Journal of Algebraic Combinatorics

, Volume 50, Issue 3, pp 281–291

# A simple counting argument of the irreducible representations of $$\mathsf {SU}(N)$$ on mixed product spaces

• J. Alcock-Zeilinger
• H. Weigert
Article

## Abstract

That the number of irreducible representations of the special unitary group $$\mathsf {SU}(N)$$ on $$V^{\otimes k}$$ (which is also the number of Young tableaux with k boxes) is given by the number of involutions in $$S_k$$ is a well-known result (see, e.g., Knuth in The art of computer programming, volume 3—sorting and searching, 2nd ed, Addison-Wesley, Boston, 1998 and other standard textbooks). In this paper, we present an alternative proof for this fact using a basis of projection and transition operators (Alcock-Zeilinger and Weigert J Math Phys 58(5):051702, 2017, J Math Phys 58(5):051703, 2017) of the algebra of invariants of $$\mathsf {SU}(N)$$ on $$V^{\otimes k}$$. This proof is shown to easily generalize to the irreducible representations of $$\mathsf {SU}(N)$$ on mixed product spaces $$V^{\otimes m}\otimes \left( V^*\right) ^{\otimes n}$$, implying that the number of irreducible representations of $$\mathsf {SU}(N)$$ on a product space $$V^{\otimes m}\otimes \left( V^*\right) ^{\otimes n}$$ remains unchanged if one exchanges factors V for $$V^*$$ and vice versa, as long as the total number of factors remains unchanged, c.f. Corollary 1.

## Keywords

Representation theory Tableaux Counting irreducible representations Group theory Special unitary group

## Mathematics Subject Classification

05E10 05E15 20C99

## Notes

### Acknowledgements

H.W. is supported by South Africa’s National Research Foundation under CPRR Grant Number 90509. J.A-Z. was supported (in sequence) by the Postgraduate Funding Office of the University of Cape Town (2014), the National Research Foundation (2015) and the Science Faculty PhD Fellowship of the University of Cape Town (2016–2017).

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