scholarly journals A Survey of Racks and Quandles: Some Recent Developments

2020 ◽  
Vol 27 (03) ◽  
pp. 509-522
Author(s):  
Mohamed Elhamdadi
Keyword(s):  
Representation Theory ◽  
Algebraic Theory ◽  
Theoretic Approach ◽  
Short Survey ◽  
Recent Developments

This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.

scholarly journals Statistics with Imprecise Probabilities—A Short Survey

2021 ◽  
pp. 67-79
Author(s):  
Thomas Augustin
Keyword(s):  
Probabilistic Models ◽  
Statistical Modelling ◽  
Imprecise Probabilities ◽  
Paradigmatic Shift ◽  
Short Survey ◽  
Recent Developments ◽  
Data Imprecision

AbstractThis chapter aims at surveying and highlighting in an introductory way some challenges and big opportunities a paradigmatic shift to imprecise probabilities could induce in statistical modelling. Working with an informal understanding of imprecise probabilities, we discuss the concepts of model imprecision and data imprecision as the two main types of imprecision in statistical modelling. Then we provide a short survey of some major developments, methodological questions and applications of imprecise probabilistic models under model imprecision in the context of different inference schools and summarize some recent developments in the area of data imprecision.

scholarly journals MODULAR DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS: A PATH THEORETIC APPROACH

2018 ◽  
Vol 6 ◽  
Author(s):  
C. BOWMAN ◽  
A. G. COX
Keyword(s):  
Representation Theory ◽  
Symmetric Groups ◽  
Upper Bounds ◽  
Linear Groups ◽  
Theoretic Approach ◽  
Strong Linkage ◽  
Cherednik Algebras ◽  
Arbitrary Characteristic ◽  
General Linear Groups ◽  
Decomposition Numbers

We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

scholarly journals On the Finiteness of Hulls

2021 ◽  
Vol 70 (2) ◽  
pp. 44-53
Author(s):  
Florian Mauer
Keyword(s):  
Representation Theory ◽  
Major Advance ◽  
Discrete Representation ◽  
Recent Developments ◽  
Breaking Work

Let β≥φΛ. Is it possible to characterize normal, almost surely semi-degenerate categories? We show that K ̄is comparable to k. The ground breaking work of P. Laplace on vectors was a major advance. Recent developments in advanced discrete representation theory[1] have raised the question of whether the re exists a sub-continuous factor.

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