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

Author: Lauren Kelly Williams
Journal: Trans. Amer. Math. Soc. 368 (2016), 1411-1433
MSC (2010): Primary 22E47; Secondary 05A19, 05C30
Published electronically: April 8, 2015
MathSciNet review: 3430368
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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.

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Additional Information

Lauren Kelly Williams
Affiliation: Department of Mathematics, University of Wisconsin–Milwaukee, P.O. Box 0413, Milwaukee, Wisconsin 53201
Address at time of publication: Department of Mathematics and Computer Systems, Mercyhurst University, 501 East 38th Street, Erie, Pennsylvania 16546

Received by editor(s): November 15, 2013
Received by editor(s) in revised form: June 18, 2014
Published electronically: April 8, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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