Invariant polynomials on tensors under the action of a product of orthogonal groups
Invariant polynomials on tensors under the action of a product of orthogonal groups
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正交群乘积作用下张量上的不变多项式
DOI：
10.1090/tran/6497
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发表时间：
2015
影响因子：
1.3
通讯作者：
L. Williams
中科院分区：
文献类型：
--
作者：
L. Williams
设K是正交群的乘积O（n1）× O（n2）× ··· × O（nr）.令V = r=1 C n i，定义每个正交因子的表示的r重张量积。我们计算一个稳定的公式的维度的K-不变代数的度d齐次多项式函数的V。要做到这一点，我们计算一个公式的匹配的数量与一个固定的置换交换。最后，我们提供了公式的不变量和描述的基础之间的双射空间的不变量和同构类的某些r-正则图的d顶点，以及一种方法相关联的每个不变量的其他组合设置，如系统发育树。
Let K be the product O(n1) × O(n2) × ··· × O(nr) of orthogonal groups. Let V = ⊗ r=1 C n i , the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of degree d homogeneous polynomial functions on V. To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain r-regular graphs on d vertices, as well as a method of associating each invariant to other combinatorial settings such as phylogenetic trees.