REMARK ON SOME COMBINATORIAL CONSTRUCTION OF RELATIVE INVARIANTS
REMARK ON SOME COMBINATORIAL CONSTRUCTION OF RELATIVE INVARIANTS
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关于相关不变量的一些组合构造的评论
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发表时间：
1981
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通讯作者：
Tsukuba J. Math
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作者：
Tsukuba J. Math
确定相对不变量的显式是一个经典问题。然而，如果它太复杂了，知道相对不变量的数学结构似乎比仅仅写下它们的所有项更重要。因此，在本文中，我们建议利用一些原理来构造§1中的相对不变量，作为例子，我们将为所有n^6(见命题4.1、4.3和4.5)构造GL(n，C)在AC“上的一些相对不变量3，包括n=6、7、8、9的所有相对不变量。这项工作是作者在访问欧洲期间完成的，他衷心感谢西德曼海姆大学的H.Popp教授和法国格勒诺布尔大学的露娜教授对他的数学激励和鼓励。作者还衷心感谢M.Sato教授对n=6.§1.设p：G-^GL(V)是定义在复数域C上的约化代数群G的有限维有理表示.若V上的齐次多项式f{x)存在有理特征标X：G->；Cx满足f{p{g)-L{g)f{x)，则称V上的齐次多项式f{x)为相对不变量。设sr(F)是V上的r次全齐次多项式，则群G作用于sr(F)为(g#)(X)=[-<；j)(p{gylx)。我们用p(R)表示这个表示。由于G是可约的，所以它是不可约表示的直和：Pm=0/Oir)-我们用W(p表示p‘p的表示空间：SR(V)=@Wir)表示。请注意，齐次多项式i)是r次相对不变量当且仅当iffix)wv对于满足dimW/f的某个W-f=L，我们称PF分解为pf1)xpir2)irx-rt-r)，当plp是p^和pF*的对称张量的不可约分量之一时，我们称pF分解为pf1)xpr2)irx-rt-r)，并记为PP^pF^xpg^.这意味着W{p中的多项式^可以从W？1*和Wir2中的多项式得到，即对某些^EfW和Dt^Wpk，对某个^EfW和Dt^Wpk，可以简化问题
Itis a classicalproblem to determine the explicitform of relativeinvariants. However, if it is too complicated, it seems more important to know the mathematical structure of relativeinvariants than just to write down the allterms of them. Hence, in thispaper, we suggest to use some principle to construct relativeinvariants in§1,and as examples, we shallconstruct some relativeinvariants 3 of GL(n, C) on AC" for alln^6 (See Propositions 4.1,4.3,and 4.5),including allrelativeinvariants for n=6, 7,8,9. This work was done while the author was visitingEurope, and he would like to express his hearty thanks to Prof. H. Popp atMannheim University in West Germany, and to Prof.D. Luna at Grenoble University in France for theirmathematical stimulation and encouragement. The author also would like to express his hearty thanks to Prof. M. Sato who kindly explained his works for n=6. §1. Let p:G-^GL(V) be a finite-dimensionalrational representation of a reductive algebraic group G, alldefined over the complex number fieldC. A homogeneous polynomial f{x) on V is called a relativeinvariant if there exists a rational character X: G->CX satisfyingf{p{g)x)―l{g)f{x) for allg^G and xe V. Now letSr(F) be the allhomogeneous polynomials of degree r on V. Then the group G acts on Sr(F) as(g#)(x)=[-<j)(p{gYlx)for 0eSr(F), g^G and jcgF. We denote thisrepresentation by p(r). Since G is reductive, it is the direct sum of irreducible representations: pm = 0/Oir)- We denote by W(p the representation space of p'p:Sr(V)=@Wir). Note that a homogeneous polynomial i fix) is a relative invariant of degree r if and only iffix)^WV forsome W-f satisfying dimW/f)=l. We say that pf decomposes to pfl)xpir2)irx-＼-rt―r)and denote thisrelationby pP^pf^Xpg^ when plpis one of the irreducible components of the symmetric tensor of p?^ and pf*. This implies that the polynomials ^ in W{p can be obtained from those in W?1* and Wir2＼i.e.,§=Y^tQt for some ^efW and dt^WpK In such a way, we can reduce the problem of