Betti numbers of holomorphic symplectic quotients
via arithmetic Fourier transform
Tamás Hausel†
Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom; and Department of Mathematics, University of Texas, Austin, TX 78712
Communicated by Robion C. Kirby, University of California, Berkeley, and approved February 24, 2006 (received for review September 27, 2005)
A Fourier transform technique is introduced for counting the
number of solutions of holomorphic moment map equations over
a finite field. This technique in turn gives information on Betti
numbers of holomorphic symplectic quotients. As a consequence,
simple unified proofs are obtained for formulas of Poincaré
polynomials of toric hyperkähler varieties (recovering results of
Bielawski–Dancer and Hausel–Sturmfels), Poincaré polynomials of
Hilbert schemes of points and twisted Atiyah–Drinfeld–Hitchin–
Manin (ADHM) spaces of instantons on ⺓2 (recovering results of
Nakajima–Yoshioka), and Poincaré polynomials of all Nakajima
quiver varieties. As an application, a proof of a conjecture of Kac
on the number of absolutely indecomposable representations of
a quiver is announced.
quiver varieties 兩 Weyl–Kac character formula
et ⺛ be a field, which will be either the complex numbers ⺓
or the finite field ⺖q in this work. Let G be a reductive
algebraic group over ⺛, g its Lie algebra. Consider a representation ␳ : G 3 GL(⺦) of G on a ⺛-vector space ⺦, inducing the
Lie algebra representation | : g 3 gl(⺦). The representation ␳
induces an action ␳ : G 3 GL(⺝) on ⺝ ⫽ ⺦ ⫻ ⺦*. The vector
space ⺝ has a natural symplectic structure, defined by the
natural pairing 具v, w典 ⫽ w(v), with v 僆 ⺦ and w 僆 ⺦*. With
respect to this symplectic form a moment map
Proposition 1. The number of solutions of the equation ␮ (v, w) ⫽
␰ over the finite field ⺖q equals
#兵共v, w兲 僆 ⺝兩␮共v, w兲 ⫽ ␰其
⫽ 兩g兩 ⫺1/2兩⺦兩F共a|兲共 ␰ 兲
⫽ 兩g兩 ⫺1兩⺦兩
F共 f兲共Y兲 ⫽ 兩g兩 ⫺1/2
f共X兲⌿共具X, Y典兲.
Proof: Using two basic properties of Fourier transform
F共F共 f兲兲共X兲 ⫽ f共⫺X兲,
for X 僆 g and
冘
⌿共具v, w典兲 ⫽ 兩V兩␦0共v兲,
[3]
w僆V*
of ␳ is given at X 僆 g by
for v 僆 V we get
[1]
#兵共v, w兲 僆 ⺝兩␮共v, w兲 ⫽ ␰其 ⫽
Let now ␰ 僆 (g*)G be a central element, then the holomorphic
symplectic quotient is defined by the affine geometric invariant
theory (GIT) quotient
⫽
冘冘
冘冘 冘
冘冘
冘冘
冘
冘
冘冘
␦␰共␮共v, w兲兲
v僆⺦ w僆⺦*
F共F共␦␰兲兲共⫺␮ 共v, w兲兲
v僆⺦ w僆⺦*
⺝兾兾兾兾␰G :⫽ 共 ␮ ⫺1共 ␰ 兲兲兾兾G,
⫽
which is the affine algebraic geometric version of the hyperkähler quotient construction of ref. 1. In particular our varieties,
in addition to the holomorphic symplectic structure, will carry a
natural hyperkähler metric, although the latter will not feature
in what follows.
Our main proposition counts rational points on the varieties
␮⫺1(␰) over the finite fields ⺖q, where q ⫽ p r is a prime power.
