数理解析研究所講究録
第 765 巻 1991 年 109-125

109

Relative invariants of the polynomial rings

over the finite and tame type quivers

小池 和彦

KOIKE KAZUHIKO

Department of Mathematics

Aoyama Gakuin University

In this note we consider the following problem. Let $F$ be one of the$D_{r},$$E_{r},\tilde{A}_{r}$ハゾ,$r’\tilde{E}r$arrows. Namely$F$type quivers with$r$

one of the Dynkin diagrams of types$V_{i}$

on each vertex in

in

$F$

$f$

is a linear homomorphism from

$V_{i}$

vertices and arbitrarily directed

$F$

$F$

, then the gragh coincide with

$A_{r},$

We take a representaion of the quiver. Here,is a directed graph without multiple edges and if we

ignore the directions of the arrows in$i$$A_{r}$$D_{r},$$F$$E_{r},\tilde{A}_{r},\tilde{D}_{r},\tilde{E}_{r}$.

, namely we put a vector spaceand put a linear homomorphism$f$on each arrowis a finite dimensional vector space over some field

For example if $F$ is an$A_{r}$$V_{i}$

to$V_{i}$

if$V_{i}arrow^{f}V_{i}$

$k$

and

.

type quiver, a representation of$F$

is givenby$V_{1}arrow f_{1}V_{2}arrow f_{2}V_{3}arrow JsV_{4}arrow f_{4}$$(F)$. $arrow\cdotarrow..$ .

$arrow^{f_{r-1}}V_{r}$Typeset by$A_{\mathcal{M}^{S- T\mathfrak{x}X}}$

110

Here

$V_{i}$

is a finite dimensional vector space over some field

is a linear endomorphism fromto$V_{i}$$V_{i}arrow^{f_{i}}V_{i+1}$if$V_{i}$

to$V_{i+1}$if$V_{i}arrow^{f_{i}}V_{i+1}$

$k$

and

and from$f_{i}$$V_{i+1}$.

For the exact definition and meanings of finite and tame type quivers,see [Kal], [Ka3], [Ka4], [Gal], [Ga2] and [B-G-P].
Let $V=\oplus_{iarrow jin}{}_{p}Hom(V_{i}, V_{i})$ and $G=GL(V_{1})\cross GL(V_{2})\cross\cdots\cross$$GL(V_{r})$. Then $G$ acts on $V$ naturally, i.e., for $g=(g_{1}, g_{2}, \cdots g_{r})\in G$ ,the action of

$G$

on

$V$

$V_{i}arrow^{f}V_{i}$is given by $g\cdot f=g_{j}fg_{i^{-1}}$ , if

For example in the case of the above$A_{r}$.

type quiver,$V= \bigoplus_{iarrow i+1inF}Hom(V_{i}, V_{i+1})\oplus\bigoplus_{iarrow i+1inF}Hom(V_{i+1}, V_{i})$

Then

$G$

acts on $V$ naturally. Let $S(V)$ be the polynomial ring over

V. The action of

$G$

on

$V$

naturally extends to the action on $S(V)$ . The

problem is:
PROBLEM. What is the relati$v^{r}e$(or absol $ute$) irivari an$ts$

in $S(V)$ withrespect to this action.?

We consider this problem for$A_{r},$$D_{r},$$E_{r},\tilde{A}_{r},\tilde{D}_{r},\tilde{E}_{r}$type quivers

with arbitrarily directed arrows.

We give answers to the above problem for the$A_{r},$$D_{r},\tilde{A}_{r},\tilde{D}_{r}$quivers with arbitrarily directed arrows in the case of
number). (The same holds for any field

For the$E_{r},\tilde{E}_{r}$$k$(complex$k=\mathbb{C}$of characteristic

$0.$

type

)

type quivers, I have not yet obtained complete answers

to the above problem.

We will show a set of generators of the relative (or absolute) invariants

in each case.

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Let $F$ be an$A_{r}$

type quivers whose arrows are directed one way,
. $..arrow^{f_{r-1}}V_{r}$ .

$V_{1}arrow^{j_{1}}V_{2}arrow^{f_{2}}V_{3}arrow^{f_{3}}$Then our theorem is given as follows.
We fix a baseof each vector space$\{e_{i}^{s}\}(1\leqq i\leqq n_{s})$$(s=1,2, \cdots r)$ denotes the dimension of$V_{s}$$V_{s},$.where$n_{s}$.

