Invariant polynomial functions on tensors under the action of a product of orthogonal groups

TL;DR

This paper develops a stable formula for the dimension of invariant polynomial functions on tensor spaces under the action of a product of orthogonal groups, linking algebraic invariants to combinatorial structures.

Contribution

It introduces a new formula for invariant dimensions, connects invariants to r-regular graphs, and relates them to combinatorial models like phylogenetic trees.

Findings

Derived a stable formula for invariant algebra dimensions

Established a bijection between invariants and r-regular graphs

Linked invariants to combinatorial structures such as phylogenetic trees

Abstract

Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V 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.