For convenience we will use the same letters ⺦, G, g, ⺝, ␰ for
the corresponding vector spaces, groups, Lie algebras, and
matrices over the finite field ⺖q. We define the function a | : g 3
⺞ 傺 ⺓ at X 僆 g as
兩g兩 ⫺1/2F共 ␦ ␰兲共X兲⌿共具X, ⫺␮ 共v, w兲典兲
v僆⺦ w僆⺦* X僆g
⫽
兩g兩 ⫺1/2F共 ␦ ␰兲共X兲
v僆⺦ X僆g
⫽
冘
⌿共⫺具|共X兲v, w典兲
w僆⺦*
兩g兩 ⫺1/2F共 ␦ ␰兲共X兲兩⺦兩 ␦ 0共|共X兲v兲
v僆⺦ X僆g
⫽
兩g兩 ⫺1/2F共 ␦ ␰兲共X兲兩⺦兩a|共X兲
X僆g
⫽
[2]
where we used the notation 兩S兩 for the number of elements in any
set S. In particular a |(X) is always a power of q. For an element
v 僆 V of any vector space we define the characteristic function
␦ v : V 3 ⺓ by ␦ v(x) ⫽ 0 unless x ⫽ v when ␦ v(v) ⫽ 1. We can
now formulate our main proposition.
6120 – 6124 兩 PNAS 兩 April 18, 2006 兩 vol. 103 兩 no. 16
冘
X僆g
␮ : ⺦ ⫻ ⺦* 3 g*,
a|共X兲 :⫽ 兩ker共|共X兲兲兩,
a|共X兲⌿共具X, ␰ 典兲.
To explain the last two terms in the proposition above, we
need to define Fourier transforms (2) of functions f : g 3 ⺓ on
the finite Lie algebra g, which here we think of as an abelian
group with its additive structure. To define this fix ⌿ : ⺖q 3 ⺓x
a nontrivial additive character, and then we define the Fourier
transform F(f) : g* 3 ⺓ at a Y 僆 g*
L
具␮共v, w兲, X典 ⫽ 具|共X兲v, w典.
冘
X僆g
兩g兩 ⫺1兩⺦兩a|共X兲
X僆g
冘
␦ ␰共Y兲⌿共具X, Y典兲
Y僆g*
Conflict of interest statement: No conflicts declared.
Abbreviation: ADHM, Atiyah–Drinfeld–Hitchin–Manin.
†E-mail:
[email protected].
© 2006 by The National Academy of Sciences of the USA
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0601337103
冘
is the h-polynomial of the dual matroid MᏮ. This way we recover
a result of refs. 4 and 5; for a more recent arithmetic proof see (6):
a|共X兲⌿共具X, ␰ 典兲
X僆g
⫽ 兩g兩 ⫺1/2兩⺦兩F共a|兲共 ␰ 兲.
Corollary 1. The Poincaré polynomial of the toric hyperkähler variety
Affine Toric Hyperkähler Varieties
We take G ⫽ ⺤d ⬵ (⺓⫻)d a torus. A vector configuration A ⫽
(a 1, . . . , a n) : ⺪n 3 ⺪d gives a representation ␳A : ⺤d 3 ⺤n 傺
GL(⺦), where ⺦ ⬵ ⺓n is an n-dimensional vector space and ⺤n
傺 GL(⺦) is a fixed maximal torus. The corresponding map on
the Lie algebras is |A : td 3 tn. The holomorphic moment map
of this action ␮A : ⺦ ⫻ ⺦* 3 (td)* is given by Eq. 1, which in this
case takes the explicit form
冘
n
␮A共v, w兲 ⫽
v iw ia i.
i⫽1
We take a generic ␰ 僆 (td)*. The affine toric hyperkähler variety
is then defined as the affine geometric invariant theory quotient:
M(␰, A) ⫽ ␮A⫺1(␰)兾兾⺤d. To use our main result, we need to
determine a|(X). Note that the natural basis e 1, . . . , e n 僆 (tn)*
gives us a collection of hyperplanes H 1, . . . , H n in td. Now for
X 僆 td we have that a|(X) ⫽ q ca(X), where ca(X) is the number
of hyperplanes that contain X. Finally, we take the intersection
lattice L(A) of this hyperplane arrangement, i.e., the set of all
subspaces of td that arise as the intersection of any collection of
our hyperplanes, with partial ordering given by containment.
