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Hultman, Axel
Publications (4 of 4) Show all publications
Hultman, A. (2016). Supersolvability and the Koszul property of root ideal arrangements. Proceedings of the American Mathematical Society, 144, 1401-1413
Open this publication in new window or tab >>Supersolvability and the Koszul property of root ideal arrangements
2016 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, p. 1401-1413Article in journal (Refereed) Published
Abstract [en]

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2016
Keywords
hyperplane arrangement, root poset, supersolvability, Koszul algebra
National Category
Discrete Mathematics Geometry
Identifiers
urn:nbn:se:liu:diva-124535 (URN)10.1090/proc/12810 (DOI)000369298400003 ()
Available from: 2016-02-02 Created: 2016-02-02 Last updated: 2017-11-30
Hultman, A. (2014). Permutation statistics of products of random permutations. Advances in Applied Mathematics, 54, 1-10
Open this publication in new window or tab >>Permutation statistics of products of random permutations
2014 (English)In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 54, p. 1-10Article in journal (Refereed) Published
Abstract [en]

Given a permutation statistic s : G(n) -greater than R, Ilk, define the mean statistic s as the class function giving the mean of a over. conjugacy classes. We describe a way to calculate the expected value of a on a product of t independently chosen elements from the uniform distribution on a union of conjugacy classes Gamma subset of G(n). In order to apply the formula, one needs to express the class function 3 as a linear combination of irreducible G(n)-characters. We provide such expressions for several commonly studied permutation statistics, including the exceedance number, inversion number, descent number, major index and k-cycle number. In particular, this leads to formulae for the expected values of said statistics.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Symmetric group characters; Random walks; Permutation statistics
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-105898 (URN)10.1016/j.aam.2013.10.003 (DOI)000332427700001 ()
Available from: 2014-04-14 Created: 2014-04-12 Last updated: 2017-12-05
Hultman, A. (2012). Criteria for rational smoothness of some symmetric orbit closures. Advances in Mathematics, 229(1), 183-200
Open this publication in new window or tab >>Criteria for rational smoothness of some symmetric orbit closures
2012 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 229, no 1, p. 183-200Article in journal (Refereed) Published
Abstract [en]

Let G be a connected reductive linear algebraic group over C with an involution theta. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities t = {theta(omega(-1))omega | omega is an element of W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph" whose vertices form a subset of t. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on t is rank symmetric. less thanbrgreater than less thanbrgreater thanIn the special case K = Sp(2n) (C), G = SL(2n) (C), we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one", needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.

Place, publisher, year, edition, pages
Elsevier, 2012
Keywords
Rational smoothness, Symmetric orbit closure, Bruhat graph
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-73094 (URN)10.1016/j.aim.2011.09.002 (DOI)000297184800007 ()
Available from: 2011-12-16 Created: 2011-12-16 Last updated: 2017-12-08
Hultman, A. (2011). Inversion arrangements and Bruhat intervals. Journal of combinatorial theory. Series A (Print), 118(7), 1897-1906
Open this publication in new window or tab >>Inversion arrangements and Bruhat intervals
2011 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 118, no 7, p. 1897-1906Article in journal (Refereed) Published
Abstract [en]

Let W be a finite Coxeter group. For a given w is an element of W, the following assertion may or may not be satisfied: (*) The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. We present a type independent combinatorial criterion which characterises the elements w is an element of W that satisfy (*). A couple of immediate consequences are derived: (1) The criterion only involves the order ideal of w as an abstract poser. In this sense, (*) is a poset-theoretic property. (2) For W of type A, another characterisation of (*), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjostrand. We obtain a short and simple proof of that result. (3) If W is a Weyl group and the Schubert variety indexed by w is an element of W is rationally smooth, then w satisfies (*).

Place, publisher, year, edition, pages
Elsevier Science B.V., Amsterdam, 2011
Keywords
Bruhat interval; Bruhat graph; Inversion arrangement; Coxeter group
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-69764 (URN)10.1016/j.jcta.2011.04.005 (DOI)000291900600001 ()
Available from: 2011-08-10 Created: 2011-08-08 Last updated: 2017-12-08
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