Since$S(V)=S( \bigoplus_{s=1}^{r-1}Hom(V_{s}, V_{s+1}))=\bigotimes_{s=1}^{r-1}S(Hom(V_{s}, V_{s+1}))$, $S(V)$ can be considered as the polynomial ring in the indeterminates$\{x_{i,j}^{(s)}\}$

where $1\leqq i\leqq n_{s+1},1\leqq j\leqq n_{s}$ , and $s=1,2,$ $\cdots r-1$ , where$\{x_{i,j}^{(s)}\}$

is the dual base of the base$\{e_{i}^{s}’\}$

denotes the dual base of the base$e_{i}^{s}\otimes e_{i}^{s+1^{*}}$$\{e_{i}^{s*}\otimes e_{i^{s+1}}\}$$\{e_{i}^{s}\}$of $Hom(V, , V_{s+1})$ . Hereof $V,$ . Namely.

In other words, if we substitute some values to$(x_{i,j}^{(s)})_{i,j}$$x_{i_{\rangle}j}^{(s)}=$corresponds to the homomorphism$f_{s}$$x_{i,j}^{(s)}’ s$, then the matrix

with respect to the above

basis.
Let $M_{s+1,s}$ be the matrix$(x_{i,j}^{(s)})_{i,j}$coefficient is the indeterminate
DEFINITION. For an$yk,$

the polynomial$P_{\ell,k}$$P_{l,k}$$l$$x_{i,j}^{(s)}.$. ($n_{s+1}\cross n_{s}$matrix whose $(i, j)$ -th)with $1\leqq k\leqq\ell\leqq r$ and $n_{k}=n_{t}$ , we defi neby$:=\det(M_{l,l-1}M_{l-1,l-2}\cdots M_{k+1,k})$and call these polynomials by determinan $tal$ invariants.

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Clearly

is a relative invariant and $P_{t,k}\neq 0$ if and only if for$P_{\ell,k}$

any $v(k<v<\ell)$ ,$n_{v}\geqq n_{k}=n_{l}$. Moreover if a pair $(k, f)$ satisfies

the condition that $n_{v}>n_{k}=n_{t}$ for any $v(k<v<f)$ , then we callthe determinantal invariant$P_{\ell,k}$primitive. Clearly any determinantal

invariant can be writen as the product of the primitive ones.
THEOREM. Let $F$ be an

Then the relati$\tau\prime e$type uiver with one-way directed arrows.

$A_{r}$$q$invariants in $S(V)$ amount to be the monom ials of

the prim tive determinantal invariants $P_{l,k}s$ . Moreover the prim itive$i$determin antal invarian$ts$

For a quiver $F$ of typeare algebraic ally independent.
$A_{r}$

with arbitrarily directed arrows, $generat_{-}ors$of the relative invariants are given as follows.
Let $p,$ $q(p<q)$ be vertices in $F$ and. . . $<u_{k}<q$ ) be the sources between$u_{1},$$p$$u_{2},$$u_{3},$

and$q$$\cdots$$u_{k}(p<u_{1}<u_{2}<$and let$v_{1},$$(p<v_{1}<v_{2}<\cdots<v_{l}<q)$ be the sinks between $p$ and$k+1$ or$k$

or $k-1.$ ) Here a vertex

all the arrows connected to(sink’$i$in a quiver$i$are started from

if all the arrows connected to

$j$

$i$

$F$

$v_{2},$$q$$v_{3},$$\cdot\cdot$$v_{l}$. ( can be$l$is called (source “ ifand a vertex$j$is called

are terminated at .
$j$

We prepare a notation. Let $u,v(u<v)$ be vertices in$F$such that there

are no sinks and sources between them. Then there are two possibilities.
(P1)$uarrow\cdotarrow\cdotarrow.$ . . $arrow v$

(P2)

$u$

\leftarrow 一一一. $arrow$ .

\leftarrow 一一--一-. . . $arrow v$

In the case of (P1), we define the matrix by$\Lambda l_{v,u}=M_{v,v-1}M_{v-1,v-2}\cdot$

4. . $M_{u+1,u}$113
and in the case of (P2), we define the matrix by$M_{u,v}=M_{u,u+1}M_{u+1,u+2}\cdots M_{v-1,v}$.

