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  • 1.
    Amini, Nima
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2019Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 50, nr 2, s. 165-190Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Branden). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

  • 2.
    Amini, Nima
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2018Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

  • 3. Bousquet-Melou, Mireille
    et al.
    Linusson, Svante
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Nevo, Eran
    On the independence complex of square grids2008Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 27, nr 4, s. 423-450Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The enumeration of independent sets of regular graphs is of interest in statistical mechanics, as it corresponds to the solution of hard-particle models. In 2004, it was conjectured by Fendley et al., that for some rectangular grids, with toric boundary conditions, the alternating number of independent sets is extremely simple. More precisely, under a coprimality condition on the sides of the rectangle, the number of independent sets of even and odd cardinality always differ by 1. In physics terms, this means looking at the hard-particle model on these grids at activity -1. This conjecture was recently proved by Jonsson. Here we produce other families of grid graphs, with open or cylindric boundary conditions, for which similar properties hold without any size restriction: the number of independent sets of even and odd cardinality always differ by 0, +/- 1, or, in the cylindric case, by some power of 2. We show that these results reflect a stronger property of the independence complexes of our graphs. We determine the homotopy type of these complexes using Forman's discrete Morse theory. We find that these complexes are either contractible, or homotopic to a sphere, or, in the cylindric case, to a wedge of spheres. Finally, we use our enumerative results to determine the spectra of certain transfer matrices describing the hard-particle model on our graphs at activity -1. These results parallel certain conjectures of Fendley et al., proved by Jonsson in the toric case.

  • 4. Bränden, Petter
    On operators on polynomials preserving real-rootedness and the Neggers-Stanley conjecture2004Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 20, nr 2, s. 119-130Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Polya frequency sequences. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for E-polynomials of series-parallel posets and column-strict labelled Ferrers posets.

  • 5.
    Goodarzi, Afshin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Cellular structure for the Herzog–Takayama resolution2014Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Herzog and Takayama constructed an explicit resolution for the ideals with a regular linear quotient. These ideals include all matroidal and stable ideals. The resolutions of matroidal and stable ideals are known to be cellular. In this note, we show that the Herzog–Takayama resolution is also cellular.

  • 6.
    Hultman, Axel
    KTH, Tidigare Institutioner                               , Matematik.
    Quotient complexes and lexicographic shellability2002Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 16, nr 1, s. 83-96Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let Pi(n,k,k) and Pi(n,k,h), h < k, denote the intersection lattices of the k-equal subspace arrangement of type D-n and the k, h-equal subspace arrangement of type B-n respectively. Denote by S-n(B) the group of signed permutations. We show that Delta(Pi(n,k,k))/S-n(B) is collapsible. For Delta(Pi(n,k,h))/S-n(B),h < k, we show the following. If n = 0 (mod k), then it is homotopy equivalent to a sphere of dimension 2n/k = 2. If n = h (mod k), then it is homotopy equivalent to a sphere of dimension 2n-h/k-1. Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of S-n(B) on the homology groups of Delta(Pi(n,k,k)) and Delta(Pi(n,k,h)) are stated. The collapsibility of Delta (Pi(n,k,k))/S-n(B) is established using a discrete Morse function. The same method is used to show that Delta(Pi(n,k,h))/S-n(B), h < k, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Bjorner and M. Wachs (Trans. Amer. Math. Soc. 4 (1996), 1299-1327) when the cell complex is an order complex of a poset.

  • 7.
    Hultman, Axel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Twisted identities in Coxeter groups2008Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 28, nr 2, s. 313-332Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Given a Coxeter system ( W, S) equipped with an involutive automorphism theta, the set of twisted identities is iota(theta) = {theta(w(-1))w vertical bar w is an element of W}. We point out how iota(theta) shows up in several contexts and prove that if there is no s is an element of S such that s theta(s) is of odd order greater than 1, then the Bruhat order on iota(theta) is a graded poset with rank function. given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or Z-acyclic. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincare series of iota(theta), i.e. its generating function with respect to rho, a factorisation phenomenon is discussed.

  • 8.
    Hultman, Axel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Vorwerk, Kathrin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Pattern avoidance and Boolean elements in the Bruhat order on involutions2009Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 30, nr 1, s. 87-102Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin paths is demonstrated.

  • 9.
    Jonsson, Jakob
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Five-torsion in the homology of the matching complex on 14 vertices2009Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 29, nr 1, s. 81-90Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    J.L. Andersen proved that there is 5-torsion in the bottom nonvanishing homology group of the simplicial complex of graphs of degree at most two on seven vertices. We use this result to demonstrate that there is 5-torsion also in the bottom nonvanishing homology group of the matching complex M-14 on 14 vertices. Combining our observation with results due to Bouc and to Shareshian and Wachs, we conclude that the case n = 14 is exceptional; for all other n, the torsion subgroup of the bottom nonvanishing homology group has exponent three or is zero. The possibility remains that there is other torsion than 3-torsion in higher-degree homology groups of M-n when n = 13 and n not equal 14.

  • 10.
    Jonsson, Jakob
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    The topology of the coloring complex2005Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 21, nr 3, s. 311-329Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In a recent paper, E. Steingrimsson associated to each simple graph G a simplicial complex G., referred to as the coloring complex of G. Certain nonfaces of &UDelta;(G) correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial χ(G) of G, and the reduced Euler characteristic is, up to sign, equal to &VERBAR;χ(G)(-1)&VERBAR; -1. We show that &UDelta;(G) is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of χ(G), and their Euler characteristics coincide with χ'(G)(0) and -χ'(G)(1), respectively. We show for a large class of graphs - including all connected graphs - that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel.

  • 11.
    Linusson, Svante
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Shareshian, John
    Welker, Volkmar
    Complexes of graphs with bounded matching size2008Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 27, nr 3, s. 331-349Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For positive integers k, n, we investigate the simplicial complex NMk(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r, s and k the simplicial complex BNMk(r, s) of all bipartite graphs G on bipartition [r] boolean OR [(s) over bar] such that there is no matching of size k in G, and obtain results similar to those obtained for NMk(n).

  • 12.
    Lundman, Anders
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    A classification of smooth convex 3-polytopes with at most 16 lattice points2013Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 37, nr 1, s. 139-165Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes and the remaining four are obtained as inverse stellar subdivisions of such polytopes. We derive a classification, up to isomorphism, of all smooth embeddings of toric threefolds in a"(TM) (N) where Na parts per thousand currency sign15. Again we have in total 103 such embeddings. Of these, 99 are projective bundles embedded in a"(TM) (N) and the remaining four are blow-ups of such toric threefolds.

  • 13.
    Tancer, Martin
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Shellability of the higher pinched Veronese posets2014Inngår i: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 40, nr 3, s. 711-742Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The pinched Veronese poset is the poset with ground set consisting of all nonnegative integer vectors of length such that the sum of their coordinates is divisible by with exception of the vector . For two vectors and in , we have if and only if belongs to the ground set of . We show that every interval in is shellable for . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for . (This also follows from a result by Conca, Herzog, Trung, and Valla.).

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