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Journal articles on the topic 'Subword complexes'

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Published: 25 May 2024

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1

Gorsky, Mikhail A. "Subword complexes and edge subdivisions." Proceedings of the Steklov Institute of Mathematics 286, no. 1 (October 2014): 114–27. http://dx.doi.org/10.1134/s0081543814060078.

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2

Knutson, Allen, and Ezra Miller. "Subword complexes in Coxeter groups." Advances in Mathematics 184, no. 1 (May 2004): 161–76. http://dx.doi.org/10.1016/s0001-8708(03)00142-7.

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3

Ceballos, Cesar, Jean-Philippe Labbé, and Christian Stump. "Subword complexes, cluster complexes, and generalized multi-associahedra." Journal of Algebraic Combinatorics 39, no. 1 (March 13, 2013): 17–51. http://dx.doi.org/10.1007/s10801-013-0437-x.

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4

Gorsky, M. A. "Subword Complexes and Nil-Hecke Moves." Modeling and Analysis of Information Systems 20, no. 6 (March 13, 2015): 121–28. http://dx.doi.org/10.18255/1818-1015-2013-6-121-128.

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5

Knutson, Allen. "Schubert Patches Degenerate to Subword Complexes." Transformation Groups 13, no. 3-4 (June 26, 2008): 715–26. http://dx.doi.org/10.1007/s00031-008-9013-1.

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6

Bergeron, Nantel, and Cesar Ceballos. "A Hopf algebra of subword complexes." Advances in Mathematics 305 (January 2017): 1163–201. http://dx.doi.org/10.1016/j.aim.2016.10.007.

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7

Gorsky, M. A. "Subword complexes and 2-truncated cubes." Russian Mathematical Surveys 69, no. 3 (June 30, 2014): 572–74. http://dx.doi.org/10.1070/rm2014v069n03abeh004903.

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8

Ceballos, Cesar, Arnau Padrol, and Camilo Sarmiento. "ν-Tamari lattices via subword complexes." Electronic Notes in Discrete Mathematics 61 (August 2017): 215–21. http://dx.doi.org/10.1016/j.endm.2017.06.041.

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9

Escobar, Laura, and Karola Mészáros. "Subword complexes via triangulations of root polytopes." Algebraic Combinatorics 1, no. 3 (2018): 395–414. http://dx.doi.org/10.5802/alco.17.

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10

Armstrong, Drew, and Patricia Hersh. "Sorting orders, subword complexes, Bruhat order and total positivity." Advances in Applied Mathematics 46, no. 1-4 (January 2011): 46–53. http://dx.doi.org/10.1016/j.aam.2010.09.006.

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11

Pilaud, Vincent, and Christian Stump. "Brick polytopes of spherical subword complexes and generalized associahedra." Advances in Mathematics 276 (May 2015): 1–61. http://dx.doi.org/10.1016/j.aim.2015.02.012.

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12

Hruska, G. Christopher, and Daniel T. Wise. "Towers, ladders and the B. B. Newman Spelling Theorem." Journal of the Australian Mathematical Society 71, no. 1 (August 2001): 53–69. http://dx.doi.org/10.1017/s1446788700002718.

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AbstractThe Spelling Theorem of B. B. Newman states that for a one-relator group (a1, … |Wn), any nontrivial word which represents the identity must contain a (cyclic) subword ofW±nlonger thanWn−1. We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classification of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entuirly along the bounday, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.
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13

Bergeron, Nantel, Cesar Ceballos, and Jean-Philippe Labbé. "Fan Realizations of Type $$A$$ A Subword Complexes and Multi-associahedra of Rank 3." Discrete & Computational Geometry 54, no. 1 (April 23, 2015): 195–231. http://dx.doi.org/10.1007/s00454-015-9691-0.

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14

Jahn, Dennis, and Christian Stump. "Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups." Mathematische Zeitschrift 304, no. 2 (May 6, 2023). http://dx.doi.org/10.1007/s00209-023-03267-w.

