| 1. |
- Hansson, Mikael, 1986-
(författare)
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Combinatorics and topology related to involutions in Coxeter groups
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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 ? S, and (ss′)m(s,s′) = e for some pairs of generators s ≠ s′ in S, where e ? 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), letℑ(θ) = {w ? 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 ? 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 ? 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 ?(θ) = {θ(w−1)w | w ? W} of twisted identities. We prove that if θ does not flip any edges with odd labels in the Coxeter graph, then ?(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.
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| 2. |
- Snellman, Jan, 1968-
(författare)
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A conjecture on poincaré-betti series of modules of differential operators on a generic hyperplane arrangement
- 2005
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Ingår i: Experimental Mathematics. - 1058-6458 .- 1944-950X. ; 14:4, s. 445-456
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Tidskriftsartikel (refereegranskat)abstract
- Holm [Holm 04, Holm 02] studied modules of higher-order differential operators (generalizing derivations) on generic (central) hyperplane arrangements. We use his results to determine the Hubert series of these modules. We also give a conjecture about the Poincaré-Betti series, these are known for the module of derivations through the work of Yuzvinsky [Yuzvinsky 91] and Rose and Terao [Rose and Terao 91 ] © A K Peters, Ltd.
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| 3. |
- Snellman, Jan, 1968-
(författare)
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A graded subring of an inverse limit of polynomial rings
- 1998
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R.Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order.If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I.In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated.The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.
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| 4. |
- Snellman, Jan, 1968-
(författare)
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Generating functions for borders
- 2007
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Annan publikation (populärvet., debatt m.m.)abstract
- We give the generating function for the index of integer lattice points, relative to a finite order ideal. The index is an important concept in the theory of border bases, an alternative to Gröbner bases.
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| 5. |
- Snellman, Jan, 1968-
(författare)
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Infinite Minkowski sums of lattice polyhedra
- 2005
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Annan publikation (populärvet., debatt m.m.)abstract
- We show that certain two-dimensional, integrally closed monomial modules can be uniquely written as a countable product of isomorphic copies of simple integrally closed monomial ideals.
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| 6. |
- Snellman, Jan, 1968-
(författare)
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Laplacians on shifted multicomplexes
- 2006
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Annan publikation (populärvet., debatt m.m.)abstract
- We define the Laplacian operator on finite multicomplexes and give a formula for its spectra in the case of shifted multicomplexes.
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| 7. |
- Snellman, Jan, 1968-, et al.
(författare)
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On the number of plane partitions and non isomorphic subgroup towers of abelian groups
- 2006
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Annan publikation (populärvet., debatt m.m.)abstract
- We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory of partitions, and for $k=2$ it was proved in Andersson-Bhowmik with another method. We give an explicit formula in terms of Young tableaux, and study the corresponding zeta-function. We give an application on the average orders of towers of abelian groups. In particular we prove that the number of isomorphism classes of ``subgroups of subgroups of ... ($k-1$ times) ... of abelian groups'' of order at most $N$ is asymptotic to $c_k N (\log N)^{k-1}$. This generalises results from Erd{\H o}s-Szekeres and Andersson-Bhowmik where the corresponding result was proved for $k=1$ and $k=2$.
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| 8. |
- Snellman, Jan, 1968-
(författare)
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Saturated chains in composition posets
- 2005
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We study three different poset structures on the set of all compositions. In the first case, the covering relation consists of inserting a part of size one to the left or to the right, or increasing the size of some part by one. The resulting poset was studied by the author in "A poset classifying non-commutative term orders", and then in "Standard paths in another composition poset" where some results about generating functions for standard paths in this poset was established. The latter article was inspired by the work of Bergeron, Bousquet-M{\'e}lou and Dulucq on "Standard paths in the composition poset", where they studied a poset where there are additional cover relations which allows the insertion of a part of size one anywhere in the composition. Finally, following a suggestion by Richard Stanley we study yet a third which is an extension of the previous two posets. This poset is related to quasi-symmetric functions. For these posets, we study generating functions for saturated chains of fixed width k. We also construct ``labeled'' non-commutative generating functions and their associated languages.
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| 9. |
- Umutabazi, Vincent, 1982-
(författare)
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Boolean complexes of involutions and smooth intervals in Coxeter groups
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation is composed of four papers in algebraic combinatorics related to Coxeter groups. By a Coxeter group, we mean a group W generated by a subset S ⊂ W such that for all s ∈ S , we have s2 = e, and (s, s′)m(s,s′) = (s′ s)m(s,s′) = e, where m(s, s′) = m(s′ s) ≥ 2 for all s ≠ s′ ≥ ∈ S . The condition m(s, s′) = ∞ is allowed and means that there is no relation between s and s′. There are some partial orders that are associated with every Coxeter group. Among them, the most notable one is the Bruhat order. Coxeter groups and their Bruhat orders have important properties that can be utilised to study Schubert varieties. In Paper I, we consider Schubert varieties that are indexed by involutions of a finite simply laced Coxeter group. We prove that the Schubert varieties which are indexed by involutions that are not longest elements of some standard parabolic subgroups are not smooth. Paper II is based on the Boolean complexes of involutions of a Coxeter group. These complexes are analogues of the Boolean complexes invented by Ragnarsson and Tenner. We use discrete Morse theory to compute the homotopy type of the Boolean complexes of involutions of some infinite Coxeter groups together with all finite Coxeter groups. In Paper III, we prove that the subposet induced by the fixed elements of any automorphism of a pircon is also a pircon. In addition, our main results are applied to the symmetric groups S 2n. As a consequence, we prove that the signed fixed point free involutions form a pircon under the dual of the Bruhat order on the hyperoctahedral group. Let W be a Weyl group and I denote a Bruhat interval in W. In Paper IV, we prove that if the dual of I is a zircon, then I is rationally smooth. After examining when the converse holds, and being influenced from conjectures by Delanoy, we are led to pose two conjectures. Those conjectures imply that for Bruhat intervals in type A, duals of smooth intervals, zircons, and being isomorphic to lower intervals are all equivalent. We have verified our conjectures in types An, n ≤ 8, by using SageMath.
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| 10. |
- Umutabazi, Vincent, 1982-
(författare)
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Smooth Schubert varieties and boolean complexes of involutions
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is composed of two papers both in algebraic combinatorics and Coxeter groups.In Paper I, we concentrate on smoothness of Schubert varieties indexed by involutions from finite simply laced types. We show that if a Schubert variety indexed by an involution of a finite and simply laced Coxeter group is smooth, then that involution must be the longest element of a parabolic subgroup.Given a Coxeter system (W, S), we introduce in Paper II the boolean complex of involutions of W as an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By using discrete Morse Theory, we compute the homotopy type for a large class of W, including all finite Coxeter groups. In all cases, the homotopy type is that of a wedge of spheres of dimension |S| − 1. In addition, we provide a recurrence formula for the number of spheres in the wedge.
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