we need the rational function f defined by
f : C3 → C(t, u, w),
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1330 F. BUTIN
(d1, d2, d3) 7→ (1− td1 + t2d2 − t3d3 + t4)−1(1− wd3 + w2d2 − w3d1 + w4)−1×
×
(
(1 + u2)(1− u2)2 − u(1− u2)2d2 + u2(d1 − ud3)(d3 − ud1)
)−1
.
According to Proposition 4, we may write
J(t, u, w) v0,0,0 =
= TΓ
(
1− tΛ(1) + t2Λ(2) − t3Λ(3) + t4
)(
1− wΛ(3) + w2Λ(2) − w3Λ(1) + w4
)
×
×
(
(1 + u2)(1− u2)2 − u(1− u2)2Λ(2) + u2(Λ(1) − uΛ(3))(Λ(3) − uΛ(1))
)
T−1
Γ PΓ(t, u, w).
We deduce that
PΓ(t, u, w) = TΓ ∆(t, u, w)T−1
Γ J(t, u, w) v0,0,0 =
= (TΓ ∆(t, u, w)TΓ) (T−2
Γ J(t, u, w) v0,0,0), (21)
where ∆(t, u, w) ∈Ml+1C(t, u, w) is the diagonal matrix defined by
∆(t, u, w)jj = f(Λ
(1)
jj , Λ
(2)
jj , Λ
(3)
jj ) = f
(
χ(gj),
1
2
(χ(gj)
2 − χ(g2
j )), χ(gj)
)
.
This last formula proves Theorem 1.
Remark 1. The Poincaré series P̂Γ(t) of the algebra of invariants C[z1, z2, z3, z4]Γ is given by
P̂Γ(t) = PΓ(t, 0, 0)0 = PΓ(0, 0, t)0.
3.3. Remark for SLnC. In this section, we consider an integer n ≥ 2 and a subgroup Γ of SLnC.
As in Section 1, let {γ0, . . . , γl} be the set of equivalence classes of irreducible finite dimensional
complex representations of Γ, where γ0 is the trivial representation. The character associated to γj is
denoted by χj .
Consider γ : Γ → SLnC the natural n-dimensional representation, and χ its character. By
complete reducibility we get the decomposition γj ⊗ γ =
⊕l
i=0 a
(1)
ij γi for every j ∈ [[0, l]], and we
set A(1) :=
(
a
(1)
ij
)
(i,j)∈[[0, l]]2
∈Ml+1N.
Let h be a Cartan subalgebra of slnC and let $1, . . . , $n−1 be the corresponding fundamental
weights, and V (p1$1 + . . . + pn−1$n−1) the simple slnC-module of highest weight p1$1 + . . .
. . . + pn−1$n−1 with p := (p1, . . . pn−1) ∈ Nn−1. Then we get an irreducible representation πp :
SLnC→ GL(V (p1$1+. . .+pn−1$n−1)). The restriction of πp to the subgroup Γ is a representation
of Γ, and by complete reducibility, we get the decomposition πp|Γ =
⊕l
i=0mi(p)γi, where the
mi(p)’s are non negative integers. Let E := (e0, . . . , el) be the canonical basis of Cl+1, and
vp :=
l∑
i=0
mi(p)ei ∈ Cl+1.
As γ0 is the trivial representation, we have v0 = e0. Let us consider the vector (with elements of
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BRANCHING LAW FOR THE FINITE SUBGROUPS OF SL4C AND THE RELATED . . . 1331
C[[t1, . . . , tn−1]] = C[[t]] as coefficients)
PΓ(t) :=
∑
p∈Nn−1
vpt
p ∈ (C[[t]])l+1,
and denote by PΓ(t)j its jth coordinate in the basis E .
Given the results from Kostant [6, 7] for SL2C and our results [2] about SL3C, we then formulate
the following conjecture:
Conjecture 1. The coefficients of the vector PΓ(t) are rational fractions in t, i.e., the formal
power series PΓ(t)i are rational functions
PΓ(t)i :=
NΓ(t)i
DΓ(t)
, i ∈ [[0, l]],
where the NΓ(t)i’s and DΓ(t) are elements of Q[t].
4. An example of explicit computation. The classification of finite subgroups of SL4C is given
in [5]. It consists in infinite series and 30 exceptional groups (types I, II, . . . , XXX). We give here
an explicit computation of PΓ(t, u, w) for one of these exceptional groups. Consider the matrices
F1 =
1 0 0 0
0 1 0 0
0 0 j 0
0 0 0 j2
, F ′2 =
1
3
3 0 0 0
0 −1 2 2
0 2 −1 2
0 2 2 −1
,
F ′3 =
1
4
−1
√
15 0 0
√
15 1 0 0
0 0 0 4
0 0 4 0
,
and the subgroup Γ = 〈F1, F
′
2, F
′
3〉 of SL4C (type II in [5]).
