Let \(\Theta\) be a Young tabreaux, then the Young projection operator \(Y_\Theta\) is \(\alpha_\Theta\cdot \mathbf{S_\Theta}\mathbf{A}_\Theta\), where \(\mathbf{S}_\Theta\) and \(\mathbf{A}_\Theta\) are symmetrizer and antisymmetrizer of \(\Theta\) and \(\alpha_\theta\) is the unique nonzero constant by which \(Y_\Theta\) becomes idempotent. If the size of Young tabreaux \(n\) is at most 4, Young projection operators classify representations of SU\((N)\) over \(V^{\oplus n}\).This follows from the following properties of Young projection operators;

–

The projections are idempotent.

–

The projections are mutually orthogonal.

–

The complete set of projection operators for SU\((N)\) over \(V^{\otimes n}\) sum up to the identity element of \(V^{\otimes n}\).

But they are not Hermitian operators in general. To construct class of Hermitian projections satisfying these properties has been done in [S. Keppeler and M. Sjödahl, ibid. 55, No. 2, 021702, 12 p. (2014; Zbl 1292.22008)], adopting birdtrack, a modernization of Penrose’ graphical method of the representation theory of semi-simple compact Lie groups. But the authors remark that the construction of this paper (KS construction, reviewed as Theorem 3) is not practical.
In this paper, by using cancellation rules and propagation rules of birdtracks [the authors, ibid. 58, No. 5, Article ID 051701, 27 p. (2017; Zbl 1367.81076)]. Reviewed as Theorem 1, Theorem 2, and Theorem 6), a practical method of construction of Hermitian Young Projection operators, by using measure of lexical disorder (MOLD) operator (Definition 3, Theorem 5) is proposed.
This paper begins with a brief history of representation theory and its use to quantum field theory (QFT). Then the authors review birdtrack and expose why Hermitian Young projection operators needed (§I and II). Generalities of Hermitian Young projection operators are studied in §III, with special attention to the third property. Review of the KS construction and how to go beyond this construction are also given in this section. Then in §IV, MOLD construction is introduced and show this gives a compact and practical algorithm to construct Hermitian Young projection operator for irreducible representations of SU\((N)\). Some detailed part of proofs are given in the Appendix.