scholarly journals Supercharacter theory for the Borel contraction of the group GL(n, Fq)

2020 ◽  
Vol 7 (65) (2) ◽  
pp. 254-268
Author(s):  
Aleksandr N. Panov ◽  
Keyword(s):  
Supercharacter Theory

Supercharacters and mixed moments of Kloosterman sums

2018 ◽  
Vol 14 (04) ◽  
pp. 1023-1032 ◽  
Author(s):  
Ángel Chávez ◽  
George Todd
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Kloosterman Sums ◽  
Fourth Degree ◽  
Mixed Power ◽  
Power Moments ◽  
Supercharacter Theory ◽  
Trace Of Frobenius

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

scholarly journals Upper and Lower Semimodularity of the Supercharacter Theory Lattices of Cyclic Groups

2013 ◽  
Vol 42 (3) ◽  
pp. 1123-1135 ◽  
Author(s):  
Samuel G. Benidt ◽  
William R. S. Hall ◽  
Anders O. F. Hendrickson
Keyword(s):  
Cyclic Groups ◽  
Supercharacter Theory

An algorithm for computing a supercharacter theory generated from a given partition

2021 ◽  
pp. 1-12
Author(s):  
Shawn T. Burkett
Keyword(s):  
Finite Group ◽  
Partial Order ◽  
Supercharacter Theory ◽  
A Finite Group

Let [Formula: see text] be a finite group. The set of all supercharacter theories of [Formula: see text] forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of [Formula: see text], with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet of two supercharacter theories.

scholarly journals Vanishing-off subgroups and supercharacter theory products

2020 ◽  
Vol 30 (05) ◽  
pp. 1057-1072
Author(s):  
Shawn T. Burkett ◽  
Mark L. Lewis
Keyword(s):  
Theoretic Characterization ◽  
Supercharacter Theory

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a [Formula: see text]-product or ∗-product from the values of its corresponding supercharacters.

scholarly journals Shell Tableaux: A Set Partition Analog of Vacillating Tableaux

10.37236/8241 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Megan Ly
Keyword(s):  
Irreducible Representations ◽  
Character Theory ◽  
Set Partition ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Weyl Duality ◽  
Supercharacter Theory ◽  
Combinatorial Representation Theory ◽  
Centralizer Algebra

Schur–Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur–Weyl duality for the group of unipotent upper triangular matrices over a finite field.  In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this paper, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analog of Young tableaux, known as shell tableaux, to index paths in this graph. 

scholarly journals Combinatorial Hopf algebra of supercharacters of type D

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Carolina Benedetti
Keyword(s):  
Hopf Algebra ◽  
Finite Field ◽  
Type A ◽  
The Other ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
Type D ◽  
International Audience ◽  
Supercharacter Theory ◽  
Triangular Group

International audience We provide a Hopf algebra structure on the supercharacter theory for the unipotent upper triangular group of type {D} over a finite field. Also, we make further comments with respect to types {B} and {C}. Type {A} was explored by M. Aguiar et. al (2010), thus this extended abstract is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types. Dotamos con una estructura de álgebra de Hopf la teoría de supercaracteres del grupo de matrices unipotentes triangulares superiores de tipo{D} sobre un cuerpo finito. Ademas, discutimos brevemente los tipos {B} y {C}. El tipo A fue explorado por M. Aguiar et al (2010), por lo tanto este resumen extendido es una contribución para entender combinatoriamente la teoría de supercaracteres de los otros tipos de Lie clásicos. Nous construisons une structure d'algèbre de Hopf sur la thérie des supercharactères du groupe de matrices unipotentes triangulaires supéieures de type {D}. Nous donnons aussi quelques commentaires à l'égard des types {B} et {C} . Le type {A} a été explorée par M. Aguiar et al. (2010), donc ce résumé étendu est une contribution à la théorie combinatoire des supercharactères pour les autres types de Lie classiques. \par

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