Here $M_{i+1,i}$ is the matrix $(x_{kl}^{(i)})(1\leqq k\leqq n_{i+1},1\leqq l\leqq n_{i})$ cor-

respoding to the element of IIom$(V_{i}, V_{i+1})^{*}$$(x_{kt}^{(i)})(1\leqq k\leqq n_{i}, 1\leqq l\leqq n_{i+1})$$Hom(V_{i+1}, V_{i})^{*}$

and $M_{1,i+1}$ is the matrix

corresponding to the element of.

Assume that the sources and the sinks between

$p$

and

$q$

are located asfollows:$p<u_{1}<v_{1}<u_{2}<\cdots<u_{k}<v_{k}<q$.

$parrow$.

$arrow\dot{u}_{1}arrow$.

$arrow v_{1}arrow u_{2}arrow.$. . $arrow v_{k}arrowarrow q$

In this case, we define the matrix $M$ as follows:$000.\cdot$$M=(M_{v_{0}}^{p_{1}u_{u^{1_{1}}}}0M_{0},,.$$M_{v_{0^{2}},u_{2}^{2}}^{v_{0^{1}},u}M0$$M_{v_{0^{2}},u_{3}^{3}}^{v_{0^{3}},u}M00$$M_{v,q}^{0}0_{k}000]$$M_{\dot{v}_{k},u_{k}}$Then $M$ is an $(n_{p}+n_{v_{1}}+n_{v_{2}}+\cdots+n_{v_{k}})\cross$ ($n_{u_{1}}+n_{u_{2}}+\cdots n_{u_{k}}$十$n_{q}$)

matrix. If $n_{p}+n_{v_{1}}+n_{v_{2}}+\cdots+n_{v_{k}}=n_{u_{1}}+n_{u_{2}}+\cdots n_{u_{k}}+n_{q}$ , we cantake the determinant of $M$ .

Clearly if $\det(M)\neq 0,$ $\det(M)$ is a relative invariant in $S(V)$ . Since theaction of$G$

on $\det(M)$ just coincides with the matrix multiplication of

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diag

$(g, g_{1},g_{2}, \cdots g_{k})$from the left and diag$(h_{1}^{-1}, h_{2}^{-1}, \cdots h_{k}^{-1}, h^{-1})$

fromthe right, where $g\in GL(V_{p}),g_{i}\in GL(V_{v_{i}}),$ $h_{i}\in GL(V_{u_{i}}),$ $h\in GL(V_{q})$and diag $(g, g_{1},g_{2}, \cdots g_{k})$ denotes the matrix whose diagonal blocks con-sist of$g,$ $g_{1},$ $g_{2},$ $\cdots g_{k}$and whose off-diagonal blocks are all

matrices.

$0$

Therefore if $\det(M)\neq 0$ , then $P_{q,p}=\det(M)$ is a relative invariant ofweight$(0,0, \cdots p\wedge 1,0, \cdots\overline{u_{1}^{-1,0}}’ v^{1_{1}}\wedge 0, \cdots v^{1_{k},0}\wedge -1,0\wedge q 0)$

We will determine when $\det(M)\neq 0$ . It is easy to see that the necessary condition for $\det(M)\neq 0$ is given by$n_{p}\leqq n_{p+1},$ $n_{p+2},$ $\cdots n_{u_{1}}$,,$n_{u_{1}}-n_{p}\leqq n_{u_{1}+1)}n_{u_{1}+2},$ $\cdots n_{v_{1}}$$n_{v_{1}}-n_{u_{1}}+n_{p}\leqq n_{v_{1}+1},$ $n_{v_{1}+2},$$n_{u_{2}}-n_{v_{1}}+n_{u_{1}}-n_{p}\leqq n_{u_{2}+1},$$n_{u_{2}+2},$$\cdots n_{u_{2}}$,$\cdots n_{v_{2}}$,$\leqq$$n_{v_{k}}-n_{u_{k}}+n_{v_{k-1}}-\cdots+n_{p}\leqq n_{v_{k}+1},$$n_{v_{k}+2},$ $\cdots n_{q}$We will define primitive determinantal invariants. A determinantal

invariant $P_{q,p}=\det(M)$ is called “primitive “ if the inequalities in theabove hold strictly.

