1,721,034 research outputs found
Generalized involution models for wreath products
No full textWe prove that if a finite group H has a generalized involution model, as defined by Bump and Ginzburg, then the wreath product H ≀ S[subscript n] also has a generalized involution model. This extends the work of Baddeley concerning involution models for wreath products. As an application, we construct a Gel’fand model for wreath products of the form A ≀ S[subscript n] with A abelian, and give an alternate proof of a recent result due to Adin, Postnikov and Roichman describing a particularly elegant Gel’fand model for the wreath product ℤ[subscript r] ≀ S[subscript n]. We conclude by discussing some notable properties of this representation and its decomposition into irreducible constituents, proving a conjecture of Adin, Postnikov and Roichman
Actions and Identities on Set Partitions
Full text linkA labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group A. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of A[superscript n] on the set of A-labeled partitions of an (n+1)-set. By investigating the orbit decomposition of various families of set partitions under this action, we derive new combinatorial proofs of Coker's identity for the Narayana polynomial and its type B analogue, and establish a number of other related identities. In return, we also prove some enumerative results concerning André and Neto's supercharacter theories of type B and D
Isomorphisms, automorphisms, and generalized involution models of projective reflection groups
Full text linkWe investigate the generalized involution models of the projective reflection groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p′, q′, n) are isomorphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if G(r, p, 1, n) ≌ G(r, 1, p, n). We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd
Demazure crystals for flagged key polynomials
No full textOne definition of key polynomials is as the weight generating functions of key tableaux. Assaf and Schilling introduced a crystal structure on key tableaux and related it to the Morse-Schilling crystal on reduced factorizations for per-mutations via the weak Edelman-Greene insertion. In this thesis, we consider generalizations of key tableaux and reduced factorizations depending on a flag. We extend the weak EG insertion to a bijection between our flagged objects and show that the recording tableau gives a crystal isomorphism. We prove that extending the Assaf-Schilling crystal operators to flagged key tableaux gives a Demazure crystal. As an application, we show that the weight generating functions of flagged key tableaux recover Reiner and Shimozono's definition of flagged key polynomials.</p
Gelfand w-graphs and perfect models
No full textA Gelfand model for an algebra is a module isomorphic to a direct sum of irreducible modules, with every isomorphism class of irreducible modules represented exactly once. We introduce and study the notion of a perfect model for a finite Coxeter group; such a model is a certain set of discrete data parametrizing a Gelfand model for the associated Iwahori-Hecke algebra. We classify which Coxeter groups have perfect models, and then describe explicit Gelfand models for the Iwahori-Hecke algebras of classical finite Coxeter groups. This simultaneously generalizes constructions of Adin, Postnikov, and Roichman (from type A to other classical types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). Our Gelfand models have interesting “canonical bases” that give rise to associated W-graphs, which we call Gelfand W-graphs. A W-graph is a kind of directed graph encoding an Iwahori-Hecke algebra module. For types B and D, we prove that these W-graphs are dual to each other, a phenomenon which does not occur in type A. The strongly connected components in a W-graph are called its cells, and the components connected by bidirected edges are called its molecules. For type A, we classify the molecules in our Gelfand W-graphs, and conjecture that in type A every molecule is a cell.</p
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Bijections between pipe dreams, plane partitions, and their symplectic variations
No full textIn 1993, Fomin and Kirillov [4] introduced a new set of diagrams called RC graphs or pipe dreams that aid in computing Schubert polynomials. Nowadays, there are many studies on different kinds of pipe dreams and their bijections with other combinatorial objects. Serrano and Stump [20] gave bijections between pipe dreams, certain north-east fillings, and k-flagged tableaux. Hamaker, Marberg, and Pawlowski [8] studied a new class of involution pipe dreams. Lam, Lee, and Shimozono [13] more recently introduced bumpless pipe dreams and Gao and Huang [6] have provided a bijection between bumpless pipe dreams and ordinary pipe dreams. As featured in the work of Knutson and Miller [12], there is a close connection between pipe dreams and the geometry of Schubert varieties. Work of Marberg and Pawlowski [16] extends this connection to involution pipe dreams. Apart from these geometric motivations, pipe dreams are also of interest on their own in combinatorics. This thesis first studies bijections between pipe dreams and a variety of other objects: for example, certain north-east chains, north-east fillings, bumpless pipe dreams, and reverse plane partitions. This expository material surveys the main results from [6] and [20]. Then we study symplectic variations of these constructions, proving some new results. Specifically, we investigate bijections between involution pipe dreams, certain reflected north-east chains and fillings, and shifted reverse plane partitions. We also study a generalization of bumpless pipe dreams for involutions. We give a conjectural bijection between these objects and involution pipe dreams, extending the correspondence of [6].</p
Orbits in the affine flag variety of type A
No full textIt is a classical result that the set K\G/B is finite, where G is a reductive algebraic group over an algebraically closed field with characteristic not equal to two, B is a Borel subgroup of G, and K = Gθis the fixed point subgroup of a holomorphic involution of G. In this thesis, we investigate the affine counterpart of the aforementioned set, where G is the general linear group over formal Laurent series, B is an Iwahori subgroup of G, and K is either the orthogonal group or the symplectic group over formal Laurent series, or GLp(ℂ((t))) × GLq(ℂ((t))). We construct explicit bijections between the double cosets and certain twisted affine involutions, or certain signed affine involutions known as affine clans. Since K\G/B can also be interpreted as the K-orbits on the affine flag variety, combinatorial descriptions of double cosets in K\G/B or orbits on the affine flag variety pave the way for exploring affine versions of previously investigated topics in the finite case. For example, the results here will make it possible in future work to study the weak order of the closure of orbits and certain cohomologies of the affine flag variety.</p
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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