The generic choice of ␰ will ensure that ␰ will not be trivial on
any subspace in the lattice L(A). Thus, for any subspace V 僆
L(A), we have from Eq. 3 that 兺X僆V ⌿ (具X, ␰ 典) ⫽ 0.
Now we can use Proposition 1. If we perform the sum we get
a combinatorial expression
#共M共␰, A兲兲 ⫽
qn⫺d
共q ⫺ 1兲d
qn⫺d
⫽
共q ⫺ 1兲d
冘
冘
is given by
Pt共M共␰, A兲兲 ⫽ h共MB兲共t 2兲,
where B is a Gale dual vector configuration of A.
Hilbert Scheme of n-Points on ⺓2 and Atiyah–Drinfeld–Hitchin–
Manin (ADHM) Spaces
Here G ⫽ GL(V), where V is an n-dimensional ⺛ vector space.
We need three types of basic representations of G. The adjoint
representation ␳ad : GL(V) 3 GL(gl(V)), the defining representation ␳ def ⫽ Id : G 3 GL(V), and the trivial representations
k
␳ triv
⫽ 1 : G 3 GL(⺛k). Fix k and n. Define ⺦ ⫽ gl(V) ⫻ V R
k
k, ⺝ ⫽ ⺦ ⫻ ⺦* and ␳ : G 3 GL(⺦) by ␳ ⫽ ␳
⺛
ad ⫻ ␳ def R ␳ triv.
Then we take the central element ␰ ⫽ Id V 僆 gl(V) and define
the twisted ADHM space as
M共n, k兲 ⫽ ⺝兾兾兾兾␰G ⫽ ␮ ⫺1共 ␰ 兲兾兾G,
where
␮共A, B, I, J兲 ⫽ 关A, B兴 ⫹ IJ,
with A, B 僆 gl(V), I 僆 Hom(⺛k, V) and J 僆 Hom(V, ⺛k).
The space M(n, k) is empty when k ⫽ 0 (the trace of a
commutator is always zero), diffeomorphic with the Hilbert
scheme of n-points on ⺓2, when k ⫽ 1, and is the twisted version
of the ADHM space (7) of U(k) Yang–Mills instantons of charge
n on ⺢4 (cf. ref. 8). By our main Proposition 1 the number of
solutions over ⺛ ⫽ ⺖q of the equation
关A, B兴 ⫹ IJ ⫽ IdV ,
a|共X兲⌿共具X, ␰典兲
X僆td
is the Fourier transform on g of the function a| (X) ⫽
兩ker(|(X))兩. First we determine a|(X) for X 僆 g ⫽ gl(V). By the
definition of | we have
␮L共A兲共V兲qca共 V兲,
V僆L共A兲
where ␮L(A) is the Möbius function of the partially ordered set
L(A), while ca(V) is the number of coatoms, i.e., hyperplanes
containing V. Because the count above is polynomial in q and the
mixed Hodge structure on M(␰, A) is pure, we get that for the
Poincaré polynomial we need to take the opposite of the count
polynomial, i.e., substitute q ⫽ 1兾t2 and multiply by t 4(n⫺d). This
way we get:
Theorem 1. The Poincaré polynomial of the toric hyperkähler variety
ker共|共X兲兲 ⫽ ker共| ad共X兲兲 ⫻ ker共| def兲 丢 ⺛ k ,
and so if a|ad(X) ⫽ 兩ker(|ad(X))兩 and a|def ⫽ 兩ker(|def)兩, then we have
a|共X兲 ⫽ a|ad共X兲a|kdef共X兲.
Now Proposition 1 gives us
#共M共n, k兲兲 ⫽
is given by
1
Pt共M共␰, A兲兲 ⫽ 2d
t 共1 ⫺ t2兲d
冘
2 n⫺ca共 V兲
␮L共A兲共V兲共t 兲
.
