Invariant polynomials on tensors under the action of a product of orthogonal groups

Abstract:

Let $K$ be the product $O_{n_1}\times O_{n_2} \times \cdots \times O_{n_r}$ of orthogonal groups. Let $V = \bigotimes _{i = 1}^r \mathbb {C}^{n_i}$, 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.