Any determinantal invariant can be decomposed into the product ofthe primitive ones.
For the cases in which the sources and sinks between

$p$

and

$q$

are

located differently, the matrix whose determinant gives a determinantal

invariant is obtained by arranging the matrices6$M_{v,u}$and$M_{v’,u}$vertically115
at the source

the matrices

are adjacent sinks to

$v$

and

$v’$

$M_{v,u}$and$AI_{v,u’}$$u$(adjacent sources to

$v.$

$u.$

) and by arranging

horizontally at the sink $v(u$ and) and by putting

$0$

$u’$

are

matrices at the other places.

The primitiveness of them is defined by a similar inequalities to theabove. (See $[K1]$ \S 4 for the details.)
In any cases the relative invariants for the$A_{r}$

type quivers are themonomials of the primitive determinantal invariants and the primitive
ones are algebraically independent.
NamelyTHEOREM. Let

$F$

be an

$A_{r}$

type quiver with arbi trarily directed arrows.

The relative invariants in $S(V)$ am ounts to the monomi als of the prim-

itive determinantal invariants$P_{\ell,k}s$. Moreover the primitive algebraic

invari ants are algebrai ally independent.
$c$

Next let $F$ be an$\tilde{A}_{r}$type quivers whose arrows are directed one way$V_{1}arrow^{f_{1}}V_{2}arrow^{f_{2}}V_{3}arrow^{j_{3}}$(F)minates$arrow^{J:-1}$$V_{i}$$\downarrow f$$J_{r}\uparrow$$\dot{V}_{r}arrow^{f_{r-1}}$$S(V)$.. .

$arrow$;$arrow\cdotsarrow^{f_{i+1}}V_{i+1}$

can also be considered as the polynomial ring in the indeter$\{x_{i,j}^{(s)}\}$

where $1\leqq i\leqq n_{s+1},1\leqq j\leqq n_{s}$ , and $s=1,2,$$\cdots r$.

We define the determinantal invariants and the primitive determinantal invariants just in the same way as the above. (Here we consider$V_{r+};=V_{i}.)$Since$\tilde{A}_{r}$type quiver has the symmetry under the cyclic per-mutations, We may assume that $n_{1}=Minimum\{n_{1}, n_{2}, \cdots n_{r}\}$ . Thenwe will define absolute invariants $\phi_{i}\in S(V)(i=1,2, \cdots n_{1})$ as follows.

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DEFINITION. Let $\phi_{i}\in S(V)(i=1,2, --, n_{1})$ be the i-th elementary

symmetric func tion of the product of matrices$M_{1,r}M_{r,r-1}M_{r-1,r-2}\cdots M_{2,1}$, namely$\det(tI_{n_{1}}-M_{1,r}M_{r,r-1}\cdots M_{2,1})=\sum_{k=0}^{n_{1}}\phi_{i}(-1)^{i}t^{n_{1}-i}$

It is easy to see that$\phi_{i}’ s$

are absolute invariants.

For a relative invariant $f\in S(V)$ , we call that$(k_{1}, k_{2}, --, k_{r})\in Z^{r}$$f$

has weight$S(V)^{\mathfrak{k}}$

$[$

$=$

if $g\cdot f=(\det g_{1})^{k_{1}}(\det g_{2})^{k_{2}}\cdots(\det g_{r})^{k_{r}}f$ where$g=(g_{1}, g_{2}, \cdots g_{r})\in G=GL(n_{1})\cross GL(n_{2})\cross\cdots GL(n_{r})$By.

, we denote the relative invariants of weight$g$.

in $S(V)$ . Here

we can state our theorem for this case.
THEOREM. Let $F$ be $an$

ノの

$r$

type quiver with one-way directed arrows.

(1)

The absol $ute$ invariants $S(V)^{G}$ is the polyn $omial$ ring oferators$\phi_{1},$$\phi_{2},$$\cdots$$\phi_{n_{1\rangle}}$gen-

namely,$S(V)^{G}=\mathbb{C}[\phi_{1}, \phi_{2}, \cdots\phi_{n_{1}}]$.

The relative invariants in $S(V)$ amount to be the monomials of(2)$\phi_{1},$$n_{1}$$\phi_{2},$$\cdots\phi_{n_{1}-1}$

invariants.
(3)$\phi_{1},$$\phi_{2},$an $dP_{ji}s$ , where$\cdots\phi_{n_{1}-1}$$P_{j,i}s$

are th primitive determin an $tal$$e$

and $P_{ji}s$ are algebraically independent.

