Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

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• 作者：Eric Marberg ; ¹ ; ^{emarberg@stanford.edu" class="auth_mail" title="E-mail the corresponding author} ; Graham White ^{grwhite@math.stanford.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E15 ; 20F55 ; 33C80
• 刊名：Advances in Applied Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：82
• 期：Complete
• 页码：129-154
• 全文大小：522 K

文摘

Let (W,S)$(W,S)$ be a Coxeter system and write P_W(q)$P_W(q)$ for its Poincaré series. Lusztig has shown that the quotient P_W(q²)/P_W(q)$P_W(q^2)/P_W(q)$ is equal to a certain power series L_W(q)$L_W(q)$, defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in W  . The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization $L_W^J(s,q)\in \mathbb{Z}[[s,q]]$ depending on a subset J⊂S$J\subset S$. This new power series specializes to L_W(q)$L_W(q)$ when s=−1$s=-1$ and is given explicitly by a sum of rational functions over the involutions which are minimal length representatives of the double cosets of the parabolic subgroup W_J$W_J$ in W. When W   is an affine Weyl group, we consider the renormalized power series $T_W(s,q)=L_W^J(s,q)/L_W(q)$ with J given by the generating set of the corresponding finite Weyl group. We show that when W is an affine Weyl group of type A  , the power series T_W(s,q)$T_W(s,q)$ is actually a polynomial in s and q with nonnegative coefficients, which turns out to be a q-analogue recently studied by Cigler of the Chebyshev polynomials of the first kind, arising in a completely different context.