On Redfield's Range-Correspondences

1966 ◽  
Vol 18 ◽  
pp. 1060-1071 ◽  
Author(s):  
H. O. Foulkes
Keyword(s):  
Symmetric Group ◽  
Scalar Product ◽  
Point Of View ◽  
Combinatorial Analysis ◽  
Important Paper ◽  
Group Reduction ◽  
Linear Graphs

In an important paper (7), long overlooked, J. H. Redfield dealt with several aspects of enumerative combinatorial analysis. In a previous paper (1) I showed the relation between a certain repeated scalar product of a set of permutation characters of a symmetric group and Redfield's composition of his group reduction functions. Here I consider, from a group representational point of view, Redfield's idea of a range-correspondence and its application to enumeration of linear graphs. The details of the application of these ideas to more general enumerations are also given.

Quadratic Surplus

2016 ◽  
Author(s):  
Alfred Galichon
Keyword(s):  
Convex Analysis ◽  
Scalar Product ◽  
Optimal Transport ◽  
Point Of View ◽  
Factorization Theorem ◽  
Polar Factorization

This chapter considers the case when the attributes are d-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it relaxes the assumption that the distribution of the firms' attributes is discrete. This setting allows us to entirely rediscover convex analysis, which is introduced from the point of view of optimal transport. As a consequence, Brenier's polar factorization theorem is given, which provides a vector extension for the scalar notions of quantile and rearrangement.

On Pólya's Theorem

1967 ◽  
Vol 19 ◽  
pp. 792-799 ◽  
Author(s):  
J. Sheehan
Keyword(s):  
Finite Groups ◽  
Scalar Product ◽  
Combinatorial Analysis ◽  
Simple Extension ◽  
Group Characters ◽  
Polya's Theorem ◽  
Representations Of Finite Groups ◽  
Pólya’S Theorem ◽  
Special Case ◽  
Superposition Theorem

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

scholarly journals Representation Stability of Power Sets and Square Free Polynomials

2015 ◽  
Vol 67 (5) ◽  
pp. 1024-1045
Author(s):  
Samia Ashraf ◽  
Haniya Azam ◽  
Barbu Berceanu
Keyword(s):  
Symmetric Group ◽  
Braid Group ◽  
Point Of View ◽  
Pure Braid Group ◽  
Representation Stability ◽  
Stability Point ◽  
Power Set ◽  
The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

scholarly journals Note on fluorescence and absorption

1905 ◽  
Vol 76 (508) ◽  
pp. 165-167 ◽  
Keyword(s):  
Chemical Composition ◽  
Chemical Change ◽  
Point Of View ◽  
The Body ◽  
Fluorescent Light ◽  
Exciting Light ◽  
Remarkable Difference ◽  
Important Paper ◽  
Physical Point ◽  
Temporary Change

In a, paper “On the Change of Absorption produced by Fluorescence,” I described the existence of a very remarkable difference in the absorption of the fluorescent light of uranium glass, when in the luminous and non-luminous states. This difference I have attributed to a temporary change in the structure or chemical composition of the body when exposed to the influence of the exciting light; and I have been led to suppose that it is due to the formation of new atomic connections which give rise to new frequencies during the period of luminosity. These I regarded as the result of the formation of unstable molecular aggregates by the more refrangible or exciting rays; and the luminosity or fluorescence, as the radiation which results from the breaking down of such unstable molecular groups. As in the case of photographic action, some chemical change appears to be produced by the blue and violet rays; the two cases differing, from the physical point of view, merely so far as the molecular aggregations, instead of remaining fixed, rapidly disintegrate, radiating intensely at the same time the energy which was stored up in their formation. Thus the luminosity itself is but the visible manifestation of the building up and breaking down of what are probably complicated molecular agglomerations. In their very interesting and important paper on this subject, Messrs. Nichols and Merritt have shown that the phenomenon of the change of absorption depends upon the intensity of the fluorescence. They find that a saturation effect takes place as the intensity of the luminosity increases, so that the change of absorption reaches a maximum with a certain intensity of the fluorescent light.

scholarly journals The combinatorics of an algebraic class of easy quantum groups

