71 research outputs found
The modern Buddhism and its followers in Orissa, by Nagendra Náth Vasu.
No full textMode of access: Internet
Qiu, R. et al. A Unified Multi-Functional Dynamic Spectrum Access Framework: Tutorial, Theory and Multi-GHzWideband Testbed. Sensors 2009, 9, 6530-6603
No full textWe found that the affiliation of author Vasu Chakravarthy was incorrect in our paper published in Sensors recently
Operators on compositions and noncommutative Schur functions
No full textIn this thesis, we study a natural noncommutative lift of the ubiquitous Schur functions, called noncommutative Schur functions. These functions were introduced by Bessenrodt, Luoto and van Willigenburg and resemble Schur functions in many regards. We prove some new results for noncommutative Schur functions that are analogues of classical results, and demonstrate that the resulting combinatorics in this setting is equally rich. First we prove a Murnaghan-Nakayama rule for noncommutative Schur functions. In other words, we give an explicit combinatorial formula for expanding the product of a noncommutative power sum symmetric function and a noncommutative Schur function in terms of noncommutative Schur functions. In direct analogy to the classical Murnaghan-Nakayama rule, the summands are computed using a noncommutative analogue of border strips, and have coefficients ±1 determined by the height of these border strips. The rule is proved by interpreting the noncommutative Pieri rules for noncommutative Schur functions in terms of box adding operators on compositions. We proceed to give a backward jeu de taquin slide analogue on semistandard reverse composition tableaux. These tableaux were first studied by Haglund, Luoto, Mason and van Willigenburg when defining quasisymmetric Schur functions. Our algorithm for performing backward jeu de taquin slides on semistandard reverse composition tableaux results in a natural operator on compositions that we call the jdt operator. This operator in turn gives rise to a new poset structure on compositions whose maximal chains we enumerate. As an application, we also give new right Pieri rules for noncommutative Schur functions that use the jdt operators, in contrast to the left Pieri rules given by Bessenrodt, Luoto and van Willigenburg.Science, Faculty ofMathematics, Department ofGraduat
Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions
No full textNon UBCUnreviewedAuthor affiliation: University of British ColumbiaGraduat
Remixed Eulerian numbers
No full text25 pages, 8 figuresInternational audienceRemixed Eulerian numbers are a polynomial -deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as -binomial coefficients and Garsia--Remmel's -hit numbers. We study their combinatorics in more depth. As polynomials in , they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At , the former recovers Postnikov's interpretation whereas the latter recovers Liu's interpretation, both of which were obtained via methods different from ours
Forest polynomials and the class of the permutahedral variety
No full textInternational audienceWe study a basis of the polynomial ring that we call forest polynomials. This family of polynomials is indexed by a combinatorial structure called indexed forests and permits several definitions, one of which involves flagged P-partitions. As such, these polynomials have a positive expansion in the basis of slide polynomials. By a novel insertion procedure that may be viewed as a generalization of the Sylvester correspondence we establish that Schubert polynomials decompose positively in terms of forest polynomials. Our insertion procedure involves a correspondence on words which allows us to show that forest polynomials multiply positively. We proceed to show that forest polynomials are a particularly convenient basis in regards to studying the quotient of the polynomial ring modulo the ideal of positive degree quasisymmetric polynomials. This aspect allows us to give a manifestly nonnegative integral description for the Schubert class expansion of the cohomology class of the permutahedral variety in terms of a parking procedure. We study the associated combinatorics in depth and introduce a multivariate extension of mixed Eulerian numbers
The permutahedral variety, mixed Eulerian numbers, and principal specializations of Schubert polynomials
No full textInternational audienceWe compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of . The description implies that the are positive for all permutations of length , thereby answering a question of Harada, Horiguchi, Masuda, and Park. We use the same expression to establish the invariance of under taking inverses and conjugation by the longest word and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the by exploiting in addition the relation to Postnikov’s divided symmetrization. Finally, we are able to give a combinatorial interpretation for when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials, and related objects. We also sketch how to extend our approach to other Lie types, highlighting an identity of Klyachko in particular
A q-deformation of an algebra of Klyachko and Macdonald's reduced word formula
No full text48 pagesInternational audienceThere is a striking similarity between Macdonald's reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a -deformation of the -invariant part of the rational cohomology ring of , which we call the -Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows. 1) A -analog of divided symmetrization (-DS) using Yang-Baxter elements in the Hecke algebra. It is a linear form that picks up coefficients in the squarefree basis. 2) A relation between -DS and the ideal of quasisymmetric polynomials involving work of Aval--Bergeron--Bergeron. 3) A family of polynomials in with nonnegative integral coefficients that specialize to Postnikov's mixed Eulerian numbers when . We refer to these new polynomials as remixed Eulerian numbers. For q>0, their normalized versions occur as probabilities in the internal diffusion limited aggregation (IDLA) stochastic process. 4) A lift of Macdonald's reduced word identity in the -Klyachko algebra. 5) The Schubert expansion of the Chow class of the standard split Deligne--Lusztig variety in type , when is a prime power
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