冘
␮L共A兲共V兲共q兲
n⫺ca共 V兲
V僆L共A兲
where
冘
n⫺d
h共MᏮ兲 ⫽
i⫽0
Hausel
兩⺦兩
兩g储G兩
V僆L共A兲
One can prove by a simple deletion contraction argument (and
it also follows from the second proof of proposition 6.3.26 of ref.
3) that for any matroid MᏭ and its dual MᏮ
1
d
q 共1 ⫺ q兲d
⫽
1
#兵共v, w兲 僆 ⺝兩 ␮ 共v, w兲 ⫽ ␰ 其
兩G兩
hi共MᏮ兲qi,
⫽ h共MᏮ兲,
⫽
兩⺦兩
兩g储G兩
冘
a|共X兲⌿共具X, ␰ 典兲
X僆g
冘
a|ad共X兲a|k def共X兲⌿共具X, ␰ 典兲.
X僆g
We will perform the sum adjoint orbit by adjoint orbit. The adjoint
orbits of gl(n), according to their Jordan normal forms, fall into
types, labeled by T(n), which stands for the set of all possible
Jordan normal forms of elements in gl(n). We denote by Treg(t)
the types of the regular (i.e., nonsingular) adjoint orbits, while
Tnil(s) ⫽ P(s) denotes the types of the nilpotent adjoint orbits,
which are just given by partitions of s. First we do the k ⫽ 0 case
where we know a priori, that the count should be 0, because the
commutator of any two matrix is always trace-free and thus
PNAS 兩 April 18, 2006 兩 vol. 103 兩 no. 16 兩 6121
MATHEMATICS
⫽ 兩g兩 ⫺1兩⺦兩
cannot equal ␰ (for almost all q). Additionally, if we separate the
nilpotent and regular parts of our adjoint orbits we get
1
0⫽
兩G兩
⫽
冘
For the grand generating function then we get
写写
⬁
k
⌽k共T兲 ⫽ ⌽nil
共T兲⌽ reg共T兲 ⫽
a|ad共X兲⌿共具X, ␰ 典兲
冘
兩ᑝ ␭兩
兩C ␭兩
n⫽s⫹t ␭ 僆Tnil共 s 兲
␶ 僆Treg共 t 兲
兩ᑝ ␶兩
⌿共具X ␶, ␰ 典兲,
兩C ␶兩
where C ␶ and, respectively, ᑝ␶ denotes the centralizer of an
element X ␶ of g of type ␶ in the adjoint representation of G,
respectively, g on g.
So if we define the generating series
冘 冘
⬁
0
共T兲 ⫽ 1 ⫹
⌽nil
s⫽1 ␭ 僆Tnil共 s 兲
1
.
共1 ⫺ T iq l兲
i⫽1 l⫽1
X僆g
冘 冘
k
兩ᑝ ␭兩 s
T ,
兩C ␭兩
Because the mixed Hodge structure is pure, and this count is
polynomial, we get the compactly supported Poincaré polynomial. To get the ordinary Poincaré polynomial, we need to
replace q ⫽ 1兾t 2 and multiply the nth term in Eq. 6 by t 4kn. This
way we get Theorem 2.
Theorem 2. The generating function of the Poincaré polynomials of
the twisted ADHM spaces are given by
冘
⬁
写写
⬁
Pt共M共k, n兲兲T n ⫽
n⫽0
k
i⫽1 b⫽1
1
.
共1 ⫺ t2共k共i⫺1兲⫹b⫺1兲T i兲
and
冘 冘
⬁
⌽reg共T兲 ⫽ 1 ⫹
t⫽1 ␶ 僆Treg共 t 兲
This result appeared as corollary 3.10 in ref. 10.
兩ᑝ ␶兩
⌿共具X ␶, ␰ 典兲T t,
兩C ␶兩
then we have
0
共T兲⌽ reg共T兲 ⫽ 1.
