数理解析研究所講究録
991 巻 1997 年 137-140

137
Coinvariant Algebras of Some Finite Groups
上智大学

彼田健– (Ken-ichi SHINODA). Recently Y.Ito and I.Nakamura [IN2], [N2] studied the Hilbert scheme of G-orbits
Hilbfor a finite groupand showed a direct correspondence between
the representation graph of (observation) and the singular fiber of the minimalresolution of(Dynkin curve). In this article we report some attempts to extend
the results to finite subgroups of, which is being studied jointly with Iku Nakamura(Hokkaido Univ.) and Yasushi Gomi(Univ.). For simplicity we take thecomplex number field
as a ground field and representations considered are complex
representations.
$0$$G\subset SL(2, \mathrm{c})$$(\mathrm{C}^{2})$$G$$\mathrm{M}_{\mathrm{C}}\mathrm{K}\mathrm{a}\mathrm{y}$$\mathrm{C}^{2}/G$$SL(3, \mathrm{c})$$\mathrm{S}_{0}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{a}$$\mathrm{C}$

1. Let be a finite group, $Irr(G)=\{\chi_{1}, \ldots, \chi_{s}\}$ be the set of all irreducible characters ofand $Irr(G)\#=Irr(G)-\{1c\}$ . Given a character of , we can form the representationgraph $\Gamma(G)=\Gamma_{\chi}(G)$ as follows: the set of vertices is $Irr(G)$ and the directed edge of
weightfromtois determined by the relation
$G$

$G$

$\chi$$m_{ij}$$\chi_{i}$$G$$\chi_{j}$$x \cdot\chi_{i}=\sum_{j=1}m_{i\chi_{j}}Sj$,$i=1,$$\ldots,$$s$.

We use the convention that a pair of opposing directed edges of weight 1 is represented
by a single edge and the weightis omitted if $m_{ij}=1$ .
$m_{ij}$Example 1. Let be the quaternion group of order 8. Then $Irr(G)$ consists of 4 linear
charcters and the character of 2-dimensional representaion. Thenis eactly theextended Dynkin diagram of type
centered at .
$G$$\Gamma_{\chi}(G)$$\chi$$D_{4}$$\chi$Example 2. Let
be the alternating group of degree 5,. Then $Irr(G)=\{1,$ $\chi=$$3_{1},3_{2},4,5\}$ , (where the characters are expressed by the degrees of theing representations), andbecomes as follows:$G$$A_{5}$$\mathrm{c}\mathrm{o}\mathrm{r}\Gamma \mathrm{e}\mathrm{S}\mathrm{p}\mathrm{o}\mathrm{n}.\mathrm{d}$$\Gamma_{\chi}(G)$2. In [M] J.

$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$stated the following which is now famous as$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$observation.

Proposition. Let
be a finite subgroup of
and be the character of theinclusion representation. Thenis an extended Dynkin diagram of typeor E.
$G$$SL(2, \mathrm{c})$$\Gamma_{\chi}(G)$$\chi$$\mathrm{A},$$\mathrm{D}$138
Conversely every such extended Dynkin diagram is obtained as a representation graph of.
a subgroup of$SL(2, \mathrm{c})$Thusofobservation establishes a bijective correspondence between subgroups
and the extended Dynkin diagramof type
and E.

$\mathrm{M}\mathrm{c}\mathrm{K}\mathrm{a}\mathrm{y}$$SL(2, \mathrm{c})$$\overline{x}_{c}$$\mathrm{A},$$G$$\mathrm{D}$3. There is another famous correspondence between subgroups
ofand theDynkin diagramand ()
extended Dynkin diagram ofisLet
and $p:Sarrow S$ be the minimal resolution of sigularity. Then the singularfiber, $p^{-1}(0)$ , is a union of projective lines, Dynkin curve of type, having intersection, where is the Cartan matrix of type
. In particular the graph obtainedby Dynkin curve as follows is the Dynkin diagram: the set of vertices is that of projective lines appearing in Dynkin curve and two lines are joined iff they meet. For details,
please see a survey article of R.
or P.
.
These two correspondences were famous, but relations between them had not beenclear. Recently an explanation of these correspondences was given by Y.Ito and I.
[IN2] and, [N2], using Hilbert schemes.
$G$$\mathrm{E}.$$\mathrm{D}$$SL(2, \mathrm{c})$$X_{G}$$\mathrm{T}\mathrm{h}\mathrm{e}$$X_{G}\mathrm{o}\mathrm{f}\underline{\mathrm{t}}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{A},$$\overline{X}_{G}.$$S=\mathrm{C}^{2}/G$$X_{G}$$C$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}-\mathit{0}$$X_{G}$$X_{G}$$\mathrm{s}_{\mathrm{t}\mathrm{e}}\mathrm{i}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}[\mathrm{S}\mathrm{t}]$$\mathrm{s}1_{\mathrm{o}\mathrm{d}_{\mathrm{o}\mathrm{w}\mathrm{y}[\mathrm{S}1]}}$$\mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}[\mathrm{I}\mathrm{N}\mathrm{l}]$$\mathrm{I}.\mathrm{N}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{m}\mathrm{u}\mathrm{r}\mathrm{a}[\mathrm{N}\mathrm{l}]$4. Let Hilb (Cm) be the Hilbert scheme ofparametrizing all the -dimensional subschemes of length and let Symm
be the n-th symmetric product of, that is,
the quotient of -copies ofby the natural action of the symmetric group of degree. There is a canonical morphism from Hilb (Cm) toassociating to each-dimensional subscheme ofits support. Let
be a finite subgroup of.
The group acts on
so that it acts naturally on both Hilb (Cm) and Symm.
Since is -equivariant, induces a morphism from the -fixed point set Hilb
to the -fixed point set Symm.
Now consider the special situation that is the order of the groupand $m=2$ .
Then Symm
is isomorphic to the quotient space
and there is a unique
irreducible component of Hilb
dominating Symm
, which we denote byHilband call it the Hilbert scheme of -orbits, following the notation and thedefinition by I.Nakamura. Notice that we have a morphism : Hilbinduced by . The following theorem is proved in a unified way.
$0$$\mathrm{C}^{m}$$n$$\mathrm{C}^{m}$$(\mathrm{c}^{m})$$\mathrm{C}^{m}$