Here l = 4,
A(1) = A(3) =
0 0 0 1 0
0 0 1 1 1
0 1 0 1 1
1 1 1 1 1
0 1 1 1 2
and A(2) =
0 1 1 0 0
1 1 0 1 2
1 0 1 1 2
0 1 1 2 2
0 2 2 2 2
,
rank(A(1)) = rank(A(2)) = 4, and the eigenvalues of A(1), A(2), A(3) are Θ(1) = Θ(3) =
= (4, 0, −1, 1, −1), Θ(2) = (6, −2, 1, 0, 1).
According to formula (21), we get
DΓ(t, u, w) = (w − 1)4 (u+ 1)3 (u− 1)5 (t− 1)4 (w2 + w + 1
) (
w4 + w3 + w2 + w + 1
)
×
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
1332 F. BUTIN
× (w + 1)2 (u4 + u3 + u2 + u+ 1
) (
u2 + u+ 1
)2 (
t2 + t+ 1
) (
t4 + t3 + t2 + t+ 1
)
(t+ 1)2 =
= (u− 1)(u+ 1)(u2 + u+ 1)D̃Γ(t)D̃Γ(u)D̃Γ(w),
with D̃Γ(t) = (t− 1)4(t+ 1)2(t2 + t+ 1)(t4 + t3 + t2 + t+ 1). Moreover,
P̂Γ(t) =
t8 − t6 + t4 − t2 + 1
t12 − 2 t10 − t9 + t8 + t7 + t5 + t4 − t3 − 2 t2 + 1
.
Because of the too big size of the numerators NΓ(t, u, w)i’s, only the denominator is given in the
text: all the numerators may be found on the web (http://math.univ-lyon1.fr/homes-www/butin/).
1. Bridgeland T., King A., Reid M. The McKay correspondence as an equivalence of derived categories // J. Amer.
Math. Soc. – 2001. – 14. – P. 535 – 554.
2. Butin F., Perets G. S. McKay correspondence and the branching law for finite subgroups of SL3C // J. Group Theory. –
2014. – 17, Issue 2. – P. 191 – 251.
3. Gomi Y., Nakamura I., Shinoda K. Coinvariant algebras of finite subgroups of SL3C // Can. J. Math. – 2004. – 56. –
P. 495 – 528.
4. Gonzalez-Sprinberg G., Verdier J.-L. Construction géométrique de la correspondance de McKay // An. sci. de
l’E. N. S., 4ème sér. – 1983. – 16, № 3. – P. 409 – 449.
5. Hanany A., He Y.-H. A monograph on the classification of the discrete subgroups of SU(4) // JHEP. – 2001. – 27.
6. Kostant B. The McKay correspondence, the coxeter element and representation theory // SMF. Astérisque, hors sér. –
1985. – P. 209 – 255.
7. Kostant B. The Coxeter element and the branching law for the finite subgroups of SU(2) // Coxeter Legacy. –
Providence, RI: Amer. Math. Soc., 2006. – P. 63–70.
Received 14.01.14,
after revision — 29.03.15
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 10
|
| id | nasplib_isofts_kiev_ua-123456789-165876 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | English |
| last_indexed | 2025-12-07T16:06:06Z |
| publishDate | 2015 |
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| record_format | dspace |
| spelling | Butin, F. 2020-02-16T20:40:05Z 2020-02-16T20:40:05Z 2015 Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials/ F. Butin // Український математичний журнал. — 2015. — Т. 67, № 10. — С. 1321–1332. — Бібліогр.: 7 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165876 512.5 en Інститут математики НАН України Український математичний журнал Статті Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials Закон галуження для скінченних підгруп SL₄C та віддповідні узагальненi поліноми Пуанкаре Article published earlier |
| spellingShingle | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials Butin, F. Статті |
| title | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials |
| title_alt | Закон галуження для скінченних підгруп SL₄C та віддповідні узагальненi поліноми Пуанкаре |
| title_full | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials |
| title_fullStr | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials |
| title_full_unstemmed | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials |
| title_short | Branching Law for the Finite Subgroups of SL₄ℂ and the Related Generalized Poincaré Polynomials |
| title_sort | branching law for the finite subgroups of sl₄ℂ and the related generalized poincaré polynomials |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/165876 |
| work_keys_str_mv | AT butinf branchinglawforthefinitesubgroupsofsl4candtherelatedgeneralizedpoincarepolynomials AT butinf zakongalužennâdlâskínčennihpídgrupsl4ctavíddpovídníuzagalʹnenipolínomipuankare |