Symmetry And Group

A characterization of a class of locally compact Abelian by Pannenberg M.

By Pannenberg M.

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B) E (c) E is a normal extraspecia1 q-subgroup of G. E/E' is a form primitive GF(q) [G]-module with kernel F(G). We will focus on the module E. without changing the module E Our object is to extend the field GF(q) or loosing the form on E. bly eeneralize our discussion, we set new hypotheses. We s~y for~ that a bilinear g on a vector space is symmetric, symplectic, or unitary. Je \Jish to extend HO~[G] K all V will K[G]-mod~~e. Assume that g : V x V ~ K u, v E V V and which is fixed by G x E G).

P(x) E and both In particular, we choose Whenever we write 0 V @ W have their Z [x) be the minimal polyno- in the Green ring. We will mention such polynomials often. 5), it follows that is algebraic. JX k[e )-module_. Then be a V {U ® W} E C(V). we will always express them in the form q,r E then ID,n. for some so that V integral. In particular, Assume that V m C* (V) generated by Thus all indecomposable sumrnands of Proposition. 6) C(V) By definition, there are modules and vn+m for some module Z-module Iv I, {w I He need to show that for lv}{v I so that Then the p p(x) q = and q(x) r r(x) Thus where have nonnegative coeffi- to have positive leading coefficient.

Additionally, i f the polynomial can be chosen so that it is monic then we call X and V respectively integral. The algebraic elements of the Green ring form a subset of it and the integral elements a subset of this set. Which subsets of the Green ring are algebraic (integral)? Of course, we only want to consider "interesting" subsets. One particularly interesting set is described as follows. We say that a k [G )-modul" sable direct summand of VI Ok V2 Ok 1I \' of V Ok Vn k[G]-modules. The value of i" i rreduci_bly generated if any indecompo- is itself isomorphic with a direct summand where the n V.

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