The Bruhat order on conjugation-invariant sets of involutions in the symmetric group

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• 作者：Mikael Hansson
• 关键词：Bruhat order ; Symmetric group ; Involution ; Conjugacy class ; Graded poset ; EL ; shellability
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：44
• 期：4
• 页码：849-862
• 全文大小：537 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192
• 卷排序：44

文摘

Let \(I_n\) be the set of involutions in the symmetric group \(S_n\), and for \(A \subseteq \{0,1,\ldots ,n\}\), let $$\begin{aligned} F_n^A=\{\sigma \in I_n\mid \sigma \text{ has } \text{ exactly } a \text{ fixed } \text{ points } \text{ for } \text{ some } a \in A\}. \end{aligned}$$We give a complete characterisation of the sets A for which \(F_n^A\), with the order induced by the Bruhat order on \(S_n\), is a graded poset. In particular, we prove that \(F_n^{\{1\}}\) (i.e. the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When \(F_n^A\) is graded, we give its rank function. We also give a short, new proof of the EL-shellability of \(F_n^{\{0\}}\) (i.e. the set of fixed-point-free involutions), recently proved by Can et al.KeywordsBruhat orderSymmetric groupInvolutionConjugacy classGraded posetEL-shellabilityMathematics Subject Classification06A0720F5520B30References1.Björner, A.: Shellable and Cohen-Macaulay partially ordered sets. Trans. Am. Math. Soc. 260, 159–183 (1980)MathSciNetCrossRefMATHGoogle Scholar2.Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)MATHGoogle Scholar3.Can, M.B., Cherniavsky, Y., Twelbeck, T.: Lexicographic shellability of the Bruhat–Chevalley order on fixed-point-free involutions. Israel J. Math. 207, 281–299 (2015)MathSciNetCrossRefMATHGoogle Scholar4.Edelman, P.H.: The Bruhat order of the symmetric group is lexicographically shellable. Proc. Am. Math. Soc. 82, 355–358 (1981)MathSciNetCrossRefMATHGoogle Scholar5.Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics, vol. 80. Academic Press, New York (1978)MATHGoogle Scholar6.Hultman, A.: Twisted identities in Coxeter groups. J. Algebraic Combin. 28, 313–332 (2008)MathSciNetCrossRefMATHGoogle Scholar7.Incitti, F.: The Bruhat order on the involutions of the symmetric group. J. Algebraic Combin. 20, 243–261 (2004)MathSciNetCrossRefMATHGoogle Scholar8.Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedicata 35, 389–436 (1990)MathSciNetCrossRefMATHGoogle Scholar9.Richardson, R.W., Springer, T.A.: Complements to: The Bruhat order on symmetric varieties. Geom. Dedicata 49, 231–238 (1994)MathSciNetCrossRefMATHGoogle Scholar10.Stanley, R.P.: Enumerative Combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49. Cambridge University Press, Cambridge (1997)Copyright information© Springer Science+Business Media New York 2016Authors and AffiliationsMikael Hansson1Email author1.Department of MathematicsLinköping UniversityLinköpingSweden About this article CrossMark Print ISSN 0925-9899 Online ISSN 1572-9192 Publisher Name Springer US About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s10801-016-0691-9_The Bruhat order on conjugation-in", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s10801-016-0691-9_The Bruhat order on conjugation-in", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips