What are generators for the kernel of the (k,r)-quotient of the Hecke algebras of type A? Are just the two projections onto the reps. corresponding to Young digarams with 1-row of length r-k+1 and that with 1-column of length k+1 enough?
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asked May 27, 2010 at 20:41
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$\begingroup$ Can you define the $(k,r)$-quotient? Also, are you really asking for generators? What's wrong with considering the image of the generators of the Hecke algebra under the canonical map? $\endgroup$– Sammy BlackCommented May 28, 2010 at 6:30
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$\begingroup$ I take it you mean quotient by the ideal generated by these two projections. Then the irreducible representations of the quotient are labelled by partitions which fit in the rectangle. If this is what you want, then the answer is Yes! $\endgroup$– Bruce WestburyCommented May 28, 2010 at 11:24
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$\begingroup$ The (k,r)-quotient is the quotient by the annihilator of the trace of the Hecke algebra specialized to $q$ an $r$th root of unity and the value of $tr(e_1)=(1-q^{1-k})/((1+q)(1-q^k)$. (As in Wenzl's 1988 Inventiones paper). Bruce is right, I want to know if the annihilator of the trace is generated by the two projections onto the (1-dimensional) representations labeled by these Young diagrams. Certainly it is known for the case $k=2$ (Temperley-Lieb) but I have never seen the general result in the literature. Do you know a reference Bruce? $\endgroup$– Eric RowellCommented May 28, 2010 at 13:56
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$\begingroup$ I edited your question for clarity based on your comment. $\endgroup$– Noah SnyderCommented May 28, 2010 at 14:36
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$\begingroup$ I have not seen this before. I take it that you have shown that the trace annihilates the two projections. Then you have a surjection from the algebra defined by these relations to the $(k,r)$-quotient. My concern is that I think the first algebra has non-zero radical and the second one doesn't. If so, then I expect the kernel is the radical. $\endgroup$– Bruce WestburyCommented May 29, 2010 at 7:05
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