As $S(V)^{G}$ mod $ule,$ $S(V)^{t}$ is a free mod le of rank one.
$u$

For the other cases in which there exist a sink or a source in theoriginal$\tilde{A}_{r}$type quiver$F$, then we have no absolute invariants other thanconstant. In this case we also can give explicit generators of the relative8

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invariants in $S(V)$ and prove that they are algebraically independent.
(See \S 5 in $[K1].$ )We will move to thequiver with$r$

$D_{r}$

andtype quivers. Let $F$ be a$\tilde{D}_{r}$$D_{r}$

type

vertices and arbitrarily directed arrows We fix a represen-tation of the quiver$F$.

For example let $F$ be a quiver in which the arrows at the branchingvertex $r-2$ are directed as follows and the other arrows are directed
arbitrarily.
ordinary at $r-2$ ( arrows started from $r-2$ to

Case

$2$

$parrow\cdotarrow\cdotarrow\cdotsarrow qarrow\cdotarrow..$$r$

and $r-1$ ). $arrow\cdotarrow\cdotarrow r-2arrow r-1$$\downarrow$

As in the$A_{r}$

type quivers,according to the distribution of the sources

and the sinks between the vertices
But as in the cases of the$A_{r}$$p$and , we must divide the cases.
$q$type quivers, a matrix whose determinant

gives a primitive invariant is obtained by arranging the matrices
and$M_{v’,u}$vertically at the sourceand by arranging the matrices
$v$

(

$u$

and

$u’$

$u$

$M_{v,u}$are adjacent sources to

$v.$

( and
$v$

and

$v’$

$M_{v,u’}$$M_{v,u}$are adjacent sinks to

$u.$

)

horizontally at the sink

) and by putting$0$matrices at theother places.

Therefore for the$D_{r}$

type quivers we only give a primitive invariant

for an exemplified case, since for the other cases, primitive invariants

are defined just in the same way.
For example in the above quiver let the sources and the sinks between

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$p$

and $r-2$ be located as follows:$p<v_{1}<u_{1}<\cdots<u_{t-1}<q<v_{t}<u_{t}<\cdots<v_{s}<u_{\theta}<r-2$ .

If $n_{u_{\epsilon}}-n_{v_{s}}+\cdots+n_{u_{1}}-n_{v_{1}}+n_{p}+n_{u_{\epsilon}}-n_{v_{s}}+\cdots+n_{u_{t}}-n_{v_{i}}+n_{q}=n_{r-1}+n_{r}$ ,then we will define the matrix $M$ in the following way.
In the case of $n_{u_{s}}-n_{v_{s}}+\cdots+n_{u_{t}}-n_{v_{t}}+n_{q}>n_{r}$ and $n_{u_{s}}-n_{v_{s}}+$. . . $+n_{u_{1}}-n_{v_{1}}+n_{p}<n_{r-1}$ , let$M=$$(M_{v_{0}}00^{1P}0000M_{v_{0}.u_{1}}000001M_{v_{s_{0}}}0^{u_{S-1}}0000M_{r,r-2}M_{r-2,u_{S}}^{u_{S}}M_{v_{0}}0’000sM_{r-1}^{M_{r,r,-2}M_{r2,u_{S}}}M_{v_{S}}^{r-2}0^{M_{u}^{-}-2,u_{S}}0000’ M_{v}.0_{u_{s-1}}00000.M_{v,u}00000_{\ell r}0.M_{v,q}^{0}0_{i}00000)$

If $n_{u_{s}}-n_{v_{\epsilon}}+\cdots+n_{u_{t}}-n_{v_{t}}+n_{q}=n_{r}$ hence $n_{u_{s}}-n_{v_{s}}+\cdots+n_{u_{1}}-$$n_{v_{1}}+n_{p}=n_{r-1}$, the situation reduces to the$A_{r}$

cases.

This $\phi_{q,p,r-1,r}=\det(M)$ is called primitive if$n_{p}<n_{p+1},$ $n_{p+2},$ $\cdots n_{v_{1}}$,$n_{v_{1}}-n_{p}<n_{v_{1}+1},$ $n_{v_{1}+2)}\cdots n_{u_{1}}$,$n_{u_{1}}-n_{v_{1}}+n_{p}<n_{u_{1}+1},$ $n_{u_{1}+2},$ $\cdots n_{v_{2}}$

,

:

$<$

.

