Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$module divides the dimension of $H$.
In which cases the conjecture is known to be true?
Let $H$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible $H$module divides the dimension of $H$. In which cases the conjecture is known to be true? 


From Shlomo Gelaki research statement (which is nice survey, by the way):



Yorck Sommerhäuser has a very nice survey about Kaplansky's conjectures. Section 6 is devoted to Kaplansky's 6th conjecture. In Sommerhäuser's survey it is mentioned that Richmond and Nichols proved that the conjecture is true if the simple module has dimension two:
In Sommerhäuser's survey it is also mentioned that Montgomery and Witherspoon proved that Kaplansky's conjecture holds if it holds for a subalgebra. In this paper Cohen, Miriam; Gelaki, Shlomo; Westreich, Sara. Hopf algebras. Handbook of algebra. Vol. 4, 173239, Handb. Algebr., 4, Elsevier/NorthHolland, Amsterdam, 2006. MR2523421 (2010j:16076), link it is written that Kaplansky's conjecture has been proved



There is a new survey on Kaplansky's sixth conjecture by L. Dai and J. Dong, available on the arxiv. Among other results, it mentions the following (always assuming $\operatorname{char} k=0$): Special primes:
Low dimension:
Particular properties:
Dividing other invariants:
For references, see the survey mentioned above. 