2014 ◽  
Vol 17 (03) ◽  
pp. 1450016 ◽  
Author(s):  
Sven Raum ◽  
Moritz Weber
Keyword(s):  
Symmetric Group ◽  
Quantum Group ◽  
Permutation Group ◽  
Free Product ◽  
Quantum Groups ◽  
Orthogonal Group ◽  
Point Of View ◽  
Algebraic Point ◽  
Matrix Quantum

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group Sn and the orthogonal group On as well as Wang's quantum permutation group [Formula: see text] and his free orthogonal quantum group [Formula: see text]. In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.

scholarly journals Universal induced characters and restriction rules for the classical groups

1990 ◽  
Vol 117 ◽  
pp. 173-205 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
General Linear Group ◽  
Hilbert Series ◽  
Point Of View ◽  
Classical Groups ◽  
Starting Point ◽  
Definition Of ◽  
Matrix Invariants

The purpose of this paper is the study of some basic properties of universal induced characters and their applications to the representation theory of the classical groups (for the definition of a universal induced character, see § 3).The starting point was the paper [F] by E. Formanek on matrix invariants. In his paper [F], Formanek has investigated the Hilbert series for the ring of matrix invariants from the point of view of the representation theory of the general linear group and the symmetric group. In this paper we shall study polynomial concomitants of a group from the same point of view.

On Redfield's Group Reduction Functions

1963 ◽  
Vol 15 ◽  
pp. 272-284 ◽  
Author(s):  
H. O. Foulkes
Keyword(s):  
Combinatorial Analysis ◽  
Permutation Groups ◽  
Group Reduction

In 1927, J. H. Redfield (12), discussed some of the links between combinatorial analysis and permutation groups, including such topics as group transitivity, the enumeration of certain geometrical configurations, and the construction of various permutation isomorphs of a given group. Except for a revision of Redfield's treatment of transitivity by D. E. Littlewood (6), this 1927 paper appears to have been overlooked. However, it has recently been described (5) as a remarkable pioneering paper which appears to contain or anticipate virtually all of the enumeration results for graphs which have been discovered and developed during the last thirty years.

scholarly journals Transitive Factorizations in the Hyperoctahedral Group

2008 ◽  
Vol 60 (2) ◽  
pp. 297-312
Author(s):  
G. Bini ◽  
I. P. Goulden ◽  
D. M. Jackson
Keyword(s):  
Symmetric Group ◽  
Moduli Space ◽  
Moduli Spaces ◽  
Point Of View ◽  
Moduli Space Of Curves ◽  
Reflection Groups ◽  
Hurwitz Numbers ◽  
Hyperoctahedral Group ◽  
Geometric Point ◽  
Enumeration Problem

AbstractThe classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type A to other finite reflection groups and, in particular, to type B. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an S2-symmetry. The type A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type B case that is studied here.

scholarly journals An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

10.37236/4567 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Yuval Filmus
Keyword(s):  
Symmetric Group ◽  
Boolean Functions ◽  
Orthogonal Basis ◽  
Point Of View ◽  
Fundamental Result ◽  
Orthogonal Representation ◽  
Low Degree ◽  
Total Influence ◽  
Multilinear Polynomials ◽  
Analysis Of Boolean Functions

We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form $\{(x_1,\ldots,x_n) \in \{0,1\}^n : \sum_i x_i = k\}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to any exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $\mathbb{R}^n \to \mathbb{R}$ which are annihilated by the differential operator $\sum_i \partial/\partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm.As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for the slice. Friedgut's theorem, a fundamental result in the analysis of Boolean functions, states that a Boolean function on the Boolean hypercube with low total influence can be approximated by a Boolean junta (a function depending on a small number of coordinates). Wimmer generalized this result to slices of the Boolean hypercube, working mostly over the symmetric group, and utilizing properties of Young's orthogonal representation. Using our basis, we show how the entire argument can be carried out directly on the slice.

scholarly journals Repeated Derivatives of Hyperbolic Trigonometric Functions and Associated Polynomials

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 138 ◽  
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Elio Sabia
Keyword(s):  
Point Of View ◽  
Combinatorial Analysis ◽  
Trigonometric Functions ◽  
Special Polynomials ◽  
Associated Polynomials ◽  
Derivatives Of

Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec ( . ) , tan ( . ) and for their hyperbolic counterparts.

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