Suppose $G$ is a finite group and $F$ is a field. Is $F[G\wr S_n]$ a cellular algebra ?. If so, what is a cellular basis for that algebra.
3 Answers
I don't believe that $F[G]$ is cellular in general. My reason (and this is not a proof) is that cellular algebras are typically quasi-hereditary (which implies finite cohomological dimension) and group algebras are not close to quasi-hereditary.
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$\begingroup$ So maybe a better question to ask is whether, if $F[G]$ is cellular, $F[G\wr S_n]$ is cellular. $\endgroup$ Nov 20, 2012 at 11:19
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$\begingroup$ That is certainly a better question. I suspect the answer is that it is. $\endgroup$ Nov 20, 2012 at 11:20
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1$\begingroup$ There is an easier argument for debunking the original question: cellular algebras are always split, but general fields are not splitting fields for general groups, so in general F[G] cannot be cellular. But I like Amritanshu's question. This seems interesting. $\endgroup$ Nov 20, 2012 at 11:58
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2$\begingroup$ Note that $F[G\wr S_n] = (F[G])\wr S_n$. In general, if $A$ is quasi-hereditary, then it is true that $A\wr S_n$ is quasi-hereditary if $n!$ is invertible in $F$ [Chunag-Tan: Representation of Wreath Product Algebras]. If $A$ is cyclic cellular (meaning $A$ is cellular and cell modules are cyclic), then so is $A\wr S_n$ (by Geetha-Goodman, in Shunsuke's answer). I suspect the same argument works if one take away cyclicity, but I have not checked through the calculation. Alternatively, I suspsect you can also follows Chuang-Tan approach and use Koenig-Xi's definition for cellular algebra. $\endgroup$ Nov 20, 2012 at 12:50
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$\begingroup$ @Johannes Good point. My feeling is that you are pointing to a deficiency in the definition of a cellular algebra. $\endgroup$ Nov 20, 2012 at 14:12
Just noticed this discussion. We invented "cyclic cellularity" precisely to make the argument for wreath products work. Our argument wouldn't work without the cyclic condition.
All standard examples of cellular algebras are in fact cyclic cellular, so one doesn't lose much by imposing this extra hypothesis.
For a special case, the implication is known. http://arxiv.org/pdf/1208.2983v1.pdf