$n_{u_{s}}-n_{v_{s}}+\cdots+n_{p}<n_{u_{S}+1},$ $n_{u_{s}+2},$ $\cdots n_{r-2}$

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and
$n_{q}<n_{q+1},$ $n_{q+2},$ $\cdots n_{v_{t}}$$n_{v_{t}}-n_{q}<n_{v_{\ell}+1},$ $n_{v_{t}+2},$:$n_{u}$。

$\cdots n_{u_{t}}$

,

:

$<$

$-n_{v_{s}}+\cdots+n_{q}<n_{u_{s}+1},$ $n_{u_{s}+2},$ $\cdots n_{r-2}$

By substituting the special values toprimitive,$\phi_{q,p,r-1,r}$$x_{i}^{(s_{j})}$, we can see easily that theis non zero..

We also define the primitive invariants

in which the sinks and sources between$p$$\phi_{q,p,r-1,r}’ s$

and

$q$

for the other cases

and $r-2$ are located in

the different ways.

Then we have
THEO REM.

The relative invari an$ts$

in $S(V)$ amount to be the$111$

onomials in all theprimitive determinantal in varian $ts\phi_{q,p,r-1,r}s,$ $P_{q,p}s$ an$d$the primitive

relative invari ants are algebraic ally independent.

We can also give explicit generators for the$D_{r}$

type quiver $F$ in which

the directions of the arrows at the branching vertex $r-2$ are different

from the above and the same theorem hold for these cases.
Let

$F$

Case

be a

$\tilde{D}_{r}$type quiver for example, given by

ordinary at the branching vertices 2 and $r-2$$1arrow 2arrow\cdotarrow\cdotarrow\cdotarrow.$$(F)$. . $arrow r-2arrow r-1$$\downarrow$$\uparrow$

0

$r$

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Let the sinks and sources between 2 and $r-2$ be located in the fol-lowing way, $2<v_{1}<u_{1}<\cdots<u_{s}<r-2$ .
If $n_{r}-n_{u_{s}}+n_{v_{s}}+\cdots-n_{u_{1}}+n_{v_{1}}+n_{r-1}-n_{u_{s}}+n_{v_{s}}+\cdots-n_{u_{1}}+n_{v_{1}}=$$n_{0}+n_{1}$, then we can define the matrix $M$ by$M=$$(00o^{1\prime}oo000000011M_{v}0^{u_{S-1}}oo000.M_{r_{0}^{-1’,u_{S}^{\epsilon_{S}}}}M-1M_{r_{0}^{v_{0}}}0s^{u_{u}}M_{v_{0’}}M_{r_{s}}^{0}0^{u_{u^{s_{s}}}}000M_{v_{s_{0^{u_{S-1}}}}}.00000.M_{1^{u}1}000000.M_{v,1}^{0}M_{v,1}0_{1}^{1}0000)$,where $M_{v_{1},1}=M_{v_{1},2}M_{2,1},$ $M_{v_{1},0}=M_{v_{1},2}M_{2,0},$ $M_{r,u_{k}}=M_{r,r-2}M_{r-2,u_{k}}$and $M_{r-1,u_{k}}=M_{r-1,r-2}M_{r-2,u_{k}}$ .

This $\phi_{0,1,r-1,r}=\det(M)$ is called primitive if$n_{2}<n_{3},$ $\cdots n_{v_{1}}$$n_{v_{1}}-n_{2}<n_{v_{1}+1},$ $n_{v_{1}+2},$ $\cdots n_{u_{1}}$,$n_{u_{1}}-n_{v_{1}}+n_{2}<n_{u_{1}+1},$ $n_{u_{1}+2},$ $\cdots n_{v_{2}}$

,

.

