scholarly journals Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

10.37236/6960 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Symmetric Group ◽  
Finite Subset ◽  
Upper Bound ◽  
Cohomology Class ◽  
Symmetric Functions ◽  
A Priori ◽  
Specht Module ◽  
Group A ◽  
Schubert Classes ◽  
Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

scholarly journals Cohomology classes of rank varieties and a counterexample to a conjecture of Liu

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Cohomology Class ◽  
Symmetric Functions ◽  
Specht Module ◽  
Nous Montrons ◽  
Group A ◽  
International Audience ◽  
Schubert Classes ◽  
Stanley Symmetric Functions ◽  
Rank Variety ◽  
Discrete Grid

International audience To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function. A chaque sous-ensemble fini de $\mathbb{N}×\mathbb{N}$ (un diagramme), on peut associer une sous-variété d’une grassmannienne complexe et une représentation d’un groupe symétrique (un module de Specht). Liu a conjecturé que la classe de cohomologie de la variété d’un diagramme est représentée par la caractéristique de Frobenius du module de Specht correspondant. Nous donnons un contre-exemple à cette conjecture.Cependant, nous montrons que dans le cas de la variété du diagramme de permutation, la classe de cohomologie conjecturée par Liu est au moins un majorant de la classe juste $\tau$ , c’est-à-dire que $\sigma - \tau$ est une combinaison linéaire non-négative des classes de Schubert. Pour ce faire, nous considérons une dégénérescence des variétés de rang de Coskun qui contient la variété appropriée d’un diagramme comme une composante irréductible. Les variétés de rang sont des exemples de variétés de positroïde, dont les classes de cohomologie sont représentées par des fonctions symétriques de Stanley affines. En effet, nous montrons que la classe de cohomologie d’une variété de rang est représentée par une fonction symétrique de Stanley ordinaire.

I.—A class of symmetric polynomials with a parameter

1970 ◽  
Vol 69 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Henry Jack
Keyword(s):  
Symmetric Group ◽  
Orthogonal Group ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Orthogonal Matrix ◽  
Explicit Representation ◽  
Symmetric Polynomials ◽  
Zonal Polynomials ◽  
Linear Combinations ◽  
Group A

SynopsisIn an attempt to evaluate the integral (5) below, using a decomposition of an orthogonal matrix (Jack 1968), the author is led to define a set of polynomials, one for each partition of an integer k, which are invariant under the orthogonal group and which depend on a real parameter α. An explicit representation of these polynomials is given in an operational form. When α = − 1, these polynomials coincide with the augmented monomial symmetric functions. When α = 1, a systematic way of taking linear combinations of these polynomials is explained and it is shown that the resulting polynomials coincide with the Schur functions from the representation theory of the symmetric group. A similar procedure in the case α = 2 then appears to give the zonal polynomials as defined by James (1964, p. 478).

Convergence of management practices in India’s IT sector with ana priorivalidation

2018 ◽  
Vol 12 (4) ◽  
pp. 402-421
Author(s):  
Jayashree Mahesh ◽  
Anil K. Bhat
Keyword(s):  
Cluster Analysis ◽  
Operations Management ◽  
Management Practices ◽  
Qualitative Interviews ◽  
A Priori ◽  
Statistical Testing ◽  
Content Type ◽  
Empirical Survey ◽  
Different Types ◽  
Group A

PurposeThe purpose of this paper is to document similarities and differences between management practices of different types of organizations in India’s IT sector through an empirical survey. The authors expected these differences to be significant enough for us to be able to groupa priorithis set of companies meaningfully through cluster analysis on the basis of the similarity of their management practices alone.Design/methodology/approachUsing a mixed-methods approach, 73 senior-level executives of companies working in India’s IT sector were approached with a pretested questionnaire to find out differences on eighteen management practices in the areas of operations management, monitoring management, targets management and talent management. The different types of organizations surveyed were small and amp; medium global multinationals, large global multinationals, small and medium Indian multinationals, large Indian multinationals and small and medium local Indian companies. The differences and similarities found through statistical testing were further validateda priorithrough cluster analysis and qualitative interviews with senior-level executives.FindingsThe management practices of multinationals in India are moving toward Western management practices, indicating that management practices converge as the organizations grow in size. Though the practices of large Indian multinationals were not significantly different from those of global multinationals, the surprising finding was that large Indian multinationals scored better than global multinationals on a few practices. The practices of small and medium Indian companies differed significantly from those of other types of organizations and hence they formed a cluster.Practical implicationsThe finding that large Indian IT multinationals have an edge over global multinationals in certain people management practices is a confirmation of the role of human resource practices in their current success and their continuing competitive advantage.Originality/valueThis is perhaps the first study of its kind to document state of specific management practices across different types of organizations in India’s IT sector and then use measures on these practices to group a priori these organizations for validation.

Stanley Symmetric Functions and Peterson Algebras

2014 ◽  
pp. 133-168 ◽  
Author(s):  
Thomas Lam ◽  
Luc Lapointe ◽  
Jennifer Morse ◽  
Anne Schilling ◽  
Mark Shimozono ◽  
...  
Keyword(s):  
Symmetric Functions ◽  
Stanley Symmetric Functions

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

AN UPPER BOUND FOR THE NUMBER OF VASSILIEV KNOT INVARIANTS

1994 ◽  
Vol 03 (02) ◽  
pp. 141-151 ◽  
Author(s):  
S. V. CHMUTOV ◽  
S. V. DUZHIN
Keyword(s):  
Upper Bound ◽  
A Priori ◽  
Knot Invariants ◽  
A Priori Bound

We prove that the number of independent Vassiliev knot invariants of order n is less than (n − 1)! — thus strengthening the a priori bound (2n − 1)!!

Sliding Mode Control With Perturbation Estimation (SMCPE) and Frequency Shaped Sliding Surfaces

1997 ◽  
Vol 119 (3) ◽  
pp. 584-588 ◽  
Author(s):  
Jairo T. Moura ◽  
Rajiv Ghosh Roy ◽  
Nejat Olgac
Keyword(s):  
Sliding Mode Control ◽  
Upper Bound ◽  
Sliding Mode ◽  
A Priori ◽  
Tracking Errors ◽  
Mode Control ◽  
Uncertain Dynamic Systems ◽  
Estimation Scheme ◽  
Frequency Components ◽  
Perturbation Estimation

Sliding Mode Control with Perturbation Estimation (SMCPE) is a recent control routine which steers uncertain dynamic systems with disturbances to follow a desired trajectory. It eliminates the conventional requirement for the knowledge of uncertainty upper bound. A perturbation estimation scheme provides a tool for robustness. This text offers an additional robustizing mechanism: selection of time-varying sliding functions utilizing frequency shaping techniques. Frequency shaping together with sliding mode control introduces a behavior for selectively penalizing tracking errors at certain frequency ranges. This combination provides two advantages concurrently: (a) It filters out certain frequency components of the perturbations therefore eliminating the possible excitation on the unmodelled dynamics, and (b) it drives the state to the desired trajectory despite perturbations. The crucial point is that a priori knowledge of the perturbation upper bound is not necessary to eliminate the perturbation effects at the designated frequencies. Numerical examples prove the effectiveness of this novel scheme.

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