$n$

$n$

$s_{ymm^{n}}(\mathrm{C}^{m})$

$\pi$

$0$

$G$

$\mathrm{C}^{m}$

$G$

$G$

$\pi$

$SL(m, \mathrm{c})$$\mathrm{C}^{m}$$(\mathrm{c}^{m})$

$G$

$\pi$

$G$

$(\mathrm{c}^{m})^{G}$$(\mathrm{C}^{m})^{G}$

$G$

$n$

$(\mathrm{C}^{2})^{G}$$\mathrm{C}^{2}/G$$(\mathrm{c}^{2})^{G}$$(\mathrm{C}^{2})^{G}$$G$$(\mathrm{C}^{2})$$(\mathrm{C}^{2})arrow \mathrm{C}^{2}/G$

$p$

$\pi$

Theorem. [IN2]. Hilb
resolution of singularity.

is nonsingular and$(\mathrm{c}^{2})$$p$: Hilb$(\mathrm{c}^{2})arrow \mathrm{C}^{2}/G$

is a minimal

5. Let $R=\mathrm{C}[x, y]$ be the ring of regular functions on
and $M$ be the maximal ideal
corresponding to the origin, that is $M=(x, y)$ . For a finite groupof order, let
be the invarint algebra of and $N$ be the ideal of generated by invariant
homogeneous polynomials of positive degree which generate. The ring $R_{G}=R/N$ iscalled the coinvariant algebra of .
We identify a -invariant -dimensional subscheme with its defining ideal of . Forwith support origin, put $V(I)=I/(MI+N)$ . Then $V(I)$ is a G-moduleand we denote its character by. Let $E$ be the exceptional set of and $Irr(E)$ be$\mathrm{C}^{2}$$G\subset SL(2, \mathrm{c})$$n$$R^{G}$$G$$R$$R^{G}$

$G$

$G$

$R$

$0$

$I\in Hilb^{G}(\mathrm{c}^{2})$$\chi_{V(I)}$

$p$

,

139

the set of irreducible components of $E$ . For $\chi\in Irr(G)\#$ , define$E(\chi)=\{I\in E|(x, xV(I))_{G}\neq 0\}$whereis the usual inner product on functions on
the following theorem is obtained.
$(, )_{G}$Theorem.

$[\mathrm{I}\mathrm{N}2],[\mathrm{N}2]$$E=$

and the map{$G$. Then by verifying every case.

$I$$\chirightarrow E(\chi)$$G$$|$-invarinant ideal of $R,$ $N\subset I\subset M,$$R/I\simeq \mathrm{C}G$}

gives a bijective correspondence between $Irr(G)\#$ and $Irr(E)$ .

$M$ and $N$ are defined similarly for6. Let be a subgroup of. $R,$
and as in 5. Now theorem 5 suggests the necessity to study$G$$SL(3, \mathrm{c})$$R^{G},$$R_{G},$$\mathrm{C}^{3}$$G$$F_{G}:=${

$I$

$|$

$G$-invarinant ideal of $R,$ $N\subset I\subset M,$$R/I\simeq \mathrm{C}G$},

which would be a fiber of the origin of the quotient spacein the Hilbert scheme of-orbits. For that purpose we need detailed structures of the coinvariant algebras.
What we have mainly obtained so far are$\mathrm{C}^{3}/G$$\mathrm{G}$$R_{G}$$\bullet$$\bullet$decomposition of(or its overalgebra) into irreducible components, particularly
for groups of orders $60(A_{5}),$ $168(PSL(2,7)),$ $108,180,216,504,648$ , and 1080,$R_{G}$

explicit determination of basis for each irreducible component above for$A_{5}$

and$PSL(2,7)$ .

As an outcome of these calculations we can show thatwhose graph is given by$F_{A_{5}}$is a union of projective lines

and a graph for $PSL(2,7)$ also can be given. Details will appear in [GNS].

References[GNS] Y.Gomi, I.Nakamura and K.Shinoda, Coinvariant algebras of some finite groups,(in preparation).

140[IN1] Y.Ito and I.Nakamura, Hilbert schemes and simple singularities, to appear in Proc.
Japan Academy.

[IN2] –, Hilbert schemes and simple singularities$A_{n}$

and$D_{n}$, (preprint).

[M]

McKay, Graphs, singularities, and finite groups, Proc. Symp. Pure Math., AMS
37(1980),183-186.

[N1]I.Nakamura, Simple singularities, McKay correspondence and Hilbert schemes of-orbits, (preprint).
$\mathrm{G}$[N2]–,[S1]P.Slodowy, Simple singularities, Springer Lecture Note 815(1980).

[St]

R.Steinberg, Kleinian singularities and unipotent elements, Proc. Symp. Pure
Math., AMS 37(1980),265-270.

Hilbert schemes and simple singulariries$E_{6},$ $E_{7}$and$E_{8},(\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t})$.