:

$<$

$n_{u_{s}}-n_{v_{s}}+\cdots+n_{2}<n_{u_{s}+1}$

Also for vertices

,

$p$

and

$q$

,

$n_{u_{s}+2},$ $\cdots n_{r-2}$.

with $u_{s}<p<v_{s+1},$ $v_{t}<q<u_{t}$we will define the matrix $M$ by $M=$

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$[Mv_{00M_{v}M_{v1}M_{v0}00}M_{v_{\theta\prime}}..\cdot 0,\cdot 0_{1},0_{1}0,,02M_{p_{\theta}},0_{u_{S-1}}.000000’0000M_{vu}^{v}M_{v,,.u_{2}}0000M_{v0}M00^{u_{u^{s_{\epsilon}}}}\cdot 00_{1},’ 0_{V}.00000000000000000000000000000000000000000000000001^{u}12111$,where and $M_{r,u_{k}}=M_{r,r-2}M_{r-2,u_{k}}$ and $M_{r-1,u_{k}}=M_{r-1,r-2}M_{r-2,u_{k}}$ .

If this matrix is a square $m$ atrix and $\det(M)\neq 0$ , then $\det(M)=$$\phi_{0,1,r-1,r,p,q}$

is a relative invariant. We also can define the primitivenessof this $\phi_{0,1,r-1,r,p,q}$ .

Then our theorem is as follows.
THEOREM. The relative invari ants in $S(V)$ amount to be the monomi-

als in all the primitive determinantal invarian $ts\phi_{q,p,r-1,r}s,$$P_{q,p}s,$$\phi_{0,1,r-1,r,p,q}s$$\phi_{0,1,p,q}s$,. The primitive relative in variants are algebraic ally

independen .
$t$These are examples of our answers to the problem. The proofs of theabove facts needs the standard monomial theory and some combinatorics

to calculate the Littlewood-Richardson coefficients explicitly for Youngdiagrams of the special shapes.
From the above the next problem comes up naturally and seems to

be interesting.
PROBLEM. For what quivers does the relative invariants $S(V)^{rel}$ havealgebraically in dependent generators? More specifically do es this condi-

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tion (having the algebraically independen generators) charact erize the$t$finite and the $tame$ type $quivers^{7}$
For the$A_{r},$$D_{r},\tilde{A}_{r},\tilde{D}_{r}$type quivers, this condition is satisfied.

We also state extentions of the original problem. Theorem comes up
naturally in the following situation.
Let

$P$

be a parabolic subgroup of GL(n) (where$n= \sum_{i=1}^{r}n_{i}$) definedby$n_{r}$. ..

$n_{2}n_{1}$.

$)n_{1}^{2}n^{r}n.\cdot$

Let $P=LU$ be a Levi decomposition of $P$ , whereof $P$ and$U$

is the unipotent radical of$n_{r}$

Let$\mathfrak{R}$

. ..

$P$

$L$

acts onjust coincides with the action of$n_{2}n_{1}$$\mathfrak{R}/[\mathfrak{R}\mathfrak{R}]$$G$

is a reductive part

. For example

be the Lie algebra corresponding toadjoint action, hence

$L$

on

$V$

$U$

. Then$L$acts on ’Yt by

by adjoint action This actionin the case of the$A_{r}$

type quiver

with one way directed arrows. So we can extend the problem as follows.

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PROBLEM 1. Let
$su$

$G$

be a semisimple Lie group and let $P$ be a arabolic$p$bgro up of G. Le $tP=LU$ is a Levi decomposition of

$P$

an le
$d$

$t\mathfrak{R}$be the Lie algebra correspon $ding$ to U. What is the relati $ve$ invarian
under the adjoint action of

$L$

on

$V=\mathfrak{R}/[\mathfrak{R}\mathfrak{R}]$It is known that the above action of
PROBLEM 1‘.

$L$

on

$V$

$ts$

?

is prehomogenius.

Consider the problem and the problem 1 over any field$k$instead of the complex fi $eld$ (or the field of characteristic ).
$0$

Especially it seems to be interesting to consider the preblem over thefinite field .
$k$

For example, let $F$ be an(F)$A_{2}$

type quiver and$k$be a finite field$V_{1}arrow^{f_{1}}V_{2}$If $\dim V_{1}=1$ , i.e., $V_{1}=k$ , then $S(V)$ is isomorphic to $S.(V_{2})$ and$G_{2}$

naturally acts on $S(V_{2})$ . It is known in this case that the absoluteinvariants$I_{1},$$I_{2},$$S(V_{2})^{G_{2}}$$\cdots I_{n_{2}}$are the polynomial ring in the Dickson’s invariants. Compared with the characteristic$0$case, (See Theorem

1) things seem to be slightly changed over a finte field,

15

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