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arXiv · 1604.08837

Representations of symmetric groups with non-trivial determinant

Arvind Ayyer·Amritanshu Prasad·Steven Spallone
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Abstract

We give a closed formula for the number of partitions $\lambda$ of $n$ such that the corresponding irreducible representation $V_\lambda$ of $S_n$ has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the $2$-core towers of such partitions. We also obtain a formula for the number of partitions of $n$ such that the associated permutation representation of $S_n$ has non-trivial determinant.

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Arvind Ayyer, Amritanshu Prasad, Steven Spallone. 2016-04-29. Representations of symmetric groups with non-trivial determinant. https://doi.org/10.1016/j.jcta.2017.03.004

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