⌽nil
0
is easy to calculate (9)
However, ⌽ nil
写写
0
共T兲 ⫽
⌽nil
⬁
⬁
i⫽1
j⫽1
1
,
共1 ⫺ T iq 1⫺j兲
[4]
Quiver Varieties of Nakajima
Here we recall the definition of the affine version of Nakajima’s
quiver varieties (11). Let Q ⫽ (V, E) be a quiver, i.e., an oriented
graph on a finite set V ⫽ {1, . . . , n} with E 傺 V ⫻ V a finite
set of oriented (perhaps multiple and loop) edges. To each vertex
i of the graph, we associate two finite dimensional ⺛ vector
spaces V i and W i. We call (v1, . . . , vn, w1, . . . , wn) ⫽ (v, w) the
dimension vector, where vi ⫽ dim(V i) and wi ⫽ dim(W i). To
these data we associate the grand vector space
⺦v,w ⫽
thus we get
写写
⬁
⌽reg共T兲 ⫽
⬁
i⫽1
共1 ⫺ T iq 1⫺j兲.
[5]
丣
共 i, j 兲 僆E
Hom共 V i, V j兲 丣
⫽
冘
the group and its Lie algebra
j⫽1
Gv ⫽
gv ⫽
兩C ␭兩
n⫽s⫹t ␭ 僆Tnil共 s 兲
GL共 V i兲
丣
gl共 V i兲,
i僆V
X僆g
冘 冘
⫻
i僆V
a|ad共X兲⌿共具X, ␰ 典兲
兩ᑝ ␭兩a|k def共X ␭兲
Hom共 V i, W i兲,
i僆V
Now the general case is easy to deal with:
1
#共M共n, k兲兲
⫽
qnk
兩G兩
丣
冘
␶ 僆Treg共 t 兲
and the natural representation
兩ᑝ ␶兩
⌿共具X, ␰ 典兲.
兩C ␶兩
␳v,w : Gv 3 GL(⺦v,w),
with derivative
Thus, if we define the grand generating function by
冘
⬁
⌽k共T兲 ⫽ 1 ⫹
n⫽1
|v,w : gv 3 gl共⺦ v,w兲.
Tn
#共M共n, k兲兲 kn ,
q
[6]
and
冘 冘
⬁
k
⌽nil
共T兲 ⫽ 1 ⫹
s⫽1 ␭ 僆Tnil共 s 兲
兩ᑝ ␭储ker共X ␭兲兩 k s
T ,
兩C ␭兩
then for the latter we get similarly to the argument for Eq. 4 in
ref. 9 that
k
⌽nil
⫽
k
⌽ nil
共T兲
⫽
写写
⬁
⬁
i⫽1
j⫽1
1
.
共1 ⫺ T iq k⫹1⫺j兲
6122 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0601337103
The action is from both left and right on the first term and from
the left on the second.
We now have Gv acting on ⺝v,w ⫽ ⺦v,w ⫻ ⺦*v,w preserving the
symplectic form with moment map ␮v,w : ⺦v,w ⫻ ⺦*v,w 3 g*v given
by Eq. 1. We take now Ev ⫽ (Id V1, . . . , Id Vn) 僆 (g*v)Gv, and define
the affine Nakajima quiver variety (11) as
⫺1
共 ␰ v兲兾兾Gv .
M共v, w兲 ⫽ ␮ v,w
Here we determine the Betti numbers of M(v, w) using our main
Proposition 1, by calculating the Fourier transform of the function a|v,w given in Eq. 2.
