Abstract
For a finite reflection group W and parabolic subgroup W J, we establish that the quotient of Poincaré polynomials \frac{W(t)}{W_J(t)}, when evaluated at t=−1, counts the number of cosets of W J in W fixed by the longest element. Our case-by-case proof relies on the work of Stembridge (Stembridge, Duke Mathematical Journal, 73 (1994), 469–490) regarding minuscule representations and on the calculations of \({\frac{{W\left( { - 1} \right)}}{{W_J \left( { - 1} \right)}}}\) of Tan (Tan, Communications in Algebra, 22 (1994), 1049–1061).
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Eng, O.D. Quotients of Poincaré Polynomials Evaluated at –1. Journal of Algebraic Combinatorics 13, 29–40 (2001). https://doi.org/10.1023/A:1008771617131
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DOI: https://doi.org/10.1023/A:1008771617131