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## Download A Filtered Category OS and Applications by Ronald S. Irving PDF

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Example text

First suppose s\ and \$2 are two reflections in W such that both s\w < w and \$2W < w. 2 of [De 2], any reflection of W which is not in Ws is the unique reflection in its Ws coset. Thus we must have si = 52. This implies that in formula (B), the sum on the right side may be taken over all elements z of the form ~sw with s a reflection and with z < w. The same is true of formula (A) as well. It only remains to verify for such a z that the respective exponents of — 1 in the two formulas have the same parity.

Then *s(io(™swy) =

We may cancel the first sum off each expression. Comparing coefficients of what remains, we obtain the formulas of (i). • Proof of equivalence of (i) and (iii). Under either the assumptions of (i) or (iii), the polynomials Qy,w(