Geodesic
Quotients of Poincaré polynomials evaluated at $-1$
Journal of Algebraic Combinatorics, Tome 13 (2001) no. 1, pp. 29-40.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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( - 1 ) $W _{ J}$ ( - 1 ) fracW$left( - 1 right)${{W\_J $left( { - 1} \right)}}}$ of Tan (Tan, Communications in Algebra, 22 (1994), 1049-1061).
Keywords: reflection groups, Poincaré polynomials, longest element, minuscule representations 
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     author = {Eng, Oliver D.},
     title = {Quotients of {Poincar\'e} polynomials evaluated at $-1$},
     journal = {Journal of Algebraic Combinatorics},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2001__13_1_a4/}
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Eng, Oliver D. Quotients of Poincaré polynomials evaluated at $-1$. Journal of Algebraic Combinatorics, Tome 13 (2001) no. 1, pp. 29-40. http://geodesic.mathdoc.fr/item/JAC_2001__13_1_a4/