Abstract
We provide a new tableau model from which one can easily deduce the characters of finite-dimensional irreducible polynomial representations of the special orthogonal group \(SO_n(\mathbb {C})\). This model originates from the representation theory of the \(\imath \)quantum group (also known as the quantum symmetric pair coideal subalgebra) of type \(\mathrm {A\!I}\) and is equipped with a combinatorial structure, which we call \(\mathrm {A\!I}\)-crystal structure. This structure enables us to describe combinatorially the tensor product of an \(SO_n(\mathbb {C})\)-module and a \(GL_n(\mathbb {C})\)-module, and the branching from \(GL_n(\mathbb {C})\) to \(SO_n(\mathbb {C})\).
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Acknowledgements
The author would like to thank the referees for careful readings and valuable comments. He is grateful to Il-Seung Jang for many helpful comments on the draft version of this paper. This work was supported by JSPS KAKENHI Grant Numbers JP20K14286 and JP21J00013.
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Watanabe, H. A new tableau model for representations of the special orthogonal group. J Algebr Comb 58, 183–230 (2023). https://doi.org/10.1007/s10801-023-01245-3
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DOI: https://doi.org/10.1007/s10801-023-01245-3