# Balanced systems of primitive idempotents in matrix algebras

## Abstract

The article develops the concept of balanced t-systems of idempotents in associative semisimple finite-dimensional algebras over the field of complex numbers 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 M{sub n}(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{sup 2}-s or s>n{sup 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 M{sub n}, n=2,3. Connections are established between conference matrices and (2n,n)-systems.

- Authors:

- M.V. Lomonosov Moscow State University, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21202927

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 191; Journal Issue: 4; Other Information: DOI: 10.1070/SM2000v191n04ABEH000471; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; MATRICES; PHOTOLUMINESCENCE

### Citation Formats

```
Ivanov, D N.
```*Balanced systems of primitive idempotents in matrix algebras*. United States: N. p., 2000.
Web. doi:10.1070/SM2000V191N04ABEH000471; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Ivanov, D N.
```*Balanced systems of primitive idempotents in matrix algebras*. United States. doi:10.1070/SM2000V191N04ABEH000471; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Ivanov, D N. Sun .
"Balanced systems of primitive idempotents in matrix algebras". United States. doi:10.1070/SM2000V191N04ABEH000471; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21202927,
```

title = {Balanced systems of primitive idempotents in matrix algebras},

author = {Ivanov, D N},

abstractNote = {The article develops the concept of balanced t-systems of idempotents in associative semisimple finite-dimensional algebras over the field of complex numbers 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 M{sub n}(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{sup 2}-s or s>n{sup 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 M{sub n}, n=2,3. Connections are established between conference matrices and (2n,n)-systems.},

doi = {10.1070/SM2000V191N04ABEH000471; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 4,

volume = 191,

place = {United States},

year = {2000},

month = {4}

}