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AbstractWe study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.
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15

Smirnov, Evgeny Yurievich, and Anna Tutubalina. "Slide complexes and subword complexes." Russian Mathematical Surveys 75, no. 6 (2020). http://dx.doi.org/10.1070/rm9981.

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16

Smirnov, Evgeny Yurievich, and Anna Alekseevna Tutubalina. "Slide polynomials and subword complexes." Sbornik: Mathematics 212, no. 10 (2021). http://dx.doi.org/10.1070/sm9477.

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17

Pilaud, Vincent, and Christian Stump. "Generalized associahedra via brick polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AR,..., Proceedings (January 1, 2012). http://dx.doi.org/10.46298/dmtcs.3021.

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International audience We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.
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18

Pilaud, Vincent, and Christian Stump. "EL-labelings and canonical spanning trees for subword complexes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.2328.

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International audience We describe edge labelings of the increasing flip graph of a subword complex on a finite Coxeter group, and study applications thereof. On the one hand, we show that they provide canonical spanning trees of the facet-ridge graph of the subword complex, describe inductively these trees, and present their close relations to greedy facets. Searching these trees yields an efficient algorithm to generate all facets of the subword complex, which extends the greedy flip algorithm for pointed pseudotriangulations. On the other hand, when the increasing flip graph is a Hasse diagram, we show that the edge labeling is indeed an EL-labeling and derive further combinatorial properties of paths in the increasing flip graph. These results apply in particular to Cambrian lattices, in which case a similar EL-labeling was recently studied by M. Kallipoliti and H. Mühle.
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19

Ceballos, Cesar, Arnau Padrol, and Camilo Sarmiento. "The ν-Tamari Lattice via ν-Trees, ν-Bracket Vectors, and Subword Complexes." Electronic Journal of Combinatorics 27, no. 1 (January 10, 2020). http://dx.doi.org/10.37236/8000.

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We give a new interpretation of the $\nu$-Tamari lattice of Préville-Ratelle and Viennot in terms of a rotation lattice of $\nu$-trees. This uncovers the relation with known combinatorial objects such as north-east fillings, \mbox{tree-like} tableaux and subword complexes. We provide a simple description of the lattice property using certain bracket vectors of $\nu$-trees, and show that the Hasse diagram of the $\nu$-Tamari lattice can be obtained as the facet adjacency graph of certain subword complex. Finally, this point of view generalizes to multi $\nu$-Tamari complexes, and gives (conjectural) insight on their geometric realizability via polytopal subdivisions of multiassociahedra.
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20

Bergeron, Nantel, Cesar Ceballos, and Jean-Philippe Labbé. "Fan realizations of type $A$ subword complexes and multi-associahedra of rank 3." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 27th..., Proceedings (January 1, 2015). http://dx.doi.org/10.46298/dmtcs.2512.

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International audience We present complete simplicial fan realizations of any spherical subword complex of type $A_n$ for $n\leq 3$. This provides complete simplicial fan realizations of simplicial multi-associahedra $\Delta_{2k+4,k}$, whose facets are in correspondence with $k$-triangulations of a convex $(2k+4)$-gon. This solves the first open case of the problem of finding fan realizations where polytopality is not known. The techniques presented in this paper work for all finite Coxeter groups and we hope that they will be useful to construct fans realizing subword complexes in general. In particular, we present fan realizations of two previously unknown cases of subword complexes of type $A_4$, namely the multi-associahedra $\Delta_{9,2}$ and $\Delta_{11,3}$. Nous construisons des éventails simpliciaux complets ayant la combinatoire des complexes de sous-mots de type $A_n$ pour $n\leq 3$. Par conséquent, nous obtenons des constructions d’éventails des multi-associaèdres $\Delta_{2k+4,k}$, dont les facettes correspondent aux $k$-triangulations d’un $(2k+4)$-gone. Cette construction confirme l’existence d’éventails ayant la combinatoire du multi-associaèdres pour une famille dont la polytopalité n’est pas confirmée. Les techniques utilisées fonctionnent pour tous les groupes de Coxeter et nous espérons qu’elles seront utiles afin de construire des éventails réalisant les complexes de sous-mots en général. En particulier, nous présentons des éventails pour deux complexes de sous-mots de type $A_4$ dont l’existence était inconnue: les multi-associaèdres $\Delta_{9,2}$ et $\Delta_{11,3}$.
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21

Bergeron, Nantel, and Cesar Ceballos. "A Hopf algebra of subword complexes (Extended abstract)." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6359.