First, we introduce, for a dimension vector w 僆 V ⺞, the
generating function
Hausel
写
T ivi
v⫽ 共 v1, . . . , vn兲 僆V ⺞ i僆V
冘
冘
···
␭ 1僆Tnil共 v1兲
a |v,w共X ␭1, . . . , X ␭n兲
兩C ␭1兩· · ·兩C ␭n兩
␭ n僆Tnil共 vn兲
,
where Tnil(s) is the set of types of nilpotent s ⫻ s matrices, where
a type is given by a partition ␭ 僆 P(s) of s, X ␭ denotes the typical
s ⫻ s nilpotent matrix in gl(s) in Jordan form of type ␭, C ␭ is the
centralizer of X ␭ under the adjoint action of GL(s) on gl(s). We
also introduce the generating function
冘
冘
⌽reg ⫽
v⫽ 共 v1, . . . ,
写
冘
T ivi
vn兲 僆V ⺞
···
␶ 1僆Treg共 v1兲
i僆V
a|v,0共X ␶1, . . . , X ␶n兲
兩C ␶1兩· · ·兩C ␶n兩
␶ n僆Treg共 vn兲
⌿共具X ␶, ␰ v典兲,
where Treg(t) is the set of types ␶, i.e., Jordan normal forms, of
a regular t ⫻ t matrix X ␶ in gl(t), C ␶ 傺 GL(t) its centralizer under
the adjoint action. Note also that for a regular element X 僆 gv,
a |v,w(X) ⫽ a |v,0(X) does not depend on w 僆 V ⺞.
Now we introduce for w 僆 V ⺞ the grand generating function
⌽共w兲 ⫽
冘
#共M共v, w兲兲
v僆V ⺞
兩gv兩
T v.
兩⺦v,w兩
[7]
As in the previous section, our main Proposition 1 implies
⌽共w兲 ⫽ ⌽ nil共w兲⌽ reg .
[8]
Finally, we note that when w ⫽ 0 we have |v,0(␰*v) ⫽ 0, where ␰*v ⫽
(Id V1, . . . , Id Vn) 僆 g, thus by Eq. 1 具␮v,0(v, 0), ␰ *v典 ⫽ 0. Because
具␰v, ␰*v典 ⫽ 兺 vi, the equation ␮v,0(v, w) ⫽ ␰v has no solutions (for
almost all q). This way we get that ⌽(0) ⫽ 1, and so Eq. 8 yields
⌽reg ⫽ 1兾⌽ nil(0), giving the result
⌽共w兲 ⫽
⌽ nil共w兲
.
⌽ nil共0兲
Therefore, it is enough to understand ⌽nil(w), which reduces to
a simple linear algebra problem of determining a|v,w (X ␭1, . . . ,
X ␭n). Putting together everything yields the following:
Theorem 3. Let Q ⫽ (V, E) be a quiver, with V ⫽ {1, . . . , n} and
E 傺 V ⫻ V, with possibly multiple edges and loops. Fix a dimension
vector w 僆 ⺞V. The Poincaré polynomials P t(M(v, w)) of the
corresponding Nakajima quiver varieties are given by the generating
function
冘
P t共M共v, w兲兲t ⫺d共v,w兲T v
v僆⺞ V
冘
Tv
v僆⺞ V
冘
冉写
␭ 1僆P共 v1兲
·
·
· 共v 兲
␭ n僆P
n
⫽
冘
冘
t ⫺2n共␭ ,␭ 兲
i
写冉
j
共 i, j 兲 僆E
i僆V
冊冉 写
t ⫺2n共␭ ,共1
i
i僆V
写写
m k共␭ i兲
t ⫺2n共␭ ,␭ 兲
i
i
写
k
t
共1 ⫺ t 2j兲
j⫽1
wi 兲兲
冊
⫺2n 共 ␭ i, ␭ j兲
冊
where d(v, w) ⫽ 2兺 (i, j)僆E vivj ⫹ 2兺 i僆V vi(wi ⫺ vi) is the dimension
of M(v, w), and T v ⫽ 兿 i僆V T ivi. P(s) stands for the set of partitions¶
of s 僆 ⺞. For two partitions ␭ ⫽ ( ␭ 1, . . . , ␭ l) 僆 P(s) and ␮ ⫽
( ␮ 1, . . . , ␮ m) 僆 P(s) we define n( ␭ , ␮ ) ⫽ 兺 i, j min( ␭ i, ␮ j), and
if we write ␭ ⫽ (1 m1( ␭ ), 2 m2( ␭ ), . . .) 僆 P(s), then we can define
l( ␭ ) ⫽ 兺 m i( ␭ ) ⫽ l the number of parts in ␭. With this notation
n( ␭ i, (1 wi)) ⫽ wil( ␭ i) in the above formula.
