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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si1.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=34dc2a1cde0148a8ad0946935eefd920" title="Click to view the MathML source">(W,S)class="mathContainer hidden">class="mathCode">$(W,S)$ be a Coxeter system and write class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si2.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=ff6a31b3ef57e178101b8345074a823b" title="Click to view the MathML source">P_W(q)class="mathContainer hidden">class="mathCode">$P_W(q)$ for its Poincaré series. Lusztig has shown that the quotient class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si3.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=2cefedacaf978c0ac66f54985d36517c" title="Click to view the MathML source">P_W(q²)/P_W(q)class="mathContainer hidden">class="mathCode">$P_W(q^2)/P_W(q)$ is equal to a certain power series class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si4.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=e55e87d356a19745d42618e1f45ed09a" title="Click to view the MathML source">L_W(q)class="mathContainer hidden">class="mathCode">$L_W(q)$, 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 class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si5.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=3e272cc0cbec231f93a00b75045bc220">class="imgLazyJSB inlineImage" height="18" width="132" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0196885816300720-si5.gif">class="mathContainer hidden">class="mathCode">$L_W^J(s,q)\in \mathbb{Z}[[s,q]]$ depending on a subset class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si6.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=a1e4256d1bf62f8947ab5df501866710" title="Click to view the MathML source">J⊂Sclass="mathContainer hidden">class="mathCode">$J\subset S$. This new power series specializes to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si4.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=e55e87d356a19745d42618e1f45ed09a" title="Click to view the MathML source">L_W(q)class="mathContainer hidden">class="mathCode">$L_W(q)$ when class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si7.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=42041ffab64db5e8ddf5285a71e821c9" title="Click to view the MathML source">s=−1class="mathContainer hidden">class="mathCode">$s=-1$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si8.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=f691b35552632dcce8ed3d40215f869a" title="Click to view the MathML source">W_Jclass="mathContainer hidden">class="mathCode">$W_J$ in W. When W   is an affine Weyl group, we consider the renormalized power series class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si9.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=3718ce6461372c18fee6cfff7c13fac3">class="imgLazyJSB inlineImage" height="18" width="196" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0196885816300720-si9.gif">class="mathContainer hidden">class="mathCode">$T_W(s,q)=L_W^J(s,q)/L_W(q)$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0196885816300720&_mathId=si10.gif&_user=111111111&_pii=S0196885816300720&_rdoc=1&_issn=01968858&md5=fe06ce6a2c1a357496b3d64a3a874aa9" title="Click to view the MathML source">T_W(s,q)class="mathContainer hidden">class="mathCode">$T_W(s,q)$ 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.