irreducible module

irreducible module

[‚ir·i¦dü·sə·bəl ′mäj·əl]
(mathematics)
A module whose only submodules are the module itself and the module that consists of the element 0.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
References in periodicals archive ?
We set Irr(H) = {[x.sub.i]}, the corresponding characters where [x.sub.i] = x[V.sub.i] and ([x.sub.1], 1) = [d.sub.i], the dimension of the irreducible module [V.sub.i].
In particular, for {[P.sub.1], ..., [P.sub.n]} a complete set of indecomposable projective modules and {[S.sub.1], ..., [S.sub.n]} their associated simple irreducible module, we have <[[P.sub.i]]>, [[S.sub.j]]> = [[delta].sub.i,j].
Let [P.sup.[lambda].sub.k] denote the irreducible module indexed by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Let [RB.sup.[lambda].sub.k] denote the irreducible module indexed by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
When [A.sub.k] is semisimple, its irreducible modules are indexed by a set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we let denote the irreducible [A.sub.k]-module labeled by [lambda].
It follows that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is irreducible and that signed conjugation produces a complete set of irreducible modules for the planar algebras.
(1) In a model representation, isotypic components are irreducible components, so projection operators map directly onto irreducible modules without being mixed up among multiple isomorphic copies of the same module.
formed as a direct sum of the Heisenberg VOA with all of its irreducible modules. M is spanned by [[PSI].sub.[alpha]] (k, 1) for all [alpha] [member of] C.
Imprimitive Irreducible Modules for Finite Quasisimple Groups
Furthermore, they are irreducible modules. We can identify which [H.sub.n](0)-module they are by looking at the action of [[pi].sub.j] on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for 1 [less than or equal to] j [less than or equal to] n - 1.
The contributors discuss known results and methods for determining root multiplicities for hyperbolic Kac-Moody algebras, results in Leibniz algebras which are analogs of corresponding results in Lie algebras, Weyl modules for quantum and hyper loop algebras, and Koszul properties of standard and irreducible modules. Other topics include supersymmetry and the modular double, toroidal Lie superalgebras and free field representations, total Frobenius-Schur indicators, and loop Grassmannians in the framework of local spaces over a curve.
The operation on the category of [H.sub.n](q) modules of tensoring with the sign module (the 1-dimensional module of [H.sub.n](q) where each [T.sub.i] acts as -1) is a functor which takes irreducible modules to irreducible modules.