Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras  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?
 A: From Shlomo Gelaki research statement (which is nice survey, by the way):

We also proved that the dimension of
  an irreducible representation of a
  semisimple Hopf algebra H, which is
  either quasitriangular or
  cotriangular, divides the dimension of
  H.
  This result partially answers a
  celebrated conjecture of Kaplansky,
  which is still open.

A: 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:

Theorem (Nichols & Richmond). The
  dimension of a semisimple Hopf algebra
  over $\mathbb{C}$ is even if the Hopf
  algebra has a simple module of
  dimension 2.

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, 173--239, Handb. Algebr., 4, Elsevier/North-Holland,
Amsterdam, 2006. MR2523421 (2010j:16076), link
it is written that Kaplansky's conjecture has been proved 


*

*if $H$ is triangular,

*if $H$ is semisolvable, 

*if $H$ is cotriangular,

*if $R(H)$ is central in $H^\*$, where $R(H)$ is the span in $H^\*$ of all the characters on $H$.
A: 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:

If a semisimple Hopf algebras has a simple module of dimension $p$, where $p=2$ or $p=3$, then its dimension is divisible by $p$.

Low dimension:

Semisimple Hopf algebras of dimension less than $60$ satisfy Kaplansky's sixth conjecture.

Particular properties:

A semisimple Hopf algebra $H$ satisfies Kaplansky's sixth conjecture if it satisfies one of the following conditions:
  
  
*
  
*It is quasitriangular.
  
*Its characters are central in $H^*.$
  
*It is semisolvable.
  

Dividing other invariants:

$H$ a semisimple Hopf algebra. $A$ a transitive $H$-module algebra (e.g. $A=\operatorname{End}_k(V)$ for a simple $H$-module $V$).  Then $\dim A$ divides $(\dim V)^2\dim H$.

For references, see the survey mentioned above.
