Quotients of Poincaré Polynomials Evaluated at -1
Oliver D. Eng
DOI: 10.1023/A:1008771617131
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( - 1 ) W J ( - 1 ) {\frac{{W\left( { - 1} \right)}}{{W_J \left( { - 1} \right)}}} of Tan (Tan, Communications in Algebra, 22 (1994), 1049-1061).
Pages: 29–40
Keywords: reflection groups; Poincaré polynomials; longest element; minuscule representations
Full Text: PDF
References
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2. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge,
1992. ENG
3. R.A. Proctor, “Bruhat lattices, plane partition generating functions, and minuscule representations,” European Journal of Combinatorics 5 (1984), 331-350.
4. J.R. Stembridge, “On minuscule representations, plane partitions, and involutions in complex Lie groups,” Duke Mathematical Journal 73 (1994), 469-490.
5. J.R. Stembridge, “Canonical bases and self-evacuating tableaux,” Duke Mathematical Journal 82 (1996), 585-606.
6. L. Tan, “On the distinguished coset representatives of the parabolic subgroups in finite Coxeter groups,” Communications in Algebra 22 (1994), 1049-1061.