Exterior power operations in the representation theory of the classical Weyl groups

1999 ◽  
Vol 27 (12) ◽  
pp. 6273-6296 ◽  
Author(s):  
John S. Bryden
Keyword(s):  
Representation Theory ◽  
Weyl Groups ◽  
Exterior Power ◽  
Power Operations

scholarly journals Exterior power operations on higher K-groups via binary complexes

2017 ◽  
Vol 2 (3) ◽  
pp. 409-450 ◽  
Author(s):  
Tom Harris ◽  
Bernhard Köck ◽  
Lenny Taelman
Keyword(s):  
Exterior Power ◽  
Binary Complexes ◽  
Power Operations

scholarly journals Lattice structure of Weyl groups via representation theory of preprojective algebras

2018 ◽  
Vol 154 (6) ◽  
pp. 1269-1305 ◽  
Author(s):  
Osamu Iyama ◽  
Nathan Reading ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Representation Theory ◽  
Weyl Group ◽  
Lattice Structure ◽  
Weak Order ◽  
Weyl Groups ◽  
Preprojective Algebra ◽  
Combinatorial Description ◽  
Irreducible Elements ◽  
Preprojective Algebras

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

Exterior power operations on higher K-theory

K-Theory ◽  
1989 ◽  
Vol 3 (3) ◽  
pp. 247-260 ◽  
Author(s):  
Daniel R. Grayson
Keyword(s):  
Exterior Power ◽  
Power Operations ◽  
K Theory

scholarly journals Two formulae for exterior power operations on higher K-groups

2019 ◽  
Vol 223 (11) ◽  
pp. 4925-4936
Author(s):  
Tom Harris ◽  
Bernhard Köck
Keyword(s):  
Exterior Power ◽  
Power Operations

scholarly journals Enumerating projective reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Riccardo Biagioli ◽  
Fabrizio Caselli
Keyword(s):  
Representation Theory ◽  
Generating Functions ◽  
Special Class ◽  
Hilbert Series ◽  
Weyl Groups ◽  
Reflection Groups ◽  
One Dimensional ◽  
Complex Reflection Groups ◽  
Major Index ◽  
International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

scholarly journals REPRESENTATIONS OF THE RENNER MONOID

2009 ◽  
Vol 19 (04) ◽  
pp. 511-525 ◽  
Author(s):  
ZHUO LI ◽  
ZHENHENG LI ◽  
YOU'AN CAO
Keyword(s):  
Representation Theory ◽  
Crucial Role ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Parabolic Subgroups ◽  
Reductive Monoids ◽  
Complete Set ◽  
Character Formulas ◽  
Renner Monoid

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

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