scholarly journals MODULAR REDUCTION OF THE STEINBERG LATTICE OF THE GENERAL LINEAR GROUP

2008 ◽  
Vol 07 (06) ◽  
pp. 793-807 ◽  
Author(s):  
FERNANDO SZECHTMAN
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Completely Reducible ◽  
Natural Filtration ◽  
Multiplicity Free ◽  
Steinberg Representation

Each factor of the natural filtration of the modular reduction of the Steinberg representation of the general linear group is shown to be completely reducible and the entire modular reduction is shown to be multiplicity free.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ivan Matić
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Discrete Series ◽  
Series Representation ◽  
General Linear ◽  
Composition Series ◽  
Induced Representations ◽  
Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

scholarly journals Triple factorisations of the general linear group and their associated geometries

2015 ◽  
Vol 469 ◽  
pp. 169-203 ◽  
Author(s):  
Seyed Hassan Alavi ◽  
John Bamberg ◽  
Cheryl E. Praeger
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear

Two-Dimensional Linear Groups Over Local Rings

1969 ◽  
Vol 21 ◽  
pp. 106-135 ◽  
Author(s):  
Norbert H. J. Lacroix
Keyword(s):  
Local Ring ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Linear Groups ◽  
Local Rings ◽  
Two Dimensional ◽  
Normal Subgroups ◽  
Field Case

The problem of classifying the normal subgroups of the general linear group over a field was solved in the general case by Dieudonné (see 2 and 3). If we consider the problem over a ring, it is trivial to see that there will be more normal subgroups than in the field case. Klingenberg (4) has investigated the situation over a local ring and has shown that they are classified by certain congruence groups which are determined by the ideals in the ring.Klingenberg's solution roughly goes as follows. To a given ideal , attach certain congruence groups and . Next, assign a certain ideal (called the order) to a given subgroup G. The main result states that if G is normal with order a, then ≧ G ≧ , that is, G satisfies the so-called ladder relation at ; conversely, if G satisfies the ladder relation at , then G is normal and has order .

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