Про спадково незвiднi унiмономiальнi зображення циклiчних p-груп над локальними кiльцями характеристики p

Abstract

The task of the description up to equivalency of the matrix representations of finite pgroups of order greater then p aver a commutative local ring of characteristics of ps (s > 0) that is not a field contains the classical unsolved problem of pair of matrices over a field. Therefore, consideration of partial cases and the study of special representation matrix representations is importent. Let K be a commutative ring with a identity, G is a finitely generated group with some fixed system of generator elements a1,...,ar. Every matrix representation of the group G over the ring K equivalent to Γ : ai → E+Mi (i = 1,...,r), where E be an identity n×n-matrix, Mi be a monomial n×n-matrix (i = 1,...,r), we call unimonomial. An reducible unimonomial representation over the ring K, in a reducible form of which on diagonal blocks at least one unimonomial representation is formed, we call hereditary reducible over the ring K. A series of unimonomial representations of a finite cyclic p-group H = haiover commutative local principle ideal ring K of characteristic p with nilpotent Jacobson radical of the degree l (1 < l < ∞) of the form a → E + 0 ... 0 1 1 ... 0 0 . . . ... . . . . . . 0 ... 1 0!· diag[ε1ts1,...,εntsn], where si ≥ 0, εi be elements from K∗ (i = 1,...,n), t be an generator element of Jacobson radical ring K. It is making up clear the criterion, when the map of the given form sets the representation of the group H (P|H|−1 j=0 si+j ≥ l (i = 1,...,n), herethe indexes are considered by the module n). It have been found the sufficient condition of hereditary irreducibility of the constructed representations ((Pn i=1 si,n) = 1,tPn i=1 si 6= 0). In addition, we can obtain the hereditary reducibility of the constructed representations in the case when (Pn i=1 si,n) > 1,Qn i=1 εi = 1). Based on the researches of Bondarenko V. M., Bortosh M. Yu. of similarity of the monomial matrices it is making up clear the criterion the equivalence of the constructed representations (the corresponding sequences (s1,...,sn) are cyclically equivalent a relevant productsQn i=1 εi are equal modulo Ann (ts), where s is the largest member of the weight sequence (s1,...,sn)). In the case of the finiteness of the ring K by computation in the GAP system it have been found the number of all, up to equivalence, constructed unimonomial hereditary irreducible matrix representations of a cyclic nontrivial p-group depending on the number of elements of the residue class field of the ring K.