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The Bruhat order on conjugation-invariant sets of involutions in the symmetric group
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 44, no 4, p. 849-862Article in journal (Refereed) Published
Abstract [en]

Let In be the set of involutions in the symmetric group Sn, and for A {0, 1, . . . , n}, let

= { In has α fixed points for some α   A}.

We give a complete characterisation of the sets A for which , with the order induced by the Bruhat order on Sn, is a graded poset. In particular, we prove that  (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When  is graded, we give its rank function. We also give a short, new proof of the EL-shellability of  (i.e., the set of fixed-point-free involutions), recently proved by Can, Cherniavsky, and Twelbeck.

Place, publisher, year, edition, pages
Springer, 2016. Vol. 44, no 4, p. 849-862
Keywords [en]
Bruhat order, symmetric group, involution, conjugacy class, graded poset, EL-shellability
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-123043DOI: 10.1007/s10801-016-0691-9ISI: 000387223300002OAI: oai:DiVA.org:liu-123043DiVA, id: diva2:876258
Note

At the time for thesis presentation publication was in status: Manuscript

Available from: 2015-12-03 Created: 2015-12-03 Last updated: 2018-05-21Bibliographically approved
In thesis
1. Generalised Ramsey numbers and Bruhat order on involutions
Open this publication in new window or tab >>Generalised Ramsey numbers and Bruhat order on involutions
2015 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers within two different areas of  combinatorics.

Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets Γ1 and Γ2 of cycles. We determine, in particular, all generalised Ramsey numbers R(Γ1, Γ2) such that Γ1 or Γ2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (Γ1, Γ2)-critical graphs. As special cases, we obtain complete characterisations of all (Cn,C3)-critical graphs for n ≥ 5, and all (Cn,C5)-critical graphs for n ≥ 6.

In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group Sn, with the order induced by the Bruhat order on Sn. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. p. 14
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1734
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123044 (URN)10.3384/lic.diva-123044 (DOI)978-91-7685-892-9 (ISBN)
Presentation
2015-12-17, Nobel (BL32), ingång 23, B-huset, Campus Valla, Linköpings universitet, Linköping, 10:15 (Swedish)
Opponent
Supervisors
Available from: 2015-12-03 Created: 2015-12-03 Last updated: 2015-12-03Bibliographically approved
2. Combinatorics and topology related to involutions in Coxeter groups
Open this publication in new window or tab >>Combinatorics and topology related to involutions in Coxeter groups
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation consists of three papers in combinatorial Coxeter group theory.

A Coxeter group is a group W generated by a set S, where all relations can be derived from the relations s2 = e for all s 𝜖 S, and (ss)m(s,s) = e for some pairs of generators s s in S, where e 𝜖 W is the identity element and m(s, s) is an integer satisfying that m(s, s) = m(s, s) ≥ 2. Two prominent examples of Coxeter groups are provided by the symmetric group Sn (i.e., the set of permutations of {1, 2, . . . , n}) and finite reflection groups (i.e., finite groups generated by reflections in some real euclidean space). There are also important infinite Coxeter groups, e.g., affine reflection groups.

Every Coxeter group can be equipped with various natural partial orders, the most important of which is the Bruhat order. Any subset of a Coxeter group can then be viewed as an induced subposet.

In Paper A, we study certain posets of this kind, namely, unions of conjugacy classes of involutions in the symmetric group. We obtain a complete classification of the posets that are pure (i.e., all maximal chains have the same length). In particular, we prove that the set of involutions with exactly one fixed point is pure, which settles a conjecture of Hultman in the affirmative. When the posets are pure, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, established by Can, Cherniavsky, and Twelbeck.

Paper B also deals with involutions in Coxeter groups. Given an involutive automorphism θ of a Coxeter system (W, S), let

ℑ(θ) = {w 𝜖 W | θ(w) = w−1}

be the set of twisted involutions. In particular, ℑ(id) is the set of ordinary involutions in W. It is known that twisted involutions can be represented by words in the alphabet  = { | s 𝜖 S}, called -expressions. If ss has finite order m(s, s), let a braid move be the replacement of  by ⋯, both consisting of m(s, s) letters. We prove a word property for ℑ(θ), for any Coxeter system (W, S) with any θ. More precisely, we provide a minimal set of moves, easily determined from the Coxeter graph of (W, S), that can be added to the braid moves in order to connect all reduced -expressions for any given w 𝜖 ℑ(θ). This improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg.

In Paper C, we investigate the topology of (the order complexes of) certain posets, called pircons. A special partial matching (SPM) on a poset is a matching of the Hasse diagram satisfying certain extra conditions. An SPM without fixed points is precisely a special matching as defined by Brenti. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Marietti’s zircons. Our main result is that every open interval in a pircon is a PL ball or a PL sphere.

An important subset of ℑ(θ) is the set 𝜄(θ) = {θ(w−1)w | w 𝜖 W} of twisted identities. We prove that if θ does not flip any edges with odd labels in the Coxeter graph, then 𝜄(θ), with the order induced by the Bruhat order on W, is a pircon. Hence, its open intervals are PL balls or spheres, which confirms a conjecture of Hultman. It is also demonstrated that Bruhat orders on Rains and Vazirani’s quasiparabolic W-sets (under a boundedness assumption) form pircons. In particular, this applies to all parabolic quotients of Coxeter groups.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2018. p. 46
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1924
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-147980 (URN)10.3384/diss.diva-147980 (DOI)9789176853344 (ISBN)
Public defence
2018-05-31, Nobel BL32, B-huset, Campus Valla, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2018-05-21 Created: 2018-05-21 Last updated: 2018-05-23Bibliographically approved

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