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International audience We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.
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22

Escobar, Laura. "Bott-Samelson Varieties, Subword Complexes and Brick Polytopes." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AT,..., Proceedings (January 1, 2014). http://dx.doi.org/10.46298/dmtcs.2448.

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International audience Bott-Samelson varieties factor the flag variety $G/B$ into a product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational; however in this paper we study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into a purely combinatorial one in terms of a subword complex. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the dual to the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg and Lange. These stories connect in a nice way: the moment polytope of a fiber of the Bott-Samelson map is the Brick polytope. In particular, we give a nice description of the toric variety of the associahedron.
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23

Escobar, Laura, and Karola Mészáros. "Toric matrix Schubert varieties and root polytopes (extended abstract)." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6405.

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International audience Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.
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24

Escobar, Laura. "Brick Manifolds and Toric Varieties of Brick Polytopes." Electronic Journal of Combinatorics 23, no. 2 (April 29, 2016). http://dx.doi.org/10.37236/5038.

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Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.
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25

Ceballos, Cesar, and Vincent Pilaud. "Cluster Algebras of Type D: Pseudotriangulations Approach." Electronic Journal of Combinatorics 22, no. 4 (December 23, 2015). http://dx.doi.org/10.37236/5282.

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We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$ gon with a small disk in the centre. This model provides convenient and uniform interpretations for clusters, cluster variables and their exchange relations, as well as for quivers and their mutations. We also present a new combinatorial interpretation of cluster variables in terms of perfect matchings of a graph after deleting two of its vertices. This interpretation differs from known interpretations in the literature. Its main feature, in contrast with other interpretations, is that for a fixed initial cluster seed, one or two graphs serve for the computation of all cluster variables. Finally, we discuss applications of our model to polytopal realizations of type $D$ associahedra and connections to subword complexes and $c$-cluster complexes.
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26

Escobar, Laura, Alex Fink, Jenna Rajchgot, and Alexander Woo. "Gröbner bases, symmetric matrices, and type C Kazhdan–Lusztig varieties." Journal of the London Mathematical Society 109, no. 2 (February 2024). http://dx.doi.org/10.1112/jlms.12856.

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AbstractWe study a class of combinatorially defined polynomial ideals that are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan–Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme‐theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan–Lusztig ideals that arise are exactly those where the opposite cell is 123‐avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley–Reisner ideals of subword complexes), and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.
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27

Ceballos, Cesar, and Vincent Pilaud. "Denominator vectors and compatibility degrees in cluster algebras of finite type." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.12795.

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We present two simple descriptions of the denominator vectors of the cluster variables of a cluster algebra of finite type, with respect to any initial cluster seed: one in terms of the compatibility degrees between almost positive roots defined by S. Fomin and A. Zelevinsky, and the other in terms of the root function of a certain subword complex. These descriptions only rely on linear algebra, and provide simple proofs of the known fact that the $d$-vector of any non-initial cluster variable with respect to any initial cluster seed has non-negative entries and is different from zero. Nous présentons deux descriptions élémentaires des vecteurs dénominateurs des algèbres amassées de type fini pour tout amas initial: l'une en termes de degrés de compatibilitié entre racines presque positives définis par S. Fomin et A. Zelevinsky, et l'autre en termes de la fonction racine d'un certain complexe de sous-mots. Ces descriptions ne reposent que sur l'algèbre linéaire et fournissent des preuves simples du fait (connu) que le $d$-vecteur de toute variable d'amas, qui n'est pas dans l'amas initial, a des entrées positives ou nulles et est différent du vecteur nul.
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