The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of S...
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The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the Landweber-Novikov algebra in complex cobordism theory;it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur Functions. We also discuss applications of our formulas to the geometry of Grassmannians. (C) 1998 Academic Press.
In this thesis, we study the representation theory of the symmetric group on a new categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained, arising...
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In this thesis, we study the representation theory of the symmetric group on a new categorification of the theory of generalized permutohedra. The vector spaces in the categorification are tightly constrained, arising as solution spaces to certain continuity relations which appeared first in Quantum Field Theory in the mid 20th century. The generators of the vector space are characteristic functions of certain polyhedral cones called plates, due to A. *** combinatorics, the Eulerian numbers count the number of permutations with a given number of ascent and descents. The classical Worpitzky identity in combinatorics expands a power $r^p$ as a sum of Eulerian numbers, with binomial coefficients. The main combinatorial result is the categorification of Worpitzky's identity to an isomorphism between two graded representations of the symmetric group, corresponding geometrically to the decomposition of a scaled simplex into a direct sum of translated unit hypersimplices. We recover the classical Worpitzky identity by evaluating the characters on the identity permutation. The character values in general can be interpreted as relative volumes of generalized *** proof involves a new algebra of commuting operators which act by translation on plates in a scaled *** show that the spectrum of this algebra is labeled by the subset $\{x\in(\mathbb{Z}\slash r)^n:\sum x_i\equiv 1\},$ on which $S_n$ acts by permuting positions. The values of the character of a permutation $\sigma$ is thus given by the number of fixed points on this set. This number is obtained by counting solutions to a certain modular Diophantine equation involving the cycle lengths of $\sigma$.
We study the dynamics of a family K-alpha of discontinuous interval maps whose (infinitely many) branches are Mobius transformations in SL(2, Z) and which arise as the critical-line case of the family of (a, b)-contin...
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We study the dynamics of a family K-alpha of discontinuous interval maps whose (infinitely many) branches are Mobius transformations in SL(2, Z) and which arise as the critical-line case of the family of (a, b)-continued fractions. We provide an explicit construction of the bifurcation locus epsilon(KU) for this family, showing it is parametrized by Farey words and it has Hausdorff dimension zero. As a consequence, we prove that the metric entropy of K-alpha is analytic outside the bifurcation set but not differentiable at points of epsilon(KU) and that the entropy is monotone as a function of the parameter. Finally, we prove that the bifurcation set is combinatorially isomorphic to the main cardioid in the Mandelbrot set, providing one more entry to the dictionary developed by the authors between continued fractions and complex dynamics.
Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, c...
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Wyckoff sequences are a way of encoding combinatorial information about crystal structures of a given symmetry. In particular, they offer an easy access to the calculation of a crystal structure's combinatorial, coordinational and configurational complexity, taking into account the individual multiplicities (combinatorial degrees of freedom) and arities (coordinational degrees of freedom) associated with each Wyckoff position. However, distinct Wyckoff sequences can yield the same total numbers of combinatorial and coordinational degrees of freedom. In this case, they share the same value for their Shannon entropy based subdivision complexity. The enumeration of Wyckoff sequences with this property is a combinatorial problem solved in this work, first in the general case of fixed subdivision complexity but non-specified Wyckoff sequence length, and second for the restricted case of Wyckoff sequences of both fixed subdivision complexity and fixed Wyckoff sequence length. The combinatorial results are accompanied by calculations of the combinatorial, coordinational, configurational and subdivision complexities, performed on Wyckoff sequences representing actual crystal structures.
In memoriam: N.G. de Bruijn. In this article we present a survey of his papers on combinatorics. The section titles show its variety. 1. Common systems of representatives 2. De Bruijn cycles 3. The De Bruijn-Erdos the...
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In memoriam: N.G. de Bruijn. In this article we present a survey of his papers on combinatorics. The section titles show its variety. 1. Common systems of representatives 2. De Bruijn cycles 3. The De Bruijn-Erdos theorem from incidence geometry 4. Bases for integers 5. The BEST theorem 6. The De Bruijn-Erdos theorem from graph theory 7. Factorizations of finite groups 8. Rooted trees in the plane 9. Permutations of a given shape 10. Covering of graphs by dimers 11. Counting (Polya's fundamental theorem, Color designs) 12. Penrose tilings (C) 2013 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bou...
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Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.
A formula for the calculation of the number of Wyckoff sequences of a given length is presented, based on the combinatorics of multisets with finite multiplicities and a generating function approach, assuming a certai...
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A formula for the calculation of the number of Wyckoff sequences of a given length is presented, based on the combinatorics of multisets with finite multiplicities and a generating function approach, assuming a certain space-group type and taking into account the number of non-fixed and fixed Wyckoff positions, respectively. The formula is applied to the 44 distinguishable combinatorial types of the 230 space-group types. A comparison is made between the calculated frequencies of occurrence of Wyckoff sequences of given space-group type and length and the observed ones for actual crystal structures, as retrieved from the Pearson's Crystal Data Crystal Structure Database for Inorganic Compounds.
This survey is mainly intended for non-specialists, though we try to include many recent developments that may interest the experts as well. We want to study "good" finite subsets of the unit sphere. To cons...
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This survey is mainly intended for non-specialists, though we try to include many recent developments that may interest the experts as well. We want to study "good" finite subsets of the unit sphere. To consider "what is good" is a part of our problem. We start with the definition of spherical t-designs on Sn-1 in R-n. After discussing some important examples, we focus on tight spherical t-designs on Sn-1. Tight t-designs have good combinatorial properties, but they rarely exist. So, we are interested in the finite subsets on Sn-1, which have properties similar to tight t-designs from the various viewpoints of algebraic combinatorics. For example, rigid t-designs, universally optimal t-codes (configurations), as well as finite sets which admit the structure of an association scheme, are among them. We will discuss various results on the existence and the non-existence of special spherical t-designs, as well as general spherical t-designs, and their constructions. We will discuss the relations between spherical t-designs and many other branches of mathematics. For example: by considering the spherical designs which are orbits of a finite group in the real orthogonal group O(n), we get many connections with group theory;by considering those which are shells of Euclidean lattices, we get many unexpected connections with number theory, such as modular forms and Lehmer's conjecture about the zeros of the Ramanujan tau function. Spherical t-designs and Euclidean t-designs are special cases of cubature formulas in approximation theory, and thus we get many connections with analysis and statistics, and in particular with orthogonal polynomials, and moment problems. Moreover, Delsarte's linear programming method and many recent generalizations, including the work of Musin and the subsequent progress in using semidefinite programming, have strong connections with geometry (in particular sphere packing problems) and the theory of optimizations. These various connections explai
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