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Balanced systems of primitive idempotents in matrix algebras
D. N. Ivanov M. V. Lomonosov Moscow State University
Abstract:
The article develops the concept of balanced
$t$-systems of idempotents in associative semisimple finite-dimensional algebras over the field of complex numbers
$\mathbb C$ this was introduced by the author as a generalization of the concept of combinatorial
$t$-schemes, which in this context corresponds to the case of commutative algebras. Balanced 2-systems are considered consisting of
$v$ primitive idempotents in the matrix algebra
$\mathrm M_n(\mathbb C)$, known as
$(v,n)$-systems. It is proved that
$(n+1,n)$-systems are unique and it is shown that there are no
$(n+s,n)$-systems with
$n>s^2-s$ or
$s>n^2-n$. The
$(q+1,n)$-systems having 2-transitive automorphism subgroup
$PSL(2,q)$,
$q$ odd, are classified. The (4,2)- and (6,3)-systems are classified. A balanced basis is constructed in the algebras
$\mathrm M_n$,
$n=2,3$. Connections are established between conference matrices and
$(2n,n)$-systems, and between suitable matrices and
$\biggl(m^2,\dfrac{m^2\pm m}2\biggr)$-systems.
UDC:
512.538+
512.542+
519.1
MSC: Primary
16P10,
05B20; Secondary
05B05,
62K10 Received: 12.05.1999
DOI:
10.4213/sm471
Fulltext:
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