scholarly journals Staircase diagrams and the enumeration of smooth Schubert varieties

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Edward Richmond ◽  
William Slofstra
Keyword(s):  
Data Structure ◽  
Finite Type ◽  
Schubert Varieties ◽  
International Audience ◽  
Combinatorial Data

International audience In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams.

scholarly journals Projective subdynamics and universal shifts

2011 ◽  
Vol DMTCS Proceedings vol. AP,... (Proceedings) ◽  
Author(s):  
Pierre Guillon
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Positive Entropy ◽  
Two Dimensional ◽  
One Dimensional ◽  
International Audience

International audience We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.

scholarly journals The Prism tableau model for Schubert polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Anna Weigandt ◽  
Alexander Yong
Keyword(s):  
Young Tableau ◽  
Schubert Varieties ◽  
Symmetric Polynomials ◽  
Schubert Polynomials ◽  
International Audience

International audience The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.

scholarly journals Analysis of some parameters for random nodes in priority trees

2008 ◽  
Vol Vol. 10 no. 2 (Analysis of Algorithms) ◽  
Author(s):  
Alois Panholzer
Keyword(s):  
Data Structure ◽  
Analysis Of Algorithms ◽  
Priority Queue ◽  
Limiting Distribution ◽  
Node Number ◽  
Random Node ◽  
International Audience ◽  
A Priority

Analysis of Algorithms International audience Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.

scholarly journals Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Hyperplane Arrangement ◽  
Schubert Variety ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Poincaré Polynomial ◽  
Generalized Flag Manifolds ◽  
Nous Montrons ◽  
International Audience

International audience We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

scholarly journals Moment graphs and KL-polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Martina Lanini
Keyword(s):  
New Techniques ◽  
International Audience ◽  
Combinatorial Data ◽  
Lusztig Polynomials

International audience Motivated by a result of Fiebig (2007), we categorify some properties of Kazhdan-Lusztig polynomials via sheaves on Bruhat moment graphs. In order to do this, we develop new techniques and apply them to the combinatorial data encoded in these moment graphs. Motivés par un résultat de Fiebig (2007), nous catégorisons certaines propriétés des polynômes de Kazhdan-Lusztig en utilisant faisceaux sur les graphes moment de Bruhat. Pour faire ça, nous développons de nouvelles techniques et les appliquons ensuite aux données combinatoires encodées dans ces graphes moment.

scholarly journals Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

scholarly journals Toric matrix Schubert varieties and root polytopes (extended abstract)

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Laura Escobar ◽  
Karola Mészáros
Keyword(s):  
Schubert Variety ◽  
Schubert Varieties ◽  
Dimensional Torus ◽  
Special Cases ◽  
Ehrhart Series ◽  
The Matrix ◽  
International Audience ◽  
Reduced Forms ◽  
The Ideal ◽  
Grothendieck Polynomials

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.

scholarly journals SOUR graphs for efficient completion

1998 ◽  
Vol Vol. 2 ◽  
Author(s):  
Christopher Lynch ◽  
Polina Strogova
Keyword(s):  
Data Structure ◽  
Term Structure ◽  
Standard Procedure ◽  
Critical Pair ◽  
Exponential Behavior ◽  
Rewriting Systems ◽  
Group Completion ◽  
International Audience ◽  
Local Graph ◽  
Rewrite Rules

International audience We introduce a data structure called \emphSOUR graphs and present an efficient Knuth-Bendix completion procedure based on it. \emphSOUR graphs allow for a maximal structure sharing of terms in rewriting systems. The term representation is a dag representation, except that edges are labelled with equational constraints and variable renamings. The rewrite rules correspond to rewrite edges, the unification problems to unification edges. The Critical Pair and Simplification inferences are recognized as patterns in the graph and are performed as local graph transformations. Our algorithm avoids duplicating term structure while performing inferences, which causes exponential behavior in the standard procedure. This approach gives a basis to design other completion algorithms, such as goal-oriented completion, concurrent completion and group completion procedures.

scholarly journals Optimal Prefix and Suffix Queries on Texts

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Maxime Crochemore ◽  
Costas S. Iliopoulos ◽  
M. Sohel Rahman
Keyword(s):  
Data Structure ◽  
Suffix Tree ◽  
Query Time ◽  
Time And Space ◽  
Matching Problem ◽  
International Audience ◽  
Tree Data ◽  
Tree Data Structure ◽  
Restricted Pattern ◽  
Time And Space Complexity

International audience In this paper, we study a restricted version of the position restricted pattern matching problem introduced and studied by Mäkinen and Navarro [Position-Restricted Substring Searching, LATIN 2006]. In the problem handled in this paper, we are interested in those occurrences of the pattern that lies in a suffix or in a prefix of the given text. We achieve optimal query time for our problem against a data structure which is an extension of the classic suffix tree data structure. The time and space complexity of the data structure is dominated by that of the suffix tree. Notably, the (best) algorithm by Mäkinen and Navarro, if applied to our problem, gives sub-optimal query time and the corresponding data structure also requires more time and space.

scholarly journals Compatibility fans realizing graphical nested complexes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Thibault Manneville ◽  
Vincent Pilaud
Keyword(s):  
Finite Type ◽  
Type B ◽  
Cluster Complexes ◽  
Explicit Constructions ◽  
International Audience ◽  
Connected Subgraphs ◽  
Generalized Associahedra

International audience Graph associahedra are polytopes realizing the nested complex N(G) on connected subgraphs of a graph G.While all known explicit constructions produce polytopes with the same normal fan, the great variety of fan realizationsof classical associahedra and the analogy between finite type cluster complexes and nested complexes incitedus to transpose S. Fomin and A. Zelevinsky's construction of compatibility fans for generalized associahedra (2003)to graph associahedra. Using a compatibility degree, we construct one fan realization of N(G) for each of its facets.Specifying G to paths and cycles, we recover a construction by F. Santos for classical associahedra (2011) and extendF. Chapoton, S. Fomin and A. Zelevinsky's construction (2002) for type B and C generalized associahedra.

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