Some Conjectures About Invariant Theory and their Applications
Some Conjectures About Invariant Theory and their Applications
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发表时间：
2001
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通讯作者：
O. Mathieu
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作者：
O. Mathieu
结果表明，各种代数计算可以归结为同一类型的计算：人们必须研究积分级数∫K f(K)g(K)dk，其中f，g是紧Lie群K上的复值K-有限函数。因此，很容易对这种积分的性质提出一个一般性的猜想，并研究这个猜想的结果。主要猜想：设K是紧连通李群，f是K上的复值K-有限函数，使得对任意n>；0，∫K f(K)dk=0。然后，对于任意K-有限函数g，当n足够大时，我们有∫Kf(K)g(K)dk=0。特别地，我们证明了主要猜想蕴含着雅可比猜想。提出了另一个非常乐观的猜想，并解释了它与等谱问题的联系。
It turns out that various algebraic computations can be reduced to the same type of computations: one has to study the series of integrals ∫ K f(k)g(k) dk, where f, g are complex valued K-finite functions on a compact Lie group K. So it is tempting to state a general conjecture about the behavior of such integrals, and to investigate the consequences of the conjecture. Main conjecture: Let K be a compact connected Lie group and let f be a complex-valued K-finite function on K such that ∫ K f(k) dk = 0 for any n > 0. Then for any K-finite function g, we have ∫ K f(k)g(k) dk = 0 for n large enough. Especially, we prove that the main conjecture implies the jacobian conjecture. Another very optimistic conjecture is proposed, and its connection to isospectrality problems is explained.