高阶transvectant是冗余的
The Higher Transvectants are Redundant
摘要：
设A，B表示一般二元形式，u_r =（A，B）_r表示它们在经典不变量理论意义下的第r个transvectant。本文对u_r之间的所有二次合偶进行了分类。因此，我们证明了每一个高阶transvectants u_r，r>1，在它可以从u_0和u_1完全恢复的意义上是冗余的。这一结果可以用不完全的Segre嵌入进行几何解释。计算依赖于SL_2-表象的柯西精确序列和角动量量子理论中的9-j符号的概念。我们给出了SL_3，g_2和S_5的显式计算例子，以表明这个结果对于其它类型的表示也有可能类似。
英文摘要：
Let A, B denote generic binary forms, and let u_r = (A,B)_r denote their r-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the u_r. As a consequence, we show that each of the higher transvectants u_r, r>1, is redundant in the sense that it can be completely recovered from u_0 and u_1. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of SL_2-representations, and the notion of a 9-j symbol from the quantum theory of angular momentum. We give explicit computational examples for SL_3, g_2 and S_5 to show that this result has possible analogues for other categories of representations.
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