Remark: This single formula encompasses a surprising amount
of combinatorics and representation theory. When v ⫽ (1, . . . ,
1), the Nakajima quiver variety is a toric hyperkähler variety,
thus Eq. 9 gives a new formula for its Poincaré polynomial, which
was given in Corollary 1. If additionally w ⫽ (1, 0, . . . , 0) then
M(v, w) is the toric quiver variety of ref. 5. Therefore, its
Poincaré polynomial, which is the reliability polynomial㛳 of the
graph underlying the quiver (5) also can be read off from the
above formula 9.
When the quiver is just a single loop on one vertex, our
formula 9 reproduces Theorem 2. When the quiver is of type A n
Nakajima (14) showed, that the Poincaré polynomials of the
quiver variety are related to the combinatorics of Youngtableaux, while in the general ADE case, Lusztig (15) conjectured a formula for the Poincaré polynomial, in terms of
formulae arising in the representation theory of quantum
groups. When the quiver is star-shaped recent work (T.H.,
unpublished work and T.H., E. Letellier, and F. Rodriguez–
Villegas, unpublished work) calculates these Poincaré polynomials using the character theory of reductive Lie algebras over
finite fields (2) and arrives at formulas determined by the
Hall–Littlewood symmetric functions (12), which arose as the
pure part of Macdonald symmetric polynomials (12). In the case
when the quiver has no loops, Nakajima (16) gives a combinatorial algorithm for all Betti numbers of quiver varieties, motivated by the representation theory of quantum loop algebras.
Finally, through the paper (17) of Crawley-Boevey and Van den
Bergh, Poincaré polynomials of quiver varieties are related to the
number of absolutely indecomposable representations of quivers
in the work of Kac (18), which were eventually completely
determined by Hua (19).
In particular, formula 9, when combined with results in refs. 19
and 11 and the Weyl–Kac character formula in the representation
theory of Kac–Moody algebras (20), yields a simple proof of
conjecture 1 of Kac (18). Consequently, formula 9 can be viewed as
a q-deformation of the Weyl–Kac character formula (20).
A detailed study of the above generating function 9, its
relationship to the wide variety of examples mentioned above,
and details of the proofs of the results of this work will appear
elsewhere.
¶The
notation for partitions is that of ref. 12.
㛳Incidentally,
the reliability polynomial measures the probability of the graph remaining
connected when each edge has the same probability of failure, a concept heavily used in
the study of reliability of computer networks (13).
,
[9]
I thank Ed Swartz for ref. 3, Fernando Rodriguez-Villegas for ref. 9,
Balázs Szendröi for ref. 10, and Hiraku Nakajima for suggestions
regarding Eq. 9. This work was supported by a Royal Society University Research Fellowship, National Science Foundation Grant DMS0305505, an Alfred P. Sloan Research Fellowship, and a Summer
Research Assignment of the University of Texas at Austin.
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2. Letellier, E. (2005) Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras (Springer, Berlin).
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Mathematics and its Applications, ed. White, N. (Cambridge Univ. Press,
Cambridge, U.K.), Vol. 40, pp. 123–225.
4. Bielawski, R. & Dancer, A. (2000) Comm. Anal. Geom. 8, 727–760.
v僆⺞V
Hausel
Tv
␭ 1僆P共 v1兲
·
·
· 共v 兲
␭ n僆P
n
写冉
i僆V
共 i, j 兲 僆E
写写
m k共␭ i兲
t ⫺2n共␭ ,␭ 兲
i
i
k
j⫽1
共1 ⫺ t 2j兲
冊
PNAS 兩 April 18, 2006 兩 vol. 103 兩 no. 16 兩 6123
MATHEMATICS
冘
⌽nil共w兲 ⫽
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Hausel