scholarly journals Characteristic-free representation theory of the general linear group II. Homological considerations

1988 ◽  
Vol 72 (2) ◽  
pp. 171-210 ◽  
Author(s):  
Kaan Akin ◽  
David A Buchsbaum
Keyword(s):  
Representation Theory ◽  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Free Representation ◽  
Group Ii

scholarly journals DEFINING RELATIONS OF LOW DEGREE OF INVARIANTS OF TWO 4 × 4 MATRICES

2009 ◽  
Vol 19 (01) ◽  
pp. 107-127 ◽  
Author(s):  
VESSELIN DRENSKY ◽  
ROBERTO LA SCALA
Keyword(s):  
Representation Theory ◽  
Linear Group ◽  
General Linear Group ◽  
Minimal Degree ◽  
General Linear ◽  
Generating Set ◽  
Defining Relations ◽  
Low Degree ◽  
Computer Calculations ◽  
Minimal Generating Set

The trace algebra Cnd over a field of characteristic 0 is generated by all traces of products of d generic n × n matrices, n, d ≥ 2. Minimal sets of generators of Cnd are known for n = 2 and 3 for any d and for n = 4 and 5 and d = 2. The explicit defining relations between the generators are found for n = 2 and any d and for n = 3, d = 2 only. Defining relations of minimal degree for n = 3 and any d are also known. The minimal degree of the defining relations of any homogeneous minimal generating set of C42 is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree ≤ 14. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.

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.

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