maps preserve the relations in Theorem 1.2 and Theorem 5.3. Hence they give representations of $A_n(Q)$ for all ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_n$ ($n = 2-\frac{1}{2}, 2, 3-\frac{1}{2}, 3, 4-\frac{1}{2}$) and for almost all ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_4$.

These representations also coincide with the ones calculated through the Murphy’s operators which are introduced in the paper [@HR] and programmed by Naruse. Moreover, the traces of the representation matrices above coincide with the “characters” which is defined by Naruse in the paper [@Na1]. This means that the representations we have presented in this note will be irreducible and define Young’s seminormal form representations of the partition algebras $A_n(Q)$.

[99]{} S. Ariki and K. Koike, A Hecke algebra of $({\mathbb Z}/r{\mathbb Z})\wr {\mathfrak S}_n$ and construction of its irreducible representations, [*Adv. in Math.*]{} **106** (1994), 216–243.

W. Doran IV and D. Wales, The partition algebra revisited, [*J. Algebra*]{} **231** (2000), 265–330.

F. M. Goodman, P. de la Harpe and V. F. R. Jones, [Coxeter Graphs and Towers of Algebras]{}, Springer-Verlag, New York, 1989.

J. J. Graham and G. I. Lehrer, Cellular algebras, [*Invent. Math.*]{} **123** (1996), 1–34.

T. Halverson and A. Ram, Partition algebras, [*European J. Combin.*]{} **26** (2005), 869–921.

V. F. R. Jones, The potts model and the symmetric group, [*Subfactors (Kyuzeso, 1993)*]{} 259–267, World Sci. Publishing, River Edge, NJ, 1994.

D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, [*Invent. Math.*]{} **53** (1979), 165–184.

M. Kosuda, Characterization for the party algebras, [*Ryukyu Math. J.*]{} **13** (2000), 7–22.

M. Kosuda, Irreducible representations of the party algebra, [*Osaka J. Math.*]{} **43** (2006), 431–474.

M. Kosuda, The standard expression for the party algebra, [*Sūrikaisekikenkyūsho Kōkyūroku*]{} **No. 1497** (2006), 5