The representation theory of the symmetric group is best understood via the Specht polynomials. In wonder how this works for other finite reflection groups, such as dihedral groups. Are the similarly nice objects?
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$\begingroup$ Since the representations of dihedral groups are easy to analyzr in classical ways, it isn't clear to me what else Specht polynomials would accomplish here. Other infinite families of reflection groups probably offer a better rationale for this machinery. $\endgroup$– Jim HumphreysCommented Dec 12, 2013 at 2:00
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Specht polynomials and Specht modules have been applied not only to the representation theory of symmetric groups, but also for other reflection groups, e.g., for octahedral groups, see for example http://www.cmi.ac.in/~pdeshpande/projects/irreps.pdf. Furthermore there are "Higher Specht Polynomials for Complex Reflection Groups", see the paper of H. Morita, and H. Yamada. Perhaps also of interest in this connection may be Garnir polynomials for finite reflection groups, which play a role for